Chaotic Systems: Theory and Applications, Selected Papers from the 2nd Chaotic Modeling and Simulation International Conference (CHAOS2009)

Chaotic Systems Theory and Applications

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Chaotic Systems Theory and Applications Selected Papers from the 2nd Chaotic Modeling and Simulation International Conference (CHAOS2009) Chania, Crete, Greece

1 - 5 June 2009

editors

Christos H Skiadas Technical University of Crete, Greece

Ioannis Dimotikalis Technological Educational Institute of Crete, Greece

,~ World Scientific NEW JERSEY· LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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CHAOTIC SYSTEMS Theory and Applications Selected Papers from the 2nd Chaotic Modeling and Simulation International Conference (CHAOS2009) Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-4299-71-8 ISBN-IO 981-4299-7 1-5

Printed in Singapore by B & Jo Enterprise Pte Ltd

v

Preface This book contains a collection of papers suggested by the scientific committee including the best papers presented in the 2nd International Conference (CHAOS2009) on Chaotic Modeling, Simulation and Applications, Chania, Crete, Greece, June 1-5,2009. The aim of the conference was to invite and bring together people working in interesting topics of chaotic modeling, nonlinear and dynamical systems and chaotic simulation. The book includes papers on various important topics of Chaotic Modeling and Simulation as: • Bifurcation and chaos in multi machine power systems, Chaos in multiplicative systems, Deterministic chaos machine, Dynamics of a bouncing ball, N-body chaos. • Spatiotemporal chaos, Chaotic mixing in the system Earth, Multiple equilibria and endogenous cycles, Growth models, EnKF and particle filters for meteorological models, 3D steady flows, Local and global Lyapunov exponents, Routh-Hurwitz conditions and the Lyapunov second method, Chaotic behavior of plasma surface interaction, Simulation of geochemical self-organization, Atmospheric pressure plasma. • On the entropy flows to disorder, Quantum scattering and transport, Electron quantum transport, Qualitative dynamics of interacting classical spins, Atomic de Broglie-wave chaos, q-deformed nonlinear maps. • A two population model for the stock market problem, Complex dynamics in an asset pricing model, Hick Samuelson Keynes dynamic economic models, Maxwell-Bloch equations as predator-prey system. • Linear communication channels and synchronization, Identifying chaotic and quasiperiodic time-series, New model of nonlinear oscillations generators, Transmission line models, Optimum CSK communication, Symmetry-break in a minimal Lorenz-like system, Longwave Marangoni convection. • Nonlinear system's synthesis, Strategies of synergetics, Synergetics control, Synergetics and multi-machine power system modification. • Sub-critical transitions in coupled map lattices, Markov processes and transition probabilities. • Music composition from the cosine law of a frequency-amplitude triangle, Composing chaotic music from a varying second order recurrence equation. • Chaotic neural networks, A novel neuro-fuzzy algorithm. We thank all the contributors to the success of the CHAOS 2009 International Conference, the committees, the plenary speakers, the reviewers and especially the authors of this volume. Special thanks to the Conference Secretary Dr. Anthi Katsirikou for her work and assistance. Finally, we would like to thank Mary Karadima, Aggeliki Oikonomou and George Matalliotakis for their valuable support. October 30, 2009

Christos H. Skiadas, Technical,Dniversity of Crete, Greece Ioannis Dimotikalis, Technological Educational Institute of Crete, Greece

vi

Honorary Committee David Ruelle Academie des Sciences de Paris Honorary Professor at the Institut des Hautes Etudes Scientifiques of Bures-sur-Yvette, France Leon O. Chua EECS Department, University of California, Berkeley, USA Editor of the International Journal of Bifurcation and Chaos Ji-Huan He Donghua University, Shanghai, China Editor of Int. Journal of Nonlinear Sciences and Numerical Simulation

International Scientific Committee C. H. Skiadas (Technical University of Crete, Chania, Greece), Chair H. Adeli (The Ohio State University, USA) N. Akhmediev (Australian National University, Australia)

M. Amabili (McGill University, Montreal, Canada) J. Awrejcewicz (Technical University of Lodz, Poland) S. Bishop (University College London, UK)

T. Bountis (University of Patras, Greece) Y. S. Boutalis (Democritus University of Thrace, Greece)

C. Chandre (Centre de Physique Theorique, Marseille, France) M. Christodoulou (Technical University of Crete, Chania, Crete, Greece) P. Commendatore (Universita di Napoli 'Federico II', Italy) D. Dhar (Tata Institute of Fundamental Research, India) J. Dimotikalis (Technological Educational Institute, Crete, Greece)

B. Epureanu (University of Michigan, Ann Arbor, MI, USA) G. Fagiolo (Sant'Anna School of Advanced Studies, Pisa, Italy) V. Grigoras (University of Iasi, Romania) L. Hong (Xi'an Jiaotong University, Xi'an, Shaanxi, China) G. Hunt (Centre for Nonlinear Mechanics, University of Bath, Bath, UK) T. Kapitaniak (Technical University of Lodz, Lodz, Poland) A. Kolesnikov (Southern Federal University, Russia) G. P. Kapoor (Indian Institute of Technology Kanpur, Kanpur, India) V. Krysko (Dept. of Math. and Modeling, Saratov State Technical University, Russia) W. Li (Northwestern Poly technical University, China)

International Scientific Committee

B. L. Lan (School of Engineering, Monash University, Selangor, Malaysia) V. J. Law (Dublin City University, Glasnevin, Dublin, Ireland) V. Lucarini (University of Reading, UK)

J. A. T. Machado (ISEP-Institute of Engineering of Porto, Porto, Portugal) W. M. Macek (Cardinal Stefan Wyszynski University, Warsaw, Poland) P. Mahanti (University of New Brunswick, Saint John, Canada) G. M. Mahmoud (Assiut University, Assiut, Egypt) P. Manneville (Laboratoire d'Hydrodynamique, Ecole Polytechnique, France) N. Mastorakis (Techn. Univ. of Sofia, Bulgaria and ASEI, Greece) A. S. Mikhailov (Fritz Haber Institute of Max Planck Society, Berlin, Germany) M. S. M. Noorani (Universiti Kebangsaan Malaysia, Malaysia) G. V. Orman (Transilvania University of Brasov, Romania) S. Panchev (Bulgarian Academy of Sciences, Bulgaria) G. Pedrizzetti (University of Trieste, Trieste, Italy) F. Pellicano (Universita di Modena e Reggio Emilia, Italy) S. V. Prants (Pacific Oceanological Institute of RAS, Vladivostok, Russia) A. G. Ramm (Kansas State University, Kansas, USA) G. Rega (University of Rome "La Sapienza", Italy) H. Skiadas (Hanover College, Hanover, USA) V. Snasel (VSB-Technical University of Ostrava, Czech) D. Sotiropoulos (Technical University of Crete, Chania, Crete, Greece) B. Spagnolo (University of Palermo, Italy)

P. D. Spanos (Rice University, Houston, TX, USA)

J. C. Sprott (University of Wisconsin, Madison, WI, USA) S. Thurner (Medical University of Vienna, Austria) D. Trigiante (Universita di Firenze, Firenze, Italy) G. Unal (Yeditepe University, Istanbul, Turkey) A. Valyaev (Nuclear Safety Institute of RAS, Russia) A. Vakakis (University of Illinois, Urbana, USA)

J. P. van der Weele (University ofPatras, Greece) M. Wiercigroch (University of Aberdeen, Aberdeen, Scotland, UK)

VII

viii

Keynote Talks Marco Amabili Department of Mechanical Engineering, McGill U ni versity, Montreal, Canada Chaotic vibrations of circular cylindrical shells: Garlekin versus reducedorder models Jan Awrejcewicz Department of Automatics and Biomechanics, Technical University of Lodz, Lodz, Poland Deterministic chaos machine: Experimental vs. numerical investigations Alfred Inselberg School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel Multidimensional visualization and its applications Pier A. Mello Institute of Physics, Universidad N acional Autonoma de Mexico, Mexico Quantum scattering and transport in classically chaotic cavities: An overview of old and new results Sergey V. Prants Pacific Oceanological Institute of the Russian Academy of Sciences, Vladivostok, Russia Quantum chaos with atoms in a laser field Alexander G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA http://www.math.ksu.edu/-ramm Acoustic and electromagnetic wave scattering by many small particles and creating materials with desired properties Giuseppe Rega Department of Structural Engineering and Geotechnical Engineering, University of Rome 'La Sapienza', Italy Experimental unfolding and theoretical model of the transition to complex dynamics in sagged cables Konstantinos S. Tsakalis Department of Electrical Engineering, Ira A. Fulton School of Engineering, Arizona State University, USA Chaos, brain and epilepsy: A bioengineering approach Alexander F. Vakakis Department of Mechanical Science and Engineering, University of Illinois, Urbana, USA Nonlinear targeted energy transfer in dynamical systems

ix

Contents Preface

v

Honorary Committee and International Scientific Committee

vi

Keynote Talks

viii

The Influence of Machine Saturation on Bifurcation and Chaos in Multimachine Power Systems

Majdi M. Alomari and Jian Cue Zhu Chaos in Multiplicative Systems

9

Dorota Aniszewska and Marek Rybaczuk Deterministic Chaos Machine: Experimental vs. Numerical Investigations

17

Jan Awrejcewicz and Crzegorz Kudra Some Issues and Results on the EnKF and Particle Filters for Meteorological Models

27

Christophe Baehr and Olivier Pannekoucke Local and Global Lyapunov Exponents in a Discrete Mass Waterwheel

35

David Becerra Alonso and Valery Tereshko Complex Dynamics in an Asset Pricing Model with Updating Wealth

43

Serena Brianzoni, Cristiana Mammana, and Elisabetta Michetti Chaotic Mixing in the System Earth: Mixing Granitic and Basaltic Liquids

51

Cristina De Campos, Werner Ertel-Ingrisch, Diego Perugini, Donald B. Dingwell, and Ciampero Poli Multiple Equilibria and Endogenous Cycles in a Non-Linear Harrodian Growth Model Pasquale Commendatore, Elisabetta Michetti, and Antonio Pinto Hick Samuelson Keynes Dynamic Economic Model with Discrete Time and Consumer Sentiment

59

67

Loretti 1. Dobrescu, Mihaela Neamtu, and Dumitru Opri§ On the Entropy Flows to Disorder

C. T. J. Dodson

75

x

Contents

Linear Communication Channel Based on Chaos Synchronization

85

Victor Grigoras and Carmen Grigoras Maxwell-Bloch Equations as Predator-Prey System A. S. Hacinliyan, 0. O. Aybar, 1. KU$beyzi, 1. Temizer, and E. E. Akkaya Identifying Chaotic and Quasiperiodic Time-Series Candidates for Efficient Nonlinear Projective Noise Reduction

93

101

Nada Jevtic and Jeffrey S. Schweitzer Chaos from the Observer's Mathematics Point of View

111

Dmitriy Khots and Boris Khots New Models of Nonlinear Oscillations Generators Alexander A. Kolesnikov Nonlinear System's Synthesis - The Central Problem of Modern Science and Technology: Synergetics Conception. Part II: Strategies of Synergetics Control Anatoly A. Kolesnikov Synergetic Approach to Traditional Control Laws Multi-Machine Power System Modification

119

128

134

Anatoly A. Kolesnikov and Andrew A. Kuzmenko Dynamic Stability Loss of Closed Circled Cylindrical Shells Estimation Using Wavelets

139

V. A. Krysko, J. Awrejcewicz, M. Zhigalov, V. Soldatov, E. S. Kuznetsova, and S. Mitskevich The Application of Multivariate Analysis Tools for Non-Invasive Performance Analysis of Atmospheric Pressure Plasma

147

V. J. Law, J. Tynan, G. Byrne, D. P. Dowling, and S. Daniels

Chaos Communication: An Overview of Exact, Optimum and Approximate Results Using Statistical Theory

155

Anthony J. Lawrance Dynamics of a Bouncing Ball

165

Shiuan-Ni Liang and Boon Leong Lan Symmetry-Break in a Minimal Lorenz-Like System Valerio Lucarini and Klaus Fraedrich

170

Contents

Sub-Critical Transitions in Coupled Map Lattices P. Manneville Quantum Scattering and Transport in Classically Chaotic Cavities: An Overview of Old and New Results

xi 185

191

Pier A. Mello, VIctor A. Gopar, and J. A. Mendez-Bermudez

Parametric Excitation of a Longwave Marangoni Convection

207

Alexander B. Mikishev, Alexander A. Nepomnyashchy, and Boris L. Smorodin

Symmetry Broken in Low Dimensional N-Body Chaos David C. Ni and Chou Hsin Chin On Markov Processes: A Survey of the Transition Probabilities and Markov Property

215

224

Gabriel V. Orman

Qualitative Dynamics of Interacting Classical Spins Leonidas Pantelidis

232

A Transmission Line Model for the Spherical Beltrami Problem

241

C. D. Papageorgiou and T. E. Raptis

Two Aspects of Optimum CSK Communication: Spreading and Decoding

249

Theodore Papamarkou

A Numerical Exploration of the Dynamical Behaviour of q-Deformed Nonlinear Maps Vinod Patidar, G. Purohit, and K. K. Sud Spatiotemporal Chaos in Distributed Systems: Theory and Practice George P. Pavlos, A. C. Iliopoulos, V. G. Tsoutsouras, L. P. Karakatsanis, and E. G. Pavlos An Enhanced Tree-Shaped Adachi-Like Chaotic Neural Network Requiring Linear-Time Computations Ke Qin and B. John Oommen Clustering of Inertial Particles in 3D Steady Flows Themistoklis Sapsis and George Haller

257

268

284

294

XII

Contents

A Two Population Model for the Stock Market Problem

302

Christos H. Skiadas

Electron Quantum Transport through a Mesoscopic Device: Dephasing and Absorption Induced by Interaction with a Complicated Background

309

Valentin V. Sokolov

Composing Chaotic Music from a Varying Second Order Recurrence Equation

320

Anastasios D. Sotiropoulos and Vaggelis D. Sotiropoulos

Music Composition from the Cosine Law of a Frequency-Amplitude Triangle

327

Vaggelis D. Sotiropoulos and Anastasios D. Sotiropoulos

Chaos in a Fractional-Order Jerk Model Using Tanh Nonlinearity

338

Banlue Srisuchinwong

Nonlinear Systems in Brunovsky Canonical Form: A Novel Neuro-Fuzzy Algorithm for Direct Adaptive Regulation with Robustness Analysis Dimitris C. Theodoridis, Yiannis Boutalis, and Manolis A. Christodoulou

346

Routh-Hurwitz Conditions and Lyapunov Second Method for a Nonlinear System Omiir Umut

362

Atomic de Broglie-Wave Chaos V. o. Vitkovsky, L. E. Konkov, and S. V. Prants Chaotic Behavior of Plasma Surface Interaction. A Table of Plasma Treatment Parameters Useful to the Restoration of Metallic Archaeological Objects

369

377

C. L. Xaplanteris and E. Filippaki Simulation of Geochemical Self-Organization: Acid Infiltration and Mineral Deposition in a Porous Ferruginous Limestone Rock

385

Farah Zaknoun, Houssam El-Rassy, Mazen AI-Ghoul, Samia Al-Joubeily, Tharwat Mokalled, and Rabih Sultan

Author Index

395

The Influence of Machine Saturation on Bifurcation and Chaos in Multimachine Power Systems Majdi M. Alomari and Jian Gue Zhu University of Technology, Sydney (UTS) P.O. Box 123, Broadway NSW 2007, Australia (e-mail: [email protected]) Abstract: A bifurcation theory is applied to the multimachine power system to investigate the effect of iron saturation on the complex dynamics of the system. The second system of the IEEE second benchmark model of Subsynchronous Resonance (SSR) is considered. The system studied can be mathematically modeled as a set of first order nonlinear ordinary differential equations with (p=X/XL ) as a bifurcation parameter. Hence, bifurcation theory can be applied to nonlinear dynamical systems, which can be written as dxldt=F(x;p). The results show that the influence of machine saturation expands the unstable region when the system loses stability at the Hopf bifurcation point at a less value of compensation. Keywords: machine saturation , Hopf bifurcation, chaos, subsynchronous resonance, damper windings.

1. Introduction Addition of series compensation in power systems, which is considered for increasing effectively the power transfer capability as well as improving the stability, introduces a problem called subsynchronous resonance (SSR). Subsynchronous resonance can be defined as electromechanical interaction between electrical resonant circuits of the transmission system and the torsional natural frequencies of the turbine-generator rotor. Where the torques produced by the interaction of the subharmonic current in the stator RLC circuit and the rotor circuits of the synchronous generator may excite one of the natural torsional modes of oscillation of the turbine-generator masses leading to shaft failure. Three types of SSR can identify the interaction of the system and the generator under the subsynchronous resonance. They have been called torsional interaction effect, induction generator effect, and transient torque effect. In this research, we focus on the torsional interaction effect, which results from the interaction of the electrical subsynchronous mode with the torsional mode. On the other hand, several methods have been used in SSR study. The most common of these methods are eigenvalue analysis, frequency scanning, and time-domain analysis. The eigenvalue analysis is used in this study. It is a very valuable technique because it provides both the frequencies of oscillation and the damping at each frequency. Abed and Varaiya [1] employed Hopf bifurcation theory to explain nonlinear oscillatory behavior in power systems. The bifurcation theorem was used by Zhu et al [2] to demonstrate the existence of a Hopf bifurcation in a single machine infinite busbar (SMIB) power system, in which the dynamics of

2

M. M. Alomari and 1. G. Zhu

the damper windings and the AVR are neglected. Tomim et al [3] proposed an index that identifies Hopf bifurcation points in power systems susceptible to subsynchronous resonance. N ayfeh et al [4] applied the bifurcation theory to a practical series capacitor compensated single machine power system, the BOARDMAN turbine-generator system. To make the analysis simple and manageable, the effect of stator and rotor iron saturation in synchronous machines was neglected in all of the bifurcation analysis given in the literature. However, the importance of saturation has been recognized by many authors [5, 6, 7]. The effect of electrical machine saturation on SSR was studied by Harb et al [8]. They concluded that the generator saturation slightly shrinks the positively damped region by shifting the Hopf bifurcation point, secondary Hopf bifurcation and bluesky catastrophe to smaller compensation level. In this research, power system dynamics has been studied using the nonlinear dynamics point of view, which utilizes the bifurcation theory. Bifurcation theory and chaos is used to investigate the complex dynamics of the considered system. The system loses stability via Hopf bifurcation point (H) and a limit cycle is born at /1=H. This limit cycle is stable if the bifurcation is supercritical and unstable if it is subcritical. The type of the Hopf bifurcation is determined by numerical integration of the system, with specific amount of initial disturbances, slightly before and after the bifurcation value. On further increase of the compensation factor, the system experiences chaos via torus attractor. Chaos is a bounded steady-state behavior that is not an equilibrium solution or a periodic solution or a quasiperiodic solution [9]. In this paper, we study the effect of iron saturation on Bifurcation and Chaos.

2. System Description The system considered is the two different machine infinite bus system, shown in Figure 1. The two machines have a common torsional mode connected to a single series compensated transmission line. The model and the parameters are provided in the second system of the IEEE second benchmark model. The electromechanical systems for the first and second units are shown in Figure. 2. The first unit consists of exciter (EX.), generator (Gen. I), low-pressure (LPl) and highpressure (HPl) turbine sections. And the second unit consists of generator (Gen.2), low-pressure (LP2) and high-pressure (HP2) turbine sections. Every LPl

M4 D4

Grn

Tran.-fonnm

Inllnite Bus

LP2 ~ K12b

~

~ MJb DJb

Fig. 1. Electrical system (Two different machine infinite bus system).

K23b

Ml Dl

D2

DJ

Iilin2l ~nth.

M2

MJ

M2b

Mlb

D2b

Dlb

Fig. 2. Electro-mechanical systems for the first and second units.

Influence of Machine Saturation on Bifurcation and Chaos

3

section has its own angular momentum constant M and damping coefficient D, and every pair of successive masses have their own shaft stiffness constant K, as shown in Figure 2. The data for electrical and mechanical system are provided in [10]. Replacement of these generators with a single equivalent generator will change the resonance characteristics and therefore is not justified. Consequently, each generator is represented in its own rotor frame of reference and suitable transformation is made.

3. Mathematical Model The mathematical model of the electrical and mechanical system will be presented in this section. Actually, the electrical system includes the dynamic nonlinear mathematical model of a synchronous generator and that of the transmission line. The generator model considered in this study includes five equations, d-axis stator winding, q-axis stator winding, d-axis rotor field winding, q-axis rotor damper winding and d-axis rotor damper winding equations. Each mass of the mechanical system can be modeled by a second order ordinary differential equation (swing equation), which is presented in state space model as two first order ordinary differential equations. Using the direct and quadrature d-q axes and Park's transformation, we can write the complete mathematical model that describes the dynamics of the system. The mathematical model of the electrical and mechanical system is provided in [II].

4. System Response in Case of Absence of Saturation In this section, we study the case of adding damper winding on sub synchronous resonance of the first generator but neglecting saturation. In case of no saturation, we set do, d l , d2, d3, qo, q" q2, and q3 equal to zero in equations (1) and (2). In this case we have 23 ordinary nonlinear differential equations. The equilibrium solution is obtained by setting the derivatives of the 23 state variables in the system equal to zero. The stability of the operating point is studied by examination of the eigenvalues of the linearized model evaluated at the operating point. The operating point stability regions in the brd1 plane together with two Hopf bifurcation points are depicted in Figure 3. We observe that the power system has a stable operating point to the left of HI::::0.212486 and to the right of H2::::0.880735, and has an unstable operating point between HI and H 2• The operating point loses stability at a Hopf bifurcation point, namely f1=H I . It regains stability at a reverse Hopf bifurcation, namely f1=H2 . In this case a pair of complex conjugate eigenvalues will transversally cross from left half to right half of the complex plane, and then back to the left half. To determine whether the limit cycles created due to the Hopf bifurcation are stable or unstable, we obtain the time response of the system by numerical integration with small disturbance slightly before HI. Figure 4 shows the response of the system with 7% initial disturbance on the speed of the generator at 1'=0.199135, which is less than HI. It can be observed that the system is unstable. Therefore, the type of this Hopf bifurcation is subcritical. So, the periodic solution emanating at the bifurcation point is unstable.

4

M. M. Alomari and 1. G. Zhu

: Stable operating point

!::\

r: ~

)I~ ~--'H-,

1-,-,

~

"'j

B~"nZ

~ 0.9

a:

: Unstable operating point Hi, H2 : Hopf bifurcation points

0.8 0.7

0 .6'----~-~-~-~-~-~-__'c=-~-~,__---'

o

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Compensation factor ~

• • • : Stable periodic orbits

1.22

o0 0

:0 1.21

: Unstable periodic orbits

IE

£

i

CF : Cyclic-fold bifurcation point

1.2

SH : Secondary Hopf bifurcation poinl

1.19

~ 1.18 ~ 1.17

H1

£ 1.16 15 ~

g>

1. 15

~ 1.14

~

CF

1.13

1.12 0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

Compensation factor V

Fig. 3. Bifurcation diagram showing variation of the first generator rotor angle 0,., with the compensation factor f1 (for the case of absence of Saturation).

90

eo

10

°O:='--'O:'":,c--c. .8-~----""::.'-----C1.4 OAC --:CO""::.6- O':: lime (sec)

1.6

1.8

Time (sec)

Fig. 4. Response of the system. Rotor speed of the generator (left) and rotor angle of the generator (right) at f1=0.199135 with 7% initial di sturbance in rotor speed of generator (for the case of absence of Saturation).

Influence of Machine Saturation on Bifurcation and Chaos

5

Locally, the system has a stable equilibrium solution and a small unstable limit cycle for values of 11 slightly before HI' As 11 increases past HI. the equilibrium solution loses stability and the small unstable limit cycle disappears. As 11 decreases below HI. the unstable limit cycle deforms and increases in size until the branch of unstable limit cycles meets a branch of stable large-amplitude limit cycles at 11=0.198526, corresponding to point CF. As the value of compensation factors decreases more, the stable and unstable limit cycles collide and exterminate each other in a cyclic-fold bifurcation. As 11 increases from CF, the stable limit cycle grows in size while remaining stable. The limit cycle undergoes a secondary Hopf bifurcation at I1=SH;.o:oO.201325, resulting in a twoperiod quasiperiodic attractor. While it oscillates with more than one incommensurate frequency slightly after the secondary Hopf bifurcation point at 11=0.201350 and obtain a period quasiperiodic (torus) attractor. As 11 increases slightly after SH up to 11=0.206433, the chaotic attractor is obtained. Figure 5 shows the two-dimensional projections and time histories of the system at different compensation factors. ~ 10015

~

(a)

1.001

iU)OO5 .; ~

1

i

~O.9995

•

If

0 .999

1.208

1.209

1.21

1. 2 11

1.212

1.2131.214

1,2151 .2161.217 Tome (.ee)

Time (.eel

(b)

"

flolor..-.gleolll'l9f1rSI9_"'W!;r I I'OO)

~1_00

~

~

1.04

1

102

(c)

;

1

~

F:

"'0.94

RolC>ran 9leoflheir
Fig. 5. Two-dimensional projection of the phases portrait onto w",- b", plane (left) and the time histories of the corresponding rotor speed of generator (right). The solution at (a) limit cycle, 11=0.198526, (b) torus-attractor, 11=0.201350 and (c) chaotic attractor, 11=0.206433 (for the case of absence of Saturation).

6

M. M. Alomari and 1. G. Zhu

5. System Response in Presence of Saturation In this section, we investigate the influence of iron saturation in both the d- and qaxes on the sub synchronous instability of the first generator. This is done by performing a bifurcation analysis in a complete model of the system. The model includes saturation but neglects the dynamics of the automatic voltage regulator (A VR) and the turbine governor. When researching iron saturation for stability studies, it is assumed that the saturation relationship between the resulting air-gap flux and the mmf under loaded conditions is the same as in no-load conditions. This allows the saturation characteristics to be represented by the open-circuit saturation curve. When saturation is taken into consideration in both the d- and q-axes, it leads to an improvement in the accuracy of the simulation models. When performing iron saturation in stability studies, the following assumptions are made [5]. First, the leakage fluxes are in air for a large portion of their paths. Therefore, they are minimally affected by saturation of the iron core. So, the leakage inductances are independent of the iron saturation. Consequently mutual flux linkages 'Pmd and 'Pmq are the only elements that saturate are the mutual flux linkages. Second, the leakage fluxes are usually small and their paths coincide with that of the main flux for only a small part of its path. So, saturation can be determined by the air-gap flux linkage only. Third and final assumption is that the saturation relationship between the resultant air-gap flux and the mmf under loaded conditions is the same as in no-load conditions. This allows the saturation characteristics to be represented by the open-circuit saturation curve. In order to obtain the generator analytical formulas for the fluxes of the first generator 'P,ndl and 'P,nqI as functions of the currents, Harb et at [8] fitted the test data which were discussed by Minnich et at [12]. The result was the following third-order polynomial functions: If/mdl =do +dl(iII -idl)+d2(iII

-idl )2 +d3(iII -idl )3 (1)

(2)

where do, dj, d2, d3, qo, qf, q2 and q3 are constants. Clearly, the mutual flux linkages of the first generator 'P,ndl and 'P,nql are nonlinear functions of the first machine currents i dI , i qI , if! ' and iQI . The flux linkages of the first generator 'P,1l and 'Pql of the d- and q-axes can be determined from the mutual flux linkages of the first generator 'P,ndl and 'PmqI as follows: If/dl =If/mdl-Xle i dl

(3)

If/ql =If/mql -X le i ql

(4)

= If/mdl If/QI = If/mql -

X le ill

(5)

X Ie i Q,

(6)

If/fl

where Xle is the leakage reactance.

Influence of Machine Saturation on Bifurcation and Chaos

7

In this formulation, the dynamics of the automatic voltage regular (A VR) and the turbine governor are neglected. The dynamics of the damper windings as well as the machine saturation on the both axes of the first generator are included. We will show that machine saturation reduces the compensation level at which subsynchronous resonance occurs. The operating point stability regions in the (jrl-fl plane together with two Hopf bifurcation points are depicted in Figure 6. We observe that the power system has a stable operating point to the left of HI~O.174653 and to the right of H2~O.856765, and has an unstable operating point between HI and H 2. The operating point loses stability at a Hopf bifurcation point where fl=H I . It regains stability at a reverse Hopf bifurcation for fl=H z. In this case a pair of complex conjugate eigenvalues will transversally cross from left half and right half of the complex plane, and then back to left half. The results show that machine saturation reduces the compensation level at which subsynchronous resonance occurs. The generator saturation slightly shrinks the positively damped region by shifting the Hopf bifurcation point to smaller compensation level.

Conclusions We have applied bifurcation theory to investigate the influence of iron saturation. The case of neglecting the saturation is considered. The results showed that as the compensation factor increases the operating point loses stability via subcritical Hopf bifurcation point. When the machine saturation is considered, the results showed that the generator saturation shifts the subcritical Hopf bifurcation to smaller value of fl and hence shrinks the stable region. In other words, the generator saturation destabilize the system by reducing the compensation level at which subsynchronous resonance occurs. It also slightly shifts the secondary Hopf bifurcation and bluesky catastrophe to smaller compensation level. 14---,----,-------,--.,----,------,--.,----,----,-----,

: Stable operating point

1.3

: Unstable operating point

Hi, H2 : Hopfbifurcation points :;;11

HI 0'1

~09 ;00.8

;;06 H2

0.5

~AOL-~0.1--,L0.2-----:0L.3-~OAc---0,L.5-----!0.~6-~0}C--C--0~.8-----!0.9;--c~nsa!ioofoctl)'~

Fig. 6. Bifurcation diagram showing variation of the first generator rotor angle 0" with the compensation factor f1 (for the case of adding saturation to the first generator).

8

M. M. Alomari and 1. G. Zhu

References [1] E. H. Abed and P. P. Varaiya, "Nonlinear Oscillations in Power Systems", Int. J. Elecr. Power and Energy System, Vol. 6, pp. 37-43, Jan., 1984. [2] W. Zhu, R.R. Mohler, R. Spce, W.A Mittelstadt and D. Maratukulman, "Hopf Bifurcation in a SMIB Power System with SSR," IEEE Trans. on Power Systems, Vol.1l, No.3, pp. 1579-1584, 1996. [3] M. A Tomim, A C. Zambroni de Souza, P. P. Carvalho Mendes, and G. LambertTorres, "Identification of Hopf Bifurcation in Power Systems Susceptible to Subsynchronous Resonance", IEEE Bologna Power Tech Conference, June 23-26, 2003, Bologna, Italy. [4]

A.H Nayfeh, AM. Harb, and C-M.Chin, AM. Hamdan and L. Mili, Application of Bifurcation Theory to Subsynchronous Resonance In Power Systems, InU.Bifurcation and Chaos,YoI.8,No.l,pp.157-l72, 1998

[5]

Kundur, P. 1994, Power System Stability and Control, Electric Power Research Institute, McGraw-Hill, New York.

[6] Harly, R. G., Limebeer, D. J., and Chirricozzi, E. , 1980, "Comparative study of the saturation methods in synchronous machine models," IEEE Proceedings, Vol. 127, Part B. [7] De Mello, F. P., and Hannet, L. N. "Representation of Saturation in Synchronous Machines," IEEE Transactions on Power Systems PWRS-l, 1986. [8] AM. Harb, L. Mili, AH Nayfeh and C-M. Chin, On the Effect of the Machine Saturation on SSR in Power Systems, Electrical Machines and Power Systems, Vo1.20, pp. 10 19-1035, 2000. [9] AH. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics, Wiley, New York, 1995. [10] IEEE SSR Working Group, "Second Benchmark Model for Computer Simulation of Sub synchronous Resonance," IEEE Trans. On Power Apparatus and Systems. Vol. PAS- 104, No.5, 1057-\064, May 1985. [II] Majdi M. Alomari and Benedykt S. Rodanski, "The Effects of Machine Components

on Bifurcation and Chaos as Applied to Multimachine System", CHAOS2008, Chaotic Modeling and Simulation International Conference, Chania, Crete, Greece, 3-6 June 2008. [12] S. H. Minnich, R. P. Schulz, D. H. Baker, D. K. Sharma, R. G. Farmer, and J. H. Fish, "Saturation Functions for Synchronous Generators from Finite Elements", IEEE Trans. Energy Conversion, vol. EC-2, December 1987.

9

Chaos In Multiplicative Systems Dorota Aniszewska and Marek Rybaczuk Institute of Materials Science and Applied Mechanics Wroclaw University of Technology 50-370 Wroclaw, Smoluchowskiego 25, Poland (e-mail: dorota. [email protected], [email protected])

Abstract: The goal of the paper is chaos examination in multiplicative systems. The paper collects results of numerical simulations as well as presentation of methods applicable in the case of multiplicative systems. Chaos examination concerns one-dimensional multiplicative version of logistic equation and multi-dimensional nonlinear system described with multiplicative derivatives. The classical Lorenz system transformed into multiplicative version was chosen for analysis of stability and chaotic behaviour. Keywords: multiplicative calculus, logistic equation, the Lorenz system, RungeKutta method, Lyapunov exponent.

1

Introduction

Main subj ect of our work is examination if chaos may occur in process of defects evolution in material. Model of fractal defects growth has been presented in Rybaczuk and Stoppel[4]. It is based on single fractal approximation and links the energy uniformly distributed over fractal , its measure VD and energy density a(D), which is essential material characteristics depending on fractal dimension: [ =

a(D)vD.

(1)

Appearance of fractals requires changes in the form of physical equations. Mathematical theory of dynamical systems describing physical processes must pay attention to physical quantities. In model (1) shift of fractal dimension means shift of physical dimension. Classic derivative based on differential quotient containing a(D + L1D) - a(D) has no physical sense. The labile physical dimension of a(D) is the essential reason for employing multiplicative calculus to formulate dynamical system of defects growth. This paper collects results of chaos examination for exemplary multiplicative systems. It is expected that presented methods are applicable to a wide variety of multiplicative dynamical systems including dynamical models of fractal defects evolution.

10

2

D. Aniszewska and M. Rybaczuk

Multiplicative calculus

Multiplicative calculus has been presented in Volterra[5]. The differential multiplicative calculus has been restored in Rybaczuk[3]' Rybaczuk and Stoppel[4]. Definition of the multiplicative derivative offunction f (x) with respect to x follows as: 1 7r

f (x) =

lim {

7rX

£->0

f (x (1 + E)) } , f(x)

(2)

,

where shift of independent variable x is multiplicative x --) x(l + E), therefore x E lR.+ and f(x) E lR.+. In Rybaczuk[3] the generalized methods to examine equations comprising quantities with labile physical dimensions are presented. All notions and theorems of the classic differential calculus have their multiplicative counterparts. To assist further applications we present some of them. If all subsequent multiplicative derivatives exist, multiplicative derivative of sum and product of functions f(x) and g(x) equals respectively: 7r f(x)

7rf(x)+g(x) 7rX

f(x) f(xJ+g(x)

7rg(x)

7rX

g(x) f(xJ+g(x)

(3)

7rX

7rf(x)g(x) 7rX

7rf(x) 7rg(x) 7rX 7rX

(4)

----

The formula for a complex function is more complicated:

7rf(g(x))

7r

f (g)

I

~ f(cc)

n ----;ex

7r

g (X )

In

~ f(g)

----:;cg

(5)

The function f(x) can have extrema in such points for which:

7rf(x) = 1 7rX

3

(6)

Lyapunov stability and Lyapunov exponent

Lyapunov type stability for system of multiplicative differential equations is introduced in Aniszewska and Rybaczuk[2] in details. In this paper it is summarized briefly. Equilibria of system of autonomous multiplicative differential equations ~ = fj (x!, ... , x n ), where j = 1, ... , n are obtained from:

(7) . · S olutlOn c1ose to fi xe dpomt Xo = multiplicative shift Ej (t):

((0) Xl

,x 2(0) , ... , ;Z;n(0)) is d escri b ed b y small

(8)

Chaos in Multiplicative Systems

Behaviour of derivative 7rE;i t ) for long time determines if equilibrium stable. Calculation of multiplicative derivatives gives following result:

11

Xo

is

(9) where

EjO)

is determined by initial conditions. Stability of fixed point

7r ~~:o)].

pends on eigenvalues of matrix [In

Xo

de-

For complex eigenvalue stability

depends on their real part. If it is negative, for long time Ej (t) ---+ 0 and Xo is stable. If it is positive, value of Ej (t) increases according to power function of time and Xo is unstable. Divergence or convergence of trajectories with close starting points proceeds according to power of time Ej :::::; EjO)t>', therefore Lyapunov exponents for multiplicative dynamical system can be defined as: \ Aj

.

1

Ej

(10)

= 11m -1- In -(0) t~oo n t E. J

and if at least one of them is positive, trajectory behaviour is chaotic.

4

Multiplicative logistic equation

X2n

= rx 1 -

x

The simplest systems employed to chaos examination are one-dimensional ones. Numerical simulations and calculation of Lyapunov exponent have been performed for multiplicative version of logistic equation, which can have a few forms. One of them follows as:

(11) where n is a step number. In case of multiplicative system, shift of step must be multiplicative: n ---+ n(l + .:1n), where .:1n = 1 for discrete system. Therefore following numbers of steps are calculated as 2n equals 1,2,4,8,16, .... This version of equation is analysed in details in Aniszewska and Rybaczuk[2]. In this paper we present another multiplicative version of the classical logistic equation, which has the form: X2n

= rx l-x

(12.)

and for this equation chaos appears for certain range of r parameter values. Phase diagrams and step series for various r values are presented in Figures 1 and 2. For r = 2 there are only two solutions and for r = 4 chaotic behaviour appears. Bifurcation diagram is shown in Figure 3(a). Calculation of Lyapunov exponent is based on two trajectories started from points with distance determined by small multiplicative shift EO. When

[2

D. Alliszewska alld M. Rybaczuk

,

if-

i

05t

oc--~-·-~

o

0.5

--- ....

~.--.-."-,----.--,-------.- . .)

1.5

2

2.5

x"

(b)

( a)

Fig. 1. Phase diagrams and step series of the multiplicative logistic equation :C2n = r::rl-~"

fix

T

= 2.

(b)

( a)

Fig. 2. Phase diagrarns and step series of the multiplicative logistic equation rxl~'J.'

for

T

:(;2",

=

= ,1.

in each k step multiplicative shift tk bet"veen points is calculated. then for many steps ,YO obtain value of Lyapunov exponent: N

\

1. "1 tk ~ n--

A= ----

InN

.

k=.l

to'

(13)

where N is number of steps. Relationship between Lyapullov exponent and parameter is presented in Figure 3(b). Slighly different version of multiplicative logistic equation has the form :r2n = (rx) 1-",. Its behaviour its similar, but chaos appears for other range of T' values. Bifurcation diagram for this version is presented in Figure 'l(a) and relationship between Lyapul10v exponent and T' parameter is shown in Figure 4(b). T

Chaos ill Multiplicative Systems

10

15

20

(a)

(b)

Fig. 3. Bifurcat.ion diagram for the multiplicative logi:>tic equation and relationship between Lyapunov exponent and r parameter.

.T2n

(b)

(a)

Fig. 4. Bifurcation dia.gram for the multiplicative logistic equa.tion and relationship between Lyapunov exponent and r parameter.

5

13

3':2",

= (r:r) [-a'

Multiplicative Lorenz system

Main goal of this paper is chaos examination in system described with multiplicative differential equations. The Lorenz system containing three nonlinear equations with three positive real parameters: d:1:

-= (JY. elt dy

--= rlt

r:D -

0';1:

'!/ - .TZ

.

dz -1 = ;r:y -Ire d

(14)

14

D. Aniszewska and M. Rybaczuk

was chosen as an exemplary chaotic dynamical system and it has been transformed into multiplicative dynamical system: addition was replaced by multiplication and multiplication by raising to suitable power. The multiplicative Lorenz system equals:

(15)

Detailed analysis of Lyapunov type stability for this system was presented in Aniszewska and Rybaczuk[2]. It provide the same results as for classical Lorenz system. The solutions of nonlinear multiplicative dynamical systems can be calculated using numerical methods therefore multiplicative RungeKutta methods presented in [1] was employed to perform numerical simulations.

0

,= ----- '000 <10" <10'

o

0

(b) r = 28

(a) r = 10 50

15

40

rf

10

N

3' 0 10

30

10

LN(Y)

o

0

(c)r=10

LN(X)

(d)r=28

Fig. 5. Trajectories of the multiplicative Lorenz system (15) for various r values in linear and logarithmic scale.

CJ

= 10, b = ~ and

Chaos in Multiplicative Systems

15

Numerical solutions of the multiplicative Lorenz system (15) revealed that chaos appears for T > 24.84 for a = 10 and b = ~. Trajectory spirals out two equilibria and creates butterfly shape attractor, which is visible in logarithmic scale. In case of multi-dimensional dynamical systems calculating Lyapunov exponents is more complicated, because local behaviour of close trajectories may vary with the direction. For calculation of the largest Lyapunov exponent of the multiplicative Lorenz system we employed multiplicative version of method introduced in [6]. For classical dynamical system the largest Lyapunov exponent equals: . 1 L(t) (16) A = t-Hx) lun -In-(-), t L 0 . where L( t) is linear extent of ellipsoid created in space by two trajectories. For multiplicative dynamical systems the largest Lyapunov exponent should be calculated as: . 1 L(t) (17) A = t~oo lun -1n- t In -( -) , L 0 . because time shift is multiplicative. Results of the largest Lyapunov exponent computation for the multiplicative Lorenz system are given in Table 1.

11-0.93

1-0.59

1- 0.34

1- 0 . 12

1

0.82

1

0.91

Table 1. The largest Lyapunov exponent of the multiplicative Lorenz system for IJ = 10, b = ~, 40 000 steps, starting time 0.01 and time step 0.01.

6

Discussion

The results presented in the paper proof that all classical conditions concerning chaotic behavior can be extended onto multiplicative systems. The paper presents exemplary multiplicative systems with chaotic behaviour. Similarly like in case of dynamical system described with classical derivatives, nonlinearity and three indepenent variables are necessary for chaos occuring in systems described with multiplicative derivatives.

References l.D.Aniszewska. Multiplicative runge-kutta methods. Nonlinear Dynamics, 50:265 - 272, 2007.

16

D. Aniszewska and M. Rybaczuk

2.D.Aniszewska and M. Rybaczuk. Lyapunov type stability and lyapunov exponent for exemplary multiplicative dynamical systems. Nonlinear Dynamics, 54:345 ~ 354, 2008. 3.M.Rybaczuk, A. Kedzia, and W. Zielinski. The concept of physical and fractal dimension II. the differential calculus in dimensional spaces. Chaos, Solitons E'3 Fractals, 12:2537~2552, 200l. 4.M.Rybaczuk and P .Stoppel. The fractal growth of fatigue defects in m aterials. International Journal of Fracture, 103:71 ~94 , 2000. 5.V. Volterra and B.Hostinsky. Herman, Paris, 1938. 6.A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano . Determining lyapunov exponents from a time series. Physica 16 D, pages 285 ~ 317, 1985.

17

Deterministic Chaos Machine: Experimental vs. Numerical Investigation Jan Awrejcewicz and Grzegorz Kudra Department of Automatics and Biomechanics Technical University of Lodz , Poland (e-mail: [email protected])

Abstract: Detenninistic chaos machine consisting of the plane triple pendulum as well as of driving and measurement subsystems is presented and studied. The pendulum-driving subsystem consists of two engines of slow alternating currents and optoelectronic commutation. In addition, a mathematical model of the experimental rig is derived as a system of three second order strongly nonlinear ODEs. Mathematical modeling includes details, taking into account some characteristic features (for example, real characteristics of joints built by the use of roller bearings) as well as some imperfections (asymmetry of the forcing) of the real system. Parameters of the model are obtained by a combination of the estimation from experimental data and direct measurements of the system's geometric and physical parameters. A few versions of the model of resistance in the joints are tested in the identification process. Good agreement between both numerical simulation results and experimental measurements have been obtained and presented. Keywords: triple pendulum, experiment, chaos, mathematical modeling, parameter estimation.

1. Introduction Development of mathematics, mechanics and related numerical methods allow more exactly model the real dynamic phenomena that are exhibited by various physical objects. Following historical overview of the natural sciences mentioned, a significant role is played by a physical pendulum, which is the very useful mechanism used in the design of various real processes. As a single or a double pendulum are often studied experimentally [5,6], a triple physical pendulum is rarely presented in literature from a point of view of real experimental object. For example, in the work [7] the triple pendulum excited by horizontal harmonic motion of the pendulum frame is presented and a few examples of chaotic attractors are reported. Experimental rigs of any pendulums are still of interest of many researchers dealing with dynamics of continuous multi degrees-of-freedom mechanical systems. The model having such a properties has been analyzed in work [3]. It consists of a chain of N identical pendulums coupled by dumped elastic joints subject to vertical sinusoidal forcing on its base. Lately, also the monograph on the pendulum has been published [4]. This is a large study on this simple system also from the historical point of view. In February, 2005, in the Department of Automatics and Biomechanics, the experimental rig of triple physical pendulum was finished and activated. This stand has been constructed and built in order to investigate experimentally various phenomena of nonlinear dynamics, including regular and chaotic motions, bifurcations, coexisting attractors, etc. In order to have more deep

18

1. Awrejcewicz and G. Kudra

insight into dynamics of the real pendulum, the corresponding mathematical model is required. In the work [1] the suitable mathematical modeling and numerical analysis have been performed, where the viscous damping in the pendulum joints (constructed by the use of rolling bearings) has been assumed. In the next step [2], we have also taken into account the dry friction in the joints with many details and variants. Here we present the model of friction taking into account only essential details.

Figure 1. Experimental rig: 1, 2,3 - links; 4 - stand; 5- rotors ; 6 - stators; 7, 8, 9 rotational potentiometers

2. Experimental setup The experimental model (see Fig. 1) of the triple physical pendulum is composed of the following main modules: pendulum, drivin·g subsystem and the measurement subsystem. The construction of the pendulum ensures the plane motion of the dynamical system. The links (1 , 2, 3) are suspended on the frame (4) and joined by the use of radial and axial needle bearings. The fust link is forced by a special directcurrent motor of our own construction with optical commutation consisting of two stators (6) and two rotors (5). The construction ensures avoiding the skewing of the structure and forming the forces and moments in planes different

Deterministic Chaos Machine

19

that the plane of the assumed pendulum motion. On the other hand the construction allows the full rotations of all the links of the pendulum. The voltage conveyed to the DC motor inductors is controlled by the use of special digital system of our own construction together with precise signal generator HAMEG. As a result the square-shape in time of the voltage signal (but with some asymmetry - see the next sections) with adjustable frequency is obtained. The measurement of the angular position of the three links is realized by the use of the precise rotational potentiometers (7, 8, 9). Then the LabView measure-programming system is used for experimental data acquisition and presentation on a computer.

y

Figure 2. Physical model of the pendulum.

3. Mathematical model In the Figure 2 the idealized physical concept of real pendulum is presented. The system is idealized since it is assumed that it is an ideally plane system of coupled links, moving in the vacuum with the assumed model of friction in joints. Each of the pendulums has a mass center lying in the lines including the joints and one of the principal central inertia axes (Zei) of each link is

20

1. Awrejcewicz and G. Kudra

perpendicular to the movement plane. The system is governed by the following set of differential equations:

where the pendulum position is described by the use of three angles If/i (i = 1,2, 3), Mit) is the external forcing of the first link, MRi (i = 1, 2, 3) are friction torques generated in the joints and where the following notation is used

M, =m,ge, +(m2 +m3)gl, ' M2 M3

= m2ge 2 + m3g12, = m3ge3,

= I, +e,2m, +1,2 (m2 +m3)' 2 2 B2 = 12 + e2 m2 + 12 m3 ' B,

B3 =13 +e32m3 ' N'2 = m2e21, + mi,12' N'3 = m3 e31, ' N 23 = m3e312,

where mj and Ii (i = 1, 2, 3) are the masses and inertia moments with respect to the central axes perpendicular to the movement plane of the corresponding links and g is gravitational acceleration. Other geometrical parameters are seen in the Figure 2. Note that there is the following dependence between introduced quantities

Deterministic Chaos Machine

21

and

The resistance torques in the pendulum joints are modeled with the assumption that they are the function s of the relative velocities of the corresponding links only (among others they are not dependent on the bearing loadings):

M RI

=I; 3.. arctan (£I'ifl) + CI'if1 ' Jr

M R2

= T2 3.. arctan (£2 ('if2 - 'ifl)) + C2('if2 - 'ifl) ,

M R3

= T33.. arctan ( £ 3 ( 'if3 -

Jr

Jr

'if2 ) ) + C3 ( 'if3 - 'if2 ) ,

and they are composed of two parts: linear (viscous) part and nonlinear part in the form of arctan function as seen above. The arctan (or tanh) function is usually used in modeling of dry friction with high value of the c parameter and its classical objective is to obtain a smooth approximation of the sign function. Although our initial goal was just a smooth approximation of sign function, the parameter estimation results (see the next section) have been shown, that smaller values of the c parameter can give better results of modeling of the investigated triple pendulum. The external input to the system is the voltage signal u controlling the direction of the constant voltage applied to the DC motor of the pendulum. The signal u in the pendulum model can be an arbitrary function of time, and in particular, it can be the same function as applied (and recorded in a file) in real system (it is useful in the parameter estimation process). On the other hand, it is possible to apply the signal u due to the following mathematical description:

U(f)={1

-I

for mod ( lOt + 90' 2Jr) ::; 2Jra for mod(lOt+90,2Jr»2Jra'

which imitates the signal applied in the real pendulum, with adjustable angular velocity w, initial phase Wo and coefficient a (in the investigated experimental pendulum there is a = 0.4927 which means an asymmetry in the forcing, as mentioned in section 2).

22

1. Awrejcewicz and C. Kudra

The relation between the control signal u and torque Me acting on the first link is modeled by the following equation

Me (t) + TM dM e = q U (t) dt

,

where q is the amplitude of the external torque corresponding to the constant voltage applied in DC motor and TM is the time-constant of the DC motor.

4. Parameter estimation and experimental vs. numerical results Although some parameters of the system (for example masses, some geometrical lengths) can be obtained from direct measurements or calculated from such measurements, it can be difficult, labour-consuming or inaccurate for some of them. Because of that we have used only values that can be got easily and accurately, that is g = 9.812 rnIs 2, II = 0.17418 m and 12 = 0.22405 m. Other model parameters are obtained from identification process, that is from experimental data of input and output signals, where the input signal is the voltage signal u controlling the direction of the constant voltage applied to the DC motor of the pendulum and the outputs are the angular positions of the pendulum links lfIi (i = 1, 2, 3). One can also note that some other special experiments might be performed in order to identify particular model parameters, but here we assume that only above mentioned experimental data from the triple pendulum is available. The model parameters are estimated by the global minimum searching of the error-function of the model and real system matching. The matching of model and real system is understood as the matching of the corresponding output signals lfIi (i = 1, 2, 3) from model integrated numerically and from the real pendulum, assuming the same inputs to both the model and real system. The sum of squares of deviations between corresponding samples of signals from model and experiment, for few different solutions, serves as a criterion function. A minimum is searched by applying the simplex method. In order to avoid the local minima, the simplex method is stopped from time to time and a random searching is then applied. After random searching the simplex method is restarted again. If we divide final value of criterion-function by the number of samples used in calculation of the error -function, we obtain average square of deviation between two signals (obtained from the model and the experiment) let us denote this parameter as Fer- Now this parameter can be used for comparison of matching of different sets of experimental data and corresponding numerical solutions.

Deterministic Chaos Machine

23

Since the amplitude of the forcing is assumed to be known before identification process (q = 1.718 Nm), the full vector of the parameters to be estimated is

and other parameters are dependent since

and

0.2 0.103 0.1 ~

0.05

~

0

eel

,-<

-~

--l _-------................... _...................

-0.15

....

........ . .-

\

-0.05 -0.1

r

I

f

J

.

.

~

..

-

\ \..

I

-0.2

o

0.2

0.4

0.6

0.8 1.2 experiment [s] simulation forcing 1.5 ,----,---,--,---,----,-"-------,

t

~

0.5

eel

~

""

.~

0 -0.5 -1

-1..5 0.8 1.2 experiment simulation forcing Figure 3. Model and real system matching for periodic solution with forcing frequency 0.85 Hz. 0

0.2

0.4

0.6

t []S

Note also that the accuracy of the measured amplitude of the external forcing (q = 1.718 Nm) is not essential for the accuracy of dynamical model since both sides of the governing equations can be divided by q eliminating in that way the q parameter. In all the identification processes, the same set of experimental solutions is used: five periodic solutions with the forcing frequencies (j = co): f = 0.2,

24

1. Awrejcewicz and G. Kudra

0.35, 0.6, 0.85 and l.IHz and one decaying solution, which starts from the periodic attractor with forcing frequency f = 0.5 Hz (after few seconds of the recorded motion, the forcing was switched off and the total length of the recorded motion was 24 sec).

~ cd

~ ...-<

~

0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0

0.5

1

1.5

2

2.5

3

3.5 4 4.5 5 experiment simulation forcing ---------

3

3.5 4 4.5 5 experiment simulation forcing ---------

t [sJ 0.4 0.3 0.2

~

cc

~ C'1

~

0.1 0 -0.1 -0.2 -0.3 -0.4 0

0.5

1.5

2

2.5

t [SJ

Figure 4. Model and real system matching for periodic solution with forcing frequency 0.2 Hz.

We have tested three versions of the model presented in this paper. Let us name them: A - the model with £1= £2= £3=10 3 and CI= C2= C3= 0; B - the model with £1= £2= £3=10 3 and C - the model with full vector of the estimated parameters. For the model A we have obtained Fer = 0.2396.10-3 rad 2, for the B model Fcr = 0.2106.10- 3 rad 2 and for the C model Fer = 0.1988.10-3 rad l . For further simulations and presented here examples the C model is used for which we have obtained the following set of parameters: BI = 0.1638 kg·m l , Bl = 0.1380 kg·m 2, B3 = 0.0161 kg·m2, MI = 8.71 Nm, M2 = 6.214 Nm, M3 = 1.106 Nm, TI = 0.0714 Nm, T2 = 0.0173 Nm, T3 = 0.0063 Nm, £ I = 10.6 s, £ 2 = 53 .9 s, £ 3 = 2l.5 s, , c I = 3.3.10-3 N·m·s, c 2 = l.3·10-3 N·m·s, c 3 = 0.9.10- 3 N·m·s, TM = 2.7·1O- 3 s.

Deterministic Chaos Machine 500

500

400

400

300

300

200

200

1/)3 100

100

-lOO

-100

-200

-21l0

25

0

<100

-:lOO - .lrr

·17r

8"

1.2rr

0

10

20

30

40

.50

60

l!h

Figure 5. Model and real system chaotic be va vi or for forcing frequency 0.73 Hz.

As seen in Figures 3 and 4 the results obtained from model fit those observed experimentally very well. As we have tested, the model can be very powerful tool used in predicting and explaining the nonlinear dynamics phenomena exhibited by the real triple physical pendulum. As an example, the model have been predicted the chaotic window observed experimentally in the range of forcing frequency f between 0.6915 Hz (for increasing frequency) and 0.7745 Hz (for decreasing frequency). In Figure 5 there are two chaotic solutions behavior for forcing frequency 0.73 Hz obtained both experimentally and numerically from the model.

5. Conclusions The main objective of the work was to find the mathematical model and their parameters which would be suitalble for numerical simulations and analysis of nonlinear dynamics phenomena exhibited by the real triple physical pendulum. The essential point of the mathematical modeling was to find mathematical description of the resistance torque present in the rotational joints of the pendulum. The model adopted here is relatively simple since the friction torques are function of the relative angular velocities of the links only, that is their dependence on bearings loadings as well as other dependencies are neglected. However, this simple model seems to be very good approximation of the real pendulum and very powerful tool that can be used in the triple pendulum dynamics prdiction and explanation.

References [I] Awrejcewicz investigation Proceedings Applications,

J., Kudra G. and Wasilewski G.: Experimental and numerical of chaotic zones exhibited by the triple physical pendulum, of the 8th Conference on Dynamical Systems - Theory and L6dz, Poland, 2005, 183-188.

26

1. Awrejcewicz and G. Kudra

[2] Awrejcewicz J. , Supel, Kudra G., Wasilewski G. and Olejnik P.: Numerical and experimental study of regular and chaotic motion of triple physical pendulum, International Journal of Bifurcation and Chaos, 18(10), 2008

[3] Galan J., Fraser W.B., Acheson OJ. and Champneys A.R.: The parametrically excited upside-down rod: an elastic jointed pendulum model, Journal of Sound and Vibration, 280, 200S, 3S9-377. [4] Baker G.L. and Blackburn J.A.: The Pendulum. A Case Study in Physics, Oxford University Press, 200S . [S]

Bishop S.R. and Sudor D.J.: The "not quite" inverted pendulum, International Journal of Bifurcation and Chaos, 9(1), 1998, 273-28S.

[6] Blackburn J.A., Zhou-Jing Y., Vik S., Smith H.J.T. and Nerenberg M.A.H.: Experimental study of chaos in a driven pendulum, Physica, 026(1-3), 1987, 38S39S . [7] Zhu Q. and Ishitobi M.: Experimental study of chaos in a driven triple pendulum, Journal of Sound and Vibration, 227(1),1999, 230-238 .

27

Some Issues and Results on the EnKF and Particle Filters for Meteorological Models Christophe Baehr 1 ,2 and Olivier Pannekoucke 1 1

2

Meteo-France / CNRS CNRM / GAME URA1357 42 Avenue G. Coriolis, 31057 Toulouse Cedex 1, France (e-mail: christophe. [email protected], olivier [email protected]) Universite de Toulouse Paul Sabatier Institut de Mathematiques 118 route de Narbonne, 31062 Toulouse Cedex 9, France

Abstract: In this paper we examine the links between Ensemble Kalman Filters (EnKF) and Particle Filters (PF). EnKF can be seen as a mean-field process with a PF approximation. We explore the problem of dimensionality on a toy model. To by-pass this difficulty, we suggest using Local Particle Filters (LPF) to catch nonlinearities and feed larger scale EnKF. To go one step forward we conclude with a real application and present the filtering of perturbed measurements of atmospheric wind in the domain of turbulence. This example is the cornerstone of the LPF for the assimilation of atmospheric turbulent wind. These local representation techniques will be used in further works to assimilate singular data of turbulence linked parameters in non-hydrostatic models. Keywords: Ensemble Kalman Filter, Particle Filter, data assimilation, mean-field process.

1

Introduction

The major problems in data assimilation for geophysical models come from nonlinearity of dynamics, non-gaussianity of perturbations and high dimensions of state space. Ensemble Kalman Filters (EnKF) was a first answer to these difficulties. For a few years some authors have tried to use Particle Filters (PF) roughly to propose an alternative strategy. But directly applied this new approach stumbles across the problem of the dimensionality. In this paper, we present the links between EnKF and PF, we also remind that the EnKF converges but tends to a particular process and we describe the dynamical system of the nonlinear filter distribution. In the case of the PF, with a modified selection step, we investigate the effect of an increasing state space dimension for a constant number of particles. Then we suggest to couple EnKF and Local Particle Filter (LPF) to propose solutions in the vicinity of strong uncertainties. The next step is the use of LPF with a stochastic representation of the medium and we present some results on the filtering

28

C. Baehr and O. Pannekoucke

of real turbulent wind measurements. In the conclusion we see the expected consequences of this work for meteorological models.

2

EnKF as a mean-field process

The nonlinear filtering process has a complete description in terms of FeynmanKac distribution (see [3]). For instance in discrete time, we consider the dynamical system for the state vector Xn E ]Rd partially observed by the process Y n : Wn ~ N(O, 1) Vn ~N(O, 1)

(1)

where An and C n are nonlinear functions of Xn (Cn is linear for EnKF), Qn and Rn are covariances of the Gaussian noises. We denote Mn the Markov Kernel of the state vector X n . The filtering problem consists in calculating the distributions fin = Law(Xn I 1[o,nj) and T)n = Law(Xn I 1[0,n-1j), where 1[o,nj is the collection {Yo, ... , Yn}. The nonlinear filter is therefore a sequential algorithm that gives the solution of the dynamic system T)n = T)n-1 K n- 1,1Jn_l' where K n- 1,1Jn_l is a (non-unique) Markov kernel representation of the filtering process. For genetic type filtering algorithm, the kernel K n- 1,1Jn_l is K n- 1,1Jn_l = Sn-1,1Jn_lMn with Sn-1,1Jn_l a (also nonunique) selection of adapted states to be defined. We call mean-field process, a process for which the evolution depends on a probability law 7fn , a priori or conditioned to the observations, for instance with the dynamical evolution X n+ 1 = F(Xn, 7fn ) where F is a nonlinear function. The filtering process is mean-field type. Now we will see the corresponding mean-field process of the EnKF. The EnKF is a clever technic to use an ensemble of state to approximate the covariance error matrix by an empirical matrix. The motivation of this approximation is the high dimensional size of the state vectors. The equations of the EnKF are described in [5], the filter has a prediction step and a corrective update. The convergence of the filter is proved in [6], but the limit process is not the filtering process. Denoting Zn the update process, it is a Markov process following the nonlinear equation Zn = An(Zn-d + ~.Wn + Gn.[Yn - Cn.An(Zn-l) - Cn.~.Wn + VRn.Vn] where Gn = Pn C,;[Cn Pn C,; + R n ]-l, Rn = lE([vIRn Vn]vIRn VI) and Pn = lE([An (Zn-1) +,;Q;, Wn][An(Zn-d +,;Q;, WnV). This mean-field process could have a particle approximation, and with a N particles system (Z~h'Si'SN computing the empirical average Zn = 1:t L~l Z~ we have Zn = An(Zn-d + G n [Yn - C n An(Zn-dL and if An is a linear function, we get exactly the Kalman estimator. The estimation is exact if the pair (Xn, Y n ) is linear and Gaussian, and that is the best linear estimator of Zn in the other cases. G n is an mean-field operator according to T)n and the EnKF filtering process approaches the dynamical equation T)n = T)n-1 Cn ,Yn,1Jn-l Mn where Cn ,Y;,,1Jn_' is a correction kernel induced by Gn[Yn - C n F(Zn-1, W n ) - Vn ].

Issues and Results on EnKF and Particle Filtersfor Meteorological Models

29

. . Cn,Y",1Jn-l (x,d Z ) -- exp ([z-X -Cn(Yn-CnX W ) dz. The For G aUSSlan nOlses, 2C2 R EnKF, as the number of elements goes to 00, tends to ~ ~ean-field process Zn different from the filtering process. With a small number of elements, the EnKF by its correction method is better than a PF, but when this number increases largely, only the PF converges to the optimal filter. All stochastic nonlinear filter have two steps, one is the prediction according to the dynamic model, the other is an update through a selection process. At this present time, no correction process is available to ensure the convergence of the nonlinear filter. The exact filter laws are not analytically known ( except in the linear Gaussian case with the Kalman estimator) and we have to use a particle approximation to learn these probability laws. To filter mean-field processes, there are various particle algorithms (see [1]). All are based on a mean-field Markovian model, a genetic selection rule and particle approximation for the filtering laws and for the mean-field laws. Now we turn the discussion onto the selection step with limited numerical ressources which is the core of the problems and the success of any particle or ensemble filter.

3

Particle filters regimes

Initially for nonlinear filters, the selection step was an Importance Sampling (IS). This kind of selection brings some difficulties with filter collapses. This is the motivation of the recent paper [8] relatively to the dimensionality. But since the late 90's, genetic selection have shown their efficiency to the filtering problems. In this selection, there is an acceptance/rejection of the state and only the rejected state are resampled. More precisely, the observational equation Yn = Cn(Xn) + $n.Vn leads to a potential function Gn ( see [3] ) which evaluates the adaptation of a state point Xn with respect to Yn . For a parameter En 2': 0 such that EnGn E [0,1]' the selection kernel is defined by Sn ,1Jn (x, dy) = EnGn(x)6x(dy) + [1- EnGn (X)]ljIn (7)n)(dy) where IjIn(7)n)(dy) = 1Jn(d,(l(j')(Y) is the resampling law. In the case of the high dimen17n n sional state space we suggest to choose for the parameter En = 1/ ess sup( G n). A small noise is added on each particle to insure the exploration of the state space, and the potential G n is corrected consequently. The use of genetic selection and this choice of the parameter En provide a very different behavior in comparison with the IS selection, especially with limited computational ressources. Snyder et al. suggest to examine the possibility of a PF collapse with wnmax , the maximum of the weight Wn = c(Cn n ). The filter is reputed to be collapsed if W,7'ax is almost surely (a.s.) equal to l. We conduct some numerical experiments using the dynamical model proposed by Lorenz in 1996 (see [7]). We used this chaotic model because we can easily increase the size d of its state space. We observe directly half of the state space and perturb the observation vector with a standard Gaussian noise. A PF using N = 1000 particles with a genetic selection filters the signal during 1460 ~

30

C. Baehr and O. Pannekoucke

time steps. We examine here the histograms of maximum of weight w:,ax. The results for dimensions d = 200,600 and 1500, presented figure 1, are quite unlike the diagrams proposed by Snyder for the IS. For a fixed number N = 1000 of particles, the collapse is reached for a higher dimension. In fact the question seems not to be on the dimension but on the existence of a critical number of particles for a given model and a given selection rule. Theoretical works are still to be done, but numerically it seems that three regimes are possible. The first one is a collapsed filter where w:,ax = 1 a.s., the filter is fully divergent. The second is a transitional regime and the filter may be locally divergent. The third is the optimal situation when the filter converges and JP'(w:,ax :::; an) = 1 where an < 1. A fourth regime occurs when all particles have a.s. the weight liN which corresponds to an ill-adapted system. In the case of Lorenz-96 model, with the chosen selection rule, the critical number of particles seems to be O(d), while Snyder et al have shown for IS an exponential critical number. Keep in mind that for other models or other selection scheme, the result could be very different.

Fig. 1. Empirical histograms of maximum of weight w~ax for a PF with 1000 particles and a genetic selection for the Lorenz-96 model with dimension d = 200 (top left), d = 600 (top right) and 1500 (bottom left). On bottom right, we summarize the 3 regimes w.r.t. to the dimension (1- collapsed filter, 2- transition regime, 3convergent filter) and a 4th regime when the system is ill-adapted.

Issues and Results on EnKF and Particle Filters/or Meteorological Models

4

31

Couple together a EnKF and a Local Particle Filter

To filter or assimilate data for meteorological system, operational forecast centers begin to use the EnKF. Joined to ensemble forecast systems, this way is very promising. But ensemble assimilation do not face rapid and local evolutions brought by nonlinear phenomena. A PF may be more efficient to catch these nonlinearities, and with an equal number of elements, PF are cheaper than EnKF. To bypass the problem of computational costs induced by PF in high dimension problems, it could be interesting to couple locally some Ensemble Filter and Local Particle Filter (LPF). This section is the presentation of a numerical example using a 1D Burgers equation. Discretized on the interval [0,1] with a spectral computation scheme, this equation could be seen as the propagation of a wave along a latitude circle with the generation of a front. The Burgers equation has been chosen for its ability to generate nonlinearities. As a reference, we consider a fine scale model all over the domain, with 361 points (blue dot on each part of fig. 2 ) and generate with it perturbed observations. Then we use an EnKF (canadian type with 2 ensembles) with 100 elements, a larger grid with 161 points and we assimilate the observations every 10 time steps. On fig. 2 top left, we examine the results after a cycle of 37 assimilations. Due to the nonlinearities, the EnKF (green curve) shows some spatial shift in its reaction, and does not follow the discontinuities, rejecting the observations too close to the front. The black dash line is the same model without assimilation. These areas are therefore where the covariance prediction error matrix has its bigger values (see the dispersion of the ensemble on fig. 2 bottom right, the green curve). Then we place a smaller domain centered on the first front (light blue longdash rectangle on top right) with a refined grid, the state vector has 161 dimensions on the intervalle [O,~] and a Limited-Area Model (LAM). There, we use a LPF with 100 particles to filter the same set of observations (green crosses). The LAM has an adapted set of parameters: time, diffusion coefficient, etc. On fig. 2, top right, we see with the red curve that the LPF fits correctly the front and perform the best assimilation in the LAM area for an equivalent cost than EnKF in this case. Then a coupling of the LFP with large scale model is performed. To feedback the information of the LAM particles to the EnKF elements, for each element of the ensemble we randomize a particle according to the a posteriori law. The result is shown on fig. 2 bottom left and we see clearly the contribution of this coupling EnKF-LPF technique. On the fig. 2 bottom right the variance error of the LPF coupled with the EnKF is largely reduced in the LAM area.

5

The filtering of atmospheric turbulent wind with Local Particle Filter

Regionalization of PF seems to be a response to the problem of dimensionality. It is possible to go one step forward with pointwise PF (one PF per grid-

32

C. Baehr and O. Pannekoucke

Fig. 2. Observed 1d Burgers system, on each part the true state is in blue dots and black dash lines are the EnKF without assimilation. The observations (green crosses) are pertubed by white noise. On top left, the green line is the EnKF estimation with 100 elements. On top right, the red line is the LPF estimation in the LAM area with 100 particles and a genetic selection. Bottom left part, in red line is LPF coupled with large scale EnKF. Bottom right part is the dispersion of the EnKF in green and the red line is the coupled LPF-EnKF in the LAM area.

point). This technique requires two ingredients. The first one is a stochastic representation of the medium, for the atmospheric wind this is a Stochastic Lagrangian Model (SLM), and then a conditioning process to a ball centered on each gridpoint. Large scale components may be learned from the ensemble assimilation system, each PF estimates subgrid quantities and uploads the information to the larger scale model or learned from observations. The PF for turbulent wind and the conditioning process are described in [1]. Here we present the result of real data numerically perturbed and filtered by a LPF with a SLM for 3D stratified turbulent flows inspired by [2]. The model has 7 dimensions (3 for the location, 3 for wind components and one for temperature). The figure shows series of horizontal wind with the perturbed signal in blue, in black the real signal and in red the denoised signal with LPF using 300 particles. On the right part of the picture we examine the energetic structure with Power Spectral Density (PSD) and see that the corrections are very impressive even if the noises are strong.

issues and Results on EnKF and Particle Filtersfor Meteorological Models

10

I

(}

10

10

10

33

1Il

i

500 1000 1500 2000 2500 3000 3500

Fig. 3. Left, series of horizontal wind (velocity (m.s - I ) vs t ime step number) and right, PSD (with a log-log scale, power (dB) vs freq uency (Hz» . The series on the top is t he U componant and the bottom the V componant. Time series are taken the 18th of July 2006 b etween 16h58 and 17hOO UTe. In light blue the perturbed signal to be denoised, in black the reference sig;l1a.l and in red the filtered with 300 particles.

T his estimation technique provides not only unperturbed states but also the estimation of quantit ies used is the dynamical model. In our case, it could be able to produce series of turbulent dissipation rate, kinetic energy, buoyancy coefficient, etc. With these estimations it is possible to close the large scale model not with empirical closures but with the observations. For a model with fully decorrelated dimensions, pointwise PF is a cheaper technique than global PF even if a pointwise filter is computed for each gridpoint.

6

Outcomes and further developments

In t his short paper we have seen t hat EnKF converges to a mean-field process which is not the filter process, while the PI" converges to the optimal 51ter. Particle F ilter is a. generic name , everything takes place in t.he stage of se1ec·· tion. We have seen that it is worthwhile to use a genetic selection instead of IS for high dimensional problems. But anyway the PF requires a lot of particles as the dimension of the state space goes to infinity. We have developed

34

C. Baehr and O. Pannekoucke

a strategy to couple together EnKF and LPF, and test it on a discretized toy model. We have pursued our investigations with Pointwise PF and seen that they are efficient for filter measurements in the domain of turbulence. Our next step will be the use of LPF, with stochastic representation or not, to assimilate data with strong nonlinearities for a barotropic model and carry information to an ensemble assimilation system. In this work we have seen that nonlinear filters have a non-unique representation with a kernel K n ,1)n ' Correction process with the kernel C n ,Yn, 1)n and genetic selection Sn ,1)n are known answers. It may have other responses, with for instance a combination of the 2 previous kernels, but also with adaptive res amp ling procedure (see [4]). There are also some strategies of piloting/tuning of the selection parameters. Find new kernels and more efficient selection rules will be the next challenges for atmospheric data assimilation.

Acknowlegments The authors thank P. Del Moral for the helpful suggestions on EnKF meanfield process interpretation and particle filter selection strategies and also thank N. Oger, F. Suzat and R. Locatelli for their collaboration in the simulation part of this work.

References l.C. Baehr. Modelisation probabiliste des ecoulements atmospheriques turbulents afin d 'en filtrer la mesure par approche particulaire. PhD thesis, Ecole Doctorale Mathematiques Appliquees Universite Toulouse III - Paul Sabatier , 200S. 2.S.K. Das and P.A. Durbin . A Lagrangian stochastic model for dispersion in stratified turbulence. Phys. of Fluids, 17:025109.1-025109.10, 2005 . 3.P. Del Moral. Feynman- Kac Formulae, Genealogical and Interacting Particle Systems with Applications. Springer-Verlag, 2004. 4.P. Del Moral, A. Doucet, and A. Jasra. On adaptive resampling procedures for sequential monte carlo methods. Technical report , INRIA, 200S. 5.G. Evensen. Data Assimilation: The Ens emble Kalman Filter. Springer-Verlag, 2006. 6.Fran<,ois Le Gland, Valerie Monbet, and Vu-Duc Tran. Large sample asymptotics for the ensemble Kalman filter. In Dan O. Crisan and Boris L. Rozovskii, editors, Handbook on Nonlinear Filtering. Oxford University Press, to appear. 7.E. N . Lorenz . Predictability: A problem partially solved (1996). In Proceedings Seminar on Predictability ECMWF, 1996. S.C. Snyder, T. Bengtsson, P. Bickel, and J. Anderson. Obstacles to highdimensional particle filtering. Mon. Wea. Rev., 136:4629-4640, 200S.

35

Local and Global Lyapunov Exponents In a Discrete Mass Waterwheel David Becerra Alonso and Valery Tereshko School of Computing University of the West of Scotland Paisley PAl 2BE United Kingdom (e-mail: [email protected]@uws.ac . uk) Abstract: A generic Jacobian is calculated to obtain the Lyapunov exponents Malkus' system. However complete, the Lyapunov exponents obtained from the Jacobian do not appropriately show the distinction between chaos and order. A further explanation for this is required. We show how the waterwheel equations, chaotic as a whole, can be decomposed into a series of convergent equations. Chaos will then come in from the transition between any two of these convergent equations. We finally use a common numerical method, not based on the Jacobian, to obtain Lyapunov exponents that properly make the distinction between chaos and order.

1

Introduction

The equations of motion for Malkus' waterwheel, obtained via analytical mechanics, were presented in [1]. The proper derivation into the Lorenz system [4] has been supported in a series of publications since the waterwheel was first proposed by Malkus in 1972 [5]. Among these it is worth citing [3], in which the Lorenz equations are obtained using a Fourier solution for the distribution of mass over the rim of the waterwheel. The equations for the position (8) and angular velocity (w) of the waterwheel are given by:

(1)

8=w .

w=

gR",£~~l mi sin fo

(8 + i~) -

vw

+ R2 ",£~~1 mi

(2)

where R is the radius of the wheel, 9 is gravity, N is the number of buckets, v is the friction coefficient, fo is the moment of inertia when no water has yet been poured on the waterwheel, and mi are the masses of water on each bucket at any given moment. All of them are parameters, except mi which also requires its own differential equation:

(3) where

q(8)

= {

~

I·f

I 8 + Z·27r . 2RI N 1< _ arcsm

otherwise

(4)

36

D. Becerra Alonso and V. Tereshko

Here we assumed that the water jet is narrower than the bucket size I, so that the latter completely defines the angle at which a bucket is exposed to the water. In (3), q represents the amount of incoming water per second and k is the leak rate of each one of the buckets.

2

Order-chaos-order transitions

In order to present a straightforward scenario of chaotic and periodic regimes being clearly differentiated, we plot a bifurcation diagram where we fix all parameters except q. As cut condition, we will extract the values of w every time a specific bucket goes through to top of the wheel (e = 0).

1.5,-----,-----,-----,-----,-----,-----,-----,-----,-----,----,

."

.

.

..----:......

-- ....----=:::-::::. ....... .-110"." ~ .... ,... "",,"

"

0.5

..,...",.,0, ....,iJ'.. ,

-r eJ ''''''

8

"""".

"'C"'''''''''':~fl- .....

:.':.;:.~!f'-~~~, ~~ :~;';~"~·~".:;",:·5·.· ( :.~ .......... - ........... .

-0.5

.. :~. :~:~·::,~~~:f;::~7:;~;~~~~{~~"'':"=·:·:::'::·~:;:~:;;.,::~~~~ ...--...... .. ........ "",

-1

~~"""'....... "'"

.......... 4"0

~... ~~...:::.:... " ........ ""'"

......

~---

-. .,.

-~

_1.5L-_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _

o

0.005

0.01

0.Q15

0.02

0.Q25

0.03

0.035

0.04

0.045

~

0.05

q Fig. 1. Bifurcation diagram with ij as bifurcation parameter, where k 10 = 0.04, v = 1.5, R = 2, 9 = 9.8 and N = 8.

0.013,

The case for N = 8 will be representative enough for the purpose of this paper. We have found that the order-chaos-order transition values of q settle to values very close to the ones found for N = 8 as we keep on increasing the number of buckets. In this particular case we can identify four very well differentiated regimes:

Local and Global Lyapunov Exponents in a Discrete Mass Waterwheel

37

• ij = 0 ---7 Trivial stationary state. • ij E (0,0.002) ---7 Periodic rotation of the waterwheel, always in the same

direction. +w values are for clockwise, and -w for counterclockwise. • ij E (0.002,0.022) ---7 Chaotic regime. • ij > 0.022 ---7 P eriodic regime where the waterwheel repeatedly goes

once clockwise and once counterclockwise. Within this regime we can easily identify 6 different frequencies. The waterwheel will settle on one of these frequencies depending on the initial conditions. As part of our analysis we wanted to look at the Lyapunov exponents for this system, as a means to systematically differentiate chaos from the other regimes. The idea was then to calculate the Jacobian, the eigenvalues, obtain the local Lyapunov exponents for a long enough time lapse, and average those to obtain a global Lyapunov exponent.

3

Approximated piecewise Jacobian

All the elements of the differential equations are derivable except q( 8). The derivative of q(8) is always zero except in the moments when a bucket stops being the highest one in favor of another bucket. In those moments the derivative of q(8) instantaneously peaks. But of course if we are going to calculate the global Lyapunov exponent as the average of all the local ones, these particular peaks would not affect this average compared to all the 8s where no transition between buckets takes place. This means that for the purpose of calculating the global Lyapunov exponent based on the average of the locals, we can propose a simplified Jacobian where the derivatives of q with respect to 8 remain equal to zero at all times. The Jacobian for equations (1), (2) and (3), and for N buckets would be an (N + 2) x (N + 2) matrix of the form: 1

0

J=

0

0 ...

d 2,2 d 2,3 d 2,4

d2,1 d 3 ,l d 4 ,l

0 0 0

dCN+2),l

...

-k 0 0 -k 0

0

0 d2,CN+2)

0 0

(5)

-k

where

(6) d2

2

,

=

-1/

-----::--=--

10

+ R2 L:i Tni

(7)

38

D. Becerra Alonso and V. Tereshko

gRsin (8 + (j - 2)~)

d 2 ,(j?3)

=

fo

+ R 2 l:i mi

R2

(-vw + gR l:i mi sin (8 + i~)) (fo

-

(8)

+ R2l:i mi)2

(9) The elements

4

d(j?3),1

would be different from zero in the unsimplified case.

Jacobian-based local Lyapunov exponents

We expect the values of Ii within the chaotic regime to be positive. For the numerical test, we choose Ii = 0.02. After ensuring that the transient is over, these are the temporal dynamics of w(t) and the local Lyapunov exponents.

8

1220

1240

1260

1280

1300

1320

1340

1360

1380

1400

t

-2.5 '::::---:-::'::-=----:~----::'::-:---_:_::"-=--~___:-"------'.---'----.l.----.J 1200 1220 1240 1260 1280 1300 1320 1340 1360 1380 1400

t

Fig. 2. wand Local Lyapunov exponents vs. time for ij ters remain as above.

= 0.02. All other parame-

Two consequences can be inferred from this result. The most direct is that the average of ALyapunov is clearly negative. We will go into detail about this on the next section. The second is that we can find an inverse proportionality between angular acceleration and the local Lyapunov exponents. Every time w hits an extreme or goes through an inflexion point, there is a corresponding peak in

Local and Global Lyapunov Exponents in a Discrete Mass Waterwheel

39

the temporal dynamics of ALyapunov. As we can see in Figure 2, the more significant peaks are those when w is going through a low slope inflexion point or an extreme of w happens close to w = o. This has an equivalent physical explanation when we think about the behaviour of the waterwheel. Higher local Lyapunov exponents imply a higher sensitive dependence to initial conditions at that particular state of the system. Malkus' waterwheel was expected to have different sensitivity depending on w. When w is close to zero, the waterwheel is slowing down, and the sum of the torques exerted by the water on each one of the buckets will decide if it keeps rotating in the same direction or not. In such a tipping point, very different future states of the system are being decided, and therefore the sensitivity to any perturbation is high. One of the things that makes Malkus' waterwheel a challenging model is that predictions beyond any of these tipping points are very difficult, and these are happening just a few seconds apart from each other.

5

Reasons for

Alocal

<

0 in chaos

The local Lyapunov exponents calculated in the previous section were the result of obtaining the eigenvalues of the Jacobian, calculating the absolute values, choosing the highest one, and applying the natural logarithm. This goes with the standard definition of Lyapunov exponent. However, the Jacobian as a derivative has a very strong limitation in this particular system when we try to use it as a means to measure the long term butterfly effect. Our equations for the waterwheel, as we presented them in Section 1, are piecewise because of q( 8). Though the complete system presents nonlinearity and chaos (with the right parameters), each one of the individual piecewise subsystems is convergent. This can also be translated into a physical explanation: as long as the top faucet keeps on filling the same bucket while the others drain, the system will be converging. At the very moment when a new bucket is the one under the faucet, the convergent state of the system is displaced, and the system will start approaching that new convergent state. It's this permanent shift of stationary states that keeps the waterwheel going. But of course that means that our derivatives are those of a convergent subsystem, part of the complete piecewise system. This makes it natural for the corresponding Lyapunov exponents to be negative. Even if we had not proposed the simplified Jacobian, the result would remain the same, because the difference would only be relevant during transitions between buckets, while the rest of the local Lyapunov exponents stay negative. It is the long term permanent transition between buckets that eventually allows the butterfly effect to take over, and no local and short term analysis will show this. In Figure 3 we can see the temporal dynamics of the system under two initial conditions separated by a distance Do = 10- 11 . Two stages are eas-

40

D. Becerra Alonso and V. Tereshko 1.5~--------~--~----~--~----~----~--~----~---,

8

-1.5 800

850

900

950

1000

1050

1100

1150

1200

1250

1300

1100

1150

1200

1250

1300

t 2 OJ

U

0

C

-2

ii5 '5

-4

ell

0

OJ

0 ....J

850

900

950

1000

1050

t Fig. 3. Butterfly effect for

q=

0.02. All other parameters remain as above.

ily recognized. While the distance D between the two systems is less than the width of a bucket, the decimal logarithm of the distance increases in an approximately linear way. The moment we have D larger than the width of a bucket, divergence is guaranteed because each system is governed by a different piecewise subsystem. This is when the butterfly effect saturates. In Figure 4 we choose a q that corresponds to a periodic state. The distance between the two initial conditions is a few orders of magnitude below the width of a bucket. So they are both under the same piecewise subsystem at practically all times. Thus, saturation of the distance happens a lot sooner and divergence never takes place.

6

Global Lyapunov exponents

Since the Jacobian-based Lyapunov exponents are not going to be good for the distinction between chaotic and periodic states, we will need to use a different method. The numerical method we will use, can be found in detail in [6, 2]. In this case, the method converges to the global Lyapunov Exponent instead of being obtained as the averages of the locals. For all the reasons presented in Section 5, it takes some time for this convergence to take place. Figure 5 shows that the numerical method for global Lyapunov exponents

Local and Global Lyapunov Exponents in a Discrete Mass Waterwheel

41

1.5

0.5

8

0 -0.5 -1 -1.5 800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1400

1500

1600

1700

1800

t -2

OJ

-4

()

C

(1j

U5

'0 0

r::il

0

-.I

-10 -12 800

900

1000

1100

1200

1300

t Fig. 4. Butterfly effect for ij = 0.05. All other parameters remain as above.

converges to positive values. It has also been verified that periodic states present negative values accordingly.

7

Conclusions

A system for Malkus' waterwheel was proposed and the difficulties of it having a piecewise term were presented. On the basis of calculating a global Lyapunov exponent based on the average of many local ones, we proposed a simplified Jacobian. A relation between the local Lyapunov exponents and the angular velocity of the waterwheel was presented, along with its physical explanation. We then found that local Lyapunov exponents under a chaotic regime do not have positive signs in this system, and exposed the reasons for this to happen. Finally, a numerical method to detect long term divergence, and identify its corresponding Lyapunov exponent was presented, and resulted in coherent results that correctly make the distinction between chaotic and periodic regimes. As we stated previously, it is the repeated top bucket transitions that create divergence in the long run. Therefore, the more buckets, the more frequency of these transitions we will have. The Lorenz equations were derived from Malkus' waterwheel in assumption of having a very high number

42

D. Becerra Alonso and V. Tereshko 1.5 1 0.5

8

0 -0.5 -1

Mw~}~\~1~~~\~1~W~~j~~~~~~

-1.5 1100

1200

1300

1400

1500

1600

1700

1800

1200

1300

1400

1500

1600

1700

1800

0.2 0.15

> c 0

:::J

0.1

C.

'"

>.

«

....I

0.05

-0.05 1100

Fig. 5. Global Lyapunov exponent obtained using [6, 2] where

q=

0.02.

of buckets. This is why Lorenz presents immediate divergence in its chaotic regime while the waterwheel takes longer. An extra nonlinearity in the mass equation could close this gap between Lorenz and the discrete waterwheel.

Bibliography [1] David Becerra and Valery Tereshko. Chaotic waterwheel: Discrete vs continuous mass representation. Chaos2008 Proceedings, 1, 2008. [2] Giancarlo Benettin, Luigi Galgani, and Jean-Marie Strelcyn. Kolmogorov entropy and numerical experiments. Phys. Rev. A, 14:2338-2345, 1976. [3] M. Kolr and G. Gumbs. Theory for the experimental observation of chaos in a rotating waterwheel. Physical Review A, 45(2):626-637, 1992. [4] E. N. Lorenz. Deterministic non-periodic flows. J. Atmas. Sci, 1963. [5] W. V. R. Malkus. Non-Periodic convection at high and low prandtl number. Mem. Soc. R. Sci. Liege, 6(IV):125, 1972. [6] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano. Determining lyapunov exponents from a time series. Physica D, 16(3):285-317, 1985.

43

Complex Dynamics in an Asset Pricing Model with Updating Wealth Serena Brianzonil, Cristiana Mammana 2 , and Elisabetta Michetti 3 1

2

3

University of Macerata Dep. of Economic and Financial Institutions 62100 Macerata, Italy (e-mail: [email protected] t) University of Macerata Dep. of Economic and Financial Institutions 62100 Macerata, Italy ( e-mail: [email protected] t) University of Macerata Dep. of Economic and Financial Institutions 62100 Macerata, Italy (e-mail: [email protected] t)

Abstract: We consider an asset pricing model with wealth dynamics and heterogeneous agents. By assuming that all agents belonging to the same group agree to share their wealth whenever an agent gets in the group (or leaves it), we develop an adaptive model which characterizes the evolution of the wealth distribution when agents switch between different trading strategies. Two groups with heterogeneous beliefs are considered: fundamentalists and chartists. The model results in a nonlinear three-dimensional dynamical system, which is studied in order to investigate complicated dynamics and to explain the effects on wealth distribution among agents in the long run. Keywords: heterogeneous agents, wealth dynamics, nonlinear dynamical systems.

1

Introduction

In recent years, seyeral models have focused on the study of the asset price dynamics and wealth distribution when the economy is populated by boundedly rational heterogeneous agents with Constant Relative Risk A version (CRRA) preferencf)s. Chiarella and He[2] study an asset pricing model with heterogeneous agents and fixed population fractions. In order to obtain a more appealing framework, Chiarella and He[3] allow agents to switch between different trading strategies. More recently, Chiarella et al.[4] consider a market maker model of asset pricing and wealth dynamics with fixed proportions of agents. A large part of contributions to the development and analysis of financiE,j models with heterogeneous agents and CRRA utility do not consider that agents can switch between different predictors. Moreover, the models which allow agents to switch between different trading strategies make the following simplified assumption: when agents switch from an old strategy to a new strategy, they agree to accept the average wealth level of agents using the new strategy.

44

S. Brianzoni, C. Mammal1{l, and E. Michetti

Motivated by such considerations, we develop a new model based upon a more realistic assumption. In fact, we assume that all agents belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it). The most important fact is that the wealth of the new group takes into account the wealth realized in the group of origin, whenever agents switch between different trading strategies. This leads the final system to a particular form, in which the average wealth of agents is defined by a continuous piecewise function and the phase space is divided into two regions. Nevertheless, our final dynamical system is three dimensional and we can find all the equilibria. We will prove that it admits two kinds of steady states, fundamental steady states (with the price being at the fundamental value) and non fundamental steady states. Several numerical simulations supplement the analysis.

2

The model

Consider an economy composed of one risky asset paying a random dividend Yt at time t and one risk free asset with constant risk free rate 7' = R - 1 > O. We denote by Pt the price (ex dividend) per share of the risky asset at time t. In order to describe the wealth's dynamics, we assume that all agents belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it). Hence, the dynamics of the wealth of investor h is described by the following equation:

where Zh,t is the fraction of wealth that agent-type h invests in the risky is the return on the risky asset at period t and Wh') t is asset, Pt = Pt+Yt-Pt-l Pt-l the average wealth of agent type h at time t defined as the total wealth of group h in the fraction of agents belonging to this group. The individual demand function Zh,t derives from the maximization problem of the expected utility of Wh,t+1, i.e. Zh,t = max Zh • t Eh,duh(Wh,Hd], where Eh,t is the belief of investor-type h about the conditional expectation, based on the available information set of past prices and dividends. Following Chiarella and He[2], the optimal (approximated) solution is given by:

where Ah is the relative risk aversion coefficient and O"~ = Va7'h,dpt+l - 7'] is the belief of investor h about the conditional variance of excess returns. In our model, different types of agents have different beliefs about future variables and prediction selection is based upon a performance measure (h,t. Let nh,t be the fraction of agents. using strategy h at time t. Hence, as in Broch and Hommes[l], the adaptation of beliefs is given by:

Complex Dynamics in Asset Pricing Model with Updating Wealth

Zt+1 =

L exp[,B(¢h,t -

45

C h )]

h

where the parameter ,B is the intensity of choice measuring how fast agents choose between different predictors and Ch ::::: 0 are the costs for strategy h. We observe that at time t + 1 agent h measures his realized performance and then chooses whether to stay in group h or to switch to another one. Hence, we measure the past performance as the personal wealth coming from the investment in the risky asset with respect to the average wealth Wh,t:

In this work, we focus on the case of a market populated by two groups of agents, i. e. h = 1,2, and assume that at any time agents can move from group i to group j, with i,j = 1,2 and i =1= j, while both movements are not simultaneously possible. We define .1nh,t+1 = nh,t+1 - nh,t as the difference in the fraction of agents of type h from time t to time t + 1. Notice that in a market with two groups of agents it follows that .1nl,t+1 = -.1n2,t+1. As a consequence, we can have two different cases:

1) .1nl,t+1 ::::: 0, if .1nl ,t+1 fraction of agents moves from group 2 to group 1 at time t + 1, 2) .1nl ,t+1 < 0, if .1nl ,t+1 fraction of agents moves from group 1 to group 2 at time t + 1. Following Brock and Hommes[l], we define the difference in fractions at d - I-mt ·th·. time t, i. e. mt = nl ,t - n2,t, so t h at nl,t -- -I +mt 2 - an n2 ,t - - 2 - ' WI

Afterwards , conditions .1nl,t+1 ::::: 0 and .1nl ,t + 1 < 0 can be replaced by mt+ I ::::: mt and mt+1 < mt, respectively. _ In order to describe the wealth dynamics of each group, we define Wh,t as the share of wealth produced by group h to the total wealth:

which we call weighted average wealth of group h. Hence, the weighted average wealth of group 1 at time t + 1 is given by:

+

WI ,t+1

=

{

.1nl,t+1 W 2,t+ 1 nl,t WI ,t+1 = nl ,t (WI ,t+1 - W 2,t+d nl ,t+1 W2 ,t+I , nl ,t+1 WI ,t+ I , if mt + 1 < mt

+

46

S. Brianzoni, C. Mammana, and E. Michetti

In a similar way we can derive the weight ed average wealth of group 2:

Finally, we define Wh ,t as the weighted average wealth of group h in the total weighted average wealth, i.e. Wh ,t = Wh ,t/ Lh Wh ,t where Wh ,t = nh ,tWh,t and h = 1,2. In the following , we will consider the dynamics of the state variable Wt := wl,t - W2 ,t, i.e. the difference in the relative weighted average wealths. As a consequence, the dynamics of the state variable Wt can be described by the following system: Fl

Wt+1 =

{

+1

if

-1

if mt+1 < mt

ffit+1 ~ mt

G

F2 G

where: FI = _21~.::n:': 1(I-Wt)[R+Z2 ,t(pt+1-r)],

i::'+

1 (1 + Wt)[R + ZI,t(Pt+1 - r)], F2 = 2 1 G = (1 - Wt)[R + Z2,t(Pt+1 - r)] + (1 + Wt)[R + Zl,t(PHl - r)]. In this work we assume that price adjustments are operated by a market maker who knows the fundamental price, see Chiarella et al.[4] . After assuming an i.i.d . dividend process and zero supply, we obtain:

Pt+l-Pt =---'-------=-Pt

l - Wt) = a (I+Wt ZI t - - - + Z2 t - - -

'2'

2

.

We analyze the case in which agents of type 1 are fundamentalists , believing that prices return to their fundamental value , while traders of type 2 are chartists, who do not take into account the fundamental value but their prediction selection is based upon a simple linear trading rule. In other words, we assume that EI ,t(pt+d = p* and E 2,t(pt+d = apt with a > O. Therefore the demand functions are given by: Zl,t

=

1 A0'2 (1

+ r)(Xt

- 1),

Z2 ,t

=

1 A0'2 [a - 1 + r(Xt - 1)],

where Xt = ~ is the fundamental price ratio. p, The final nonlinear dynamical system T is written in t erms of the state variables Xt , mt and Wt:

= h(Xt, Wt) = tanh {,6/2 [,\~2 (Xt - a) [2'\"'0'2 (Xt - 2 + a + 2r(Xt - 1) +(Xt - a)Wt ) + r(Xt - 1)]- en,

mt+1

Complex Dynamics in Asset Pricing Model with Updating Wealth

with:

F _ 1 -

p.2

-

-4(1- w t){R+6 [a-Hr(xt- 1)][ ~ (Xt -2+a+2r(xt- 1)+(Xt -a)w,)+r(xt- 1)]} (1-m,)[exp{!3[",!2 (Xt-a)[2",Qa2 (xt-2+a+2r(xt-1)+(xt-a)w,)+r(xt-1)]-C]}+1]' 4(1+wt){ R+~ (l+r )(xt-1) [~(Xt -2+a+2r(xt -1 )+(Xt -a)w')+r(xt -1)]} /\17 2,\q (l+m,)[exp{ -!3[ (Xt- a )[ (Xt -2+a+2r(xt-1)+(xt -a)w,)+r(xt-1)]-C]}+1] '

2>.:2

",!2

G = 2R+ >'~2 [xt-2+a+2r(xt-l)+(xt-a)Wt]· +(Xt - a)Wt) + r(xt - 1)].

3

47

[2>''''0"2

(Xt - 2 + a + 2r(xt - 1)

Results

Our model admits two types of steady states:

- fundamental steady states characterized by x = 1, i.e. by the price being at the fundamental value, - non fundamental steady states for which x =I 1. More precisely, for a =I 1 the fundamental steady state E f of the system is such that wf = 1 and there exists a non fundamental steady state Enf such that wnf = -1, xnf = l-;:-a + 1. Observe that at the fundamental (non-fundamental) equilibrium the total wealth is owned by fundamentalists (chartists). When a = 1 the fixed point Enf becomes a fundamental steady state. More precisely, every point E = (1, tanh{ - cf}, w) is a fundamental equilibrium, i.e. the long-run wealth distribution at a fundamental steady state is given by any constant w E [-1,1]. In other words, a continuum of steady states exists: they are located in a one-dimensional subset (a straight line) of the phase space. In this case the steady state wealth distribution which is reached in the long run by the system strictly depends on the initial condition. Summarizing, the following lemma deals with the existence of the steady states: Lemma 1. The number of the steady states of the system T depends on the parameter a:

1. Let a =I 1, then: • for a < 1+r there exist two steady states: the fundamental equilibrium Ef =

(X f

=

1, rn

f = tanh { -

c:} ,W 1) f =

and the non-fundamental equilibrium l-a Enf= ( --+I,tanh

r

{f3[ l+r - -1- ( I - a ) 2]}) -C ,-1 2

'\0'2

r

48

S. Brianzoni, C. Mammana, and E. Michetti

1

(a)

(b)

1\~~~\ o

-1L-______________~

60

o

60

Fig. 1. State variable Wt versus time for an i.c. Xo = 0.8, rna = 1 and Wo = 1 and parameter values a = 0.5, ).. = 1, (J2 = 1, r = 0.02, C = 0.5, (3 = 1. (a) a = 0.1. Periodical fluctuations. (b) a = 0.5. Convergence .

• for a ~ 1 + r the fundamental steady state E f is unique 2. Let a = 1, then: • EveTY point E = (1, tanh { - Cf}, ill) is a fundamental equilibrium. Now we move to the study of the asymptotic dynamics of the system by using numerical simulations. We first consider the extreme case in which, at the initial time, all agents are fundamentalists and they own the total wealth. Moreover, we focus on the more interesting a's range of values, i.e. a < 1 + r. In Figure 1 (a) we present the trajectory of the state variable Wt while assuming that at the initial time 'rna = Wa = 1 and the price is above its fundamental value. We choose two different values of a and observe that if a is low enough the system fluctuates (a 12-period cycle is observed) while, as a increases the long run dynamics becomes simpler (Wt --+ 1 after a long transient) . Numerical simulations show the existence of a value ii such that for all a E (ii, 1 +r) the system converges to Ef, while more complex dynamics is exhibited if a < ii, i.e. for a low enough (see Figure 2). Notice that the system preserves similar features, i.e. the stabilizing effect of a in the long run, also in the other extreme case Wa = 'rna = -1, i.e. at the initial time the market is dominated by chartists who own all the wealth. In Figure 2 we observe a succession of periodic windows which occur at border collision bifurcations (see Hommes and Nusse[5] and Nusse and Yorke[6]). These periodic windows have the following properties: (i) the width of the periodic window reduces monotonically as the parameter a increases; (ii) the windows are characterized by the same periodicity; (iii) for a > ii, the periodic windows terminate in convergence. In order to consider also the role of the parameter (3, in Figure 3 we present a two-dimensional bifurcation diagram in the parameters' plane (a, (3) in the extreme case 'rna = Wa = 1 and the price .TO above the fundamental value.

Complex Dynamics ill Asset Pricing Model with Updating Wealth

a

49

a

a

Fig. 2. Bifurcation diagrams of the state variables w.r.L a. Initial condition and parameter values as in Figure 1.

Rkh dynamics is exhibited and the final behaviour increases in complexity as (3 increases (according to what happens in asset pricing models with heterogeneous agents and adapti veness). Other numerical simulations confirm the evidence that the systems is lIlore cornplicated for high values of /3 and low values of a.

.~jg~

I Jj:;

I.

.

G-~'

c--';r. ~<

"; ..fl

r.;. ..', (~.~

(:,,~

cc...o

C-~

c;. ..';;

'J

Fig.3 . Two dimensional bifurcation diagran:l in parameters' plane (a, ,(1) for i.e. Xo = 0.8, ?no = 'Wo = 1. The other parameter values as in Figure 1.

In order to focus on the role of the initial condition, we fix a low enough. [n Figure 4 we present the basins of attraction in the plane (rno,1uo), being the initial price below the fundamental. Observe that if we move parameter 3, the structure of the basin also undergoes to a change. More precisely, it seems to become more complex aN (] increases, confirming the evideJlc:e previously obtained. In fact the struetme of the basin becomes fractal providing the strong dependence of the final d.ynamics w.r.t. the initial conditions (in

50

S. Brianzoni, C. Mammana, aml E. Michetti

panel (b) ouside the stability region the system may converge to a IO-period cyele or to a more complex attractor).

(a)

(b)

Fig.4. Basins of attraction in the plane (mo,100) for :ro = 1.3, a other parameter values as in Figure 1. (a) (j = 1.8, (b) ;3 = 2.

,=

0.1 and the

In summmary, our model characterizes the evolution of the distribution of wealth in a framework with heterogeneous beliefs and adaptiven€ss. Analytical and numerical results show coexistence of attractors, sensitive dependence on initial eonditions and complicated dynamics. Complexity is mainly due to border collision bifurcations whieh are involved by the wealth dynamies.

References l.'vV.A. Brock, and C.H. Hommes, Heterogeneous beliefs and routes to chaos in a simple a..'lset pricing model. JO'li.rnal of Economic Dynam'ic8 and Co'nt1'01, 22: 12:'\5-1271.1998.

2.0. Chiarella, and X. He, Asset price and wealth dynamies under heterogeneous expc;ctations. Q1UJ,TI,titati'Ve P'i'fl,a1l.ce, 1(5):509-526, 2001. 3.C. Chiarella, and X. He. An adaptive model on asset pricing a.nd wealth dynamics with het.erogeneous trading strategies. vVorking paper no. 84, School of Finance and Economics, UTS, 2002. 4.C. Chiarella, R. Dieci, and L. Gardini. Asset price and wealth dynamics in a financial market with heterogeneous agents. Jov,7'nal of Ec07l0'lnic DynaTYIiic8 and Control, :30:1755-1786, 2006. 5.C.H. Homme", and H. E. Nusse. Period three to period two bifurcation for piece~ wise linear modeL JO'U.r'7I,al of Econormcs. 54:157-169, 1991. D.H. E. Nmlse, and .J. A. Yorke. Border-collision Bifurcations for Piecewise Smooth One-dimensional J\faps. International Journal of B~f1trCl1t.ion a.nd Chaos: .5:189207, 1995.

51

Chaotic Mixing in the System Earth: Mixing Granitic and Basaltic Liquids Cristina De Camposl, Werner Ertel-Ingrisch l, Diego Perugini2 , Donald B. DingweU l and Giampero Poli 2 lEarth and Environ. Sc., LMU - Univ. of Munich Theresienstr. 411III, D-80333, Munich, Germany (e-mail: [email protected]) 2Dep t. of Earth Sciences, Univ. of Perugia Piazza Universita, 06100 Perugia, Italy Abstract: A widely debated question in Geosciences is if and how viscous magmas with extreme viscosity contrast mix under natural conditions. Chaotic mixing in magma chambers is thought to play a central role not only in determining the timing and dynamics of volcanic eruptions but may be of equal relevance for the evolutionary history of our planet. To date the dynamics of chaotic mixing have been investigated mostly both in analogue systems and numerical simulations. Here we report the first experimental simulation of chaotic mixing dynamics in molten silicates of geologic relevance and at geologically relevant temperatures (up to 1,700°C). A newly developed device for the in-situ experimental simulation of chaotic dynamics has been successfully developed and employed for this purpose. This device was designed in full awareness of the importance of chaotic dynamics for mixing processes; and earlier studies evidencing that chaotic dynamics equally control magma mixing processes in nature. Keywords: chaotic flow simulation, chaotic mixing, journal bearing system, newly designed experimental device, high viscosity silicate melts.

1. Introduction Magmatic rocks are known to be formed by cooling of natural silicate melts or magmas of wide compositional diversity. Primitive Earth is thought to have been covered by at least one giant magma ocean of more than 2,000 km depth, at temperatures far exceeding 2,000°C (Righter and Drake, 1999)[1]. At those temperatures magmas are predominantly multi-component viscous fluids behaving according to fluid dynamics (Reese and Solomatov, 2006)[2]. Since Hadean times, different compositional magma layers must have been constantly stirred and mingled together by thermal and/or compositional convection, at different scales (Abe, 1997)[3]. With decreasing temperature, magmas started to differentiate under wide range of crystallization rates and conditions. Even to the present day, this process controls heat budget and changes in chemical composition during magma segregation and crystallization. From this point of view, magma mixing is one of the most active processes in the Earth, its

52

C. De Campos et al.

range of activity spanning from the core-mantle boundary via mantle plumes up to the Earth's surface, with volcanic eruptions, under varying length- and timescales. From the physics of mixing we know that mixing is mainly two variables dependent: convection and diffusion (Ottino, 1989)[4]. Since Aref (1984)[5] demonstrated that processes involving chaotic mixing may run under absolutely laminar conditions, this is a decisive theoretical foundation for the mixing of high viscosity molten silicate melts in nature. As a consequence, viscous fluids flowing along simple periodic patterns can generate just enough chaos for efficient mixing (Ottino, 1990[6]; Funakoshi, 2008[7]). The first purpose of this study was to build a new high temperature experimental device allowing the first in-situ chaotic mixing experiments at geologically relevant temperatures (1,300 to 1,700 0c) and experimentally reproducible conditions. Applying this device, a systematic study of the chaotic mixing between synthetic basalt inside a granitic starting glass was initiated. Full chemical analysis of the resultant solidified materials was performed by electron microprobe analysis (EMPA). The obtained results point towards a clear increase in mixing efficiency due to combination of convection and chaotic dynamics, despite very low diffusion coefficients. This process can be an efficient way of mixing molten silicate melts over filament or lamellae dimensions.

2. The Journal Bearing System (JBS) 2.1. The original model, set up and starting materials The journal bearing or eccentric cylinder geometry proposed by Swanson and Ottino (1990)[8] is shown in Figure 1. It consists in principle of a cylindrical crucible, holding the melt of interest, and a cylindrical spindle, which is located off centre to enable stirring. The basic geometry is determined by two parametrers: a) the ratio of the radii of the two cylinders, r = rinlrout and the eccentricity ratio to the outer cylinder £ = elrout ; b) the discontinuous stirring protocol and its velocities. In order to generate chaos in a flow, the streamlines, and not just the velocity must be time dependent. This is the reason why Swanson and Ottino (1990) proposed to move both cylinders in opposite directions. Final experimental conditions are r = 1/3, £ = 0.3, and the viscosity contrast between glycerin and dye is ;:::: O.

Chaotic Mixing in S.vstem Earth

53

In this Oliginal model the velocity field is derived from the stream function: ~

a~ ~

-=V

dt

a~

=--'-=V x

ay , dt

Y

ax

And the final equation for the mixing in the flow is then determined by the integrated velocity field (the so-caned motion):

a( rp I v nu) dy --,d"- = -'--'-'-; -- = a( tp I v out) dB ay dB ax

--'---_.::=.-

where e cylinder.

IS

the angular rotation and

Vout

the velocity of the outer

The working fluid in Swanson and Ottino (1990)[8J was glycerin which has a viscosity around 0.7 Pa. and a density of 1.2 g/cm3 . The resultant Reynold number reported by the authors is 0.8. For the basic experiments, fluorescent dye with a very small diff11sion coefficient (lO-8cm2/s) has been dissolved in the glycerin to highlight the flow lines. Experimental results were then recorded on color slide film. Figure 2 depicts the resultant image for Bill:::: 270°.

Hgure 1. Basic geometry of JBS after Swanson and Ottino ([990)[8J.

Figure 2. How lines for Oin=270° (Swanson and Ottino, 1990)[8).

54

C. De Campos et al.

2.2. Our set up and experimental end-member compositions Our self-made device has built according to the basic geometry for the JBS, and is shown in Figures 3 and 4. Starting materials consisted of: 1) hapJogranite (HPG8), and 2) a basaltic composition due to the 1 atm eutectic of Anorthite-Oiopside (An-Oi). Mean values of end-member compositions are reproduced in Table 1. Experiments were performed with the conditions given by Swanson and Ottino (1990)[8]: (r = 1/3 and € = 0.3; b), and a discontinuous velocity protocol, presented in Table 2. Calculated viscosity values for the stmting materials (after Giordanno et aI. , 2008)[9] at 1,400°C are: 1) haplogranite (HPG8) ;:::; 15.102 Pa s (log a =3.19) and 2) An-Oi ;:::; 7. 103 Pa s (log a =-0.15), so that the viscosity contrast averages ;:::; 2.103 , e.g. , over three orders of magnitude higher than in the original JBS experiment. A representative streamline distribution through a cross section of the obtained glass cylinder observed in section 5 (20 mm above the lower end of the glass cylinder) is shown in Figure 5.

Figure 3. Starting vertical geometry of JBS in our experiment with sampling location. Table 1: Composition of starting materials (end-members) in weight % (normalized) SiOz

Ah03

Na20

K 20

CaO

MgO

L: (Rb+Sr+Ba +Zr+RE)

%

HPG8

70.33

12.43

7.70

9.50

>0.02

>0.02

- 0.10

9 5

AnOi

48.50

16.31

>0.1

>0.5

23 .72

11.33

-

5

Hy

69.25

12.61

7.34

9.05

1.26

0.56

»0.10

*

Chaotic Mixing in System Earth

55

Table 2: Protocol for the first chaotic mixing experiment with silicate melts Obs: final wobble of the platin/ceramic spindle was 0.057 mm (eccentricity of the spindle). Spindle changes direction. Crucible rotates only in one direction.

e out

Time

Tin °C

14:27

Heating started

15:53

1400°C±

0°

+720°

16:07

1400"C±

-2160"

0°

16:27

1400°C±

-2160° +1440°

2 rotations in 40 min -7

VOUl 0.05

rpm

17:07

1400°C±

+2160 0

6 rotations in 20 min -7

Vom 0.33

rpm

17:07

1400"C±

00

+1440 0 full speed down with T. All rotation stopps.

17:07

1400°C±

00

+1440 0

Sin

Observations towards 1400°C full speed

00

2 rotations in 40 min -7 vou , 0.05 rpm 6 rotations in 20 min -7

VOUl 0.33

rpm

full speed down with T. Rotation stopps.

Final remarks: no leaking or visible deformation in the crucible. Easy to remove it from the pedestal. Total duration: 1 hour and 54 min. Error. 6 min.

Figure 4. Stmting geometry of JBS, our expo(horiz. crosssection).

Figure 5. Flow Jines from this work; 6>oUl=1440°.

3. Sample Preparation and Preliminary Electron Microprobe (EMP) Data The experiment was stopped by termination of the stirring protocol, switching off furnace power and cooling to room temperature. Crucible and spindle were removed from the furnace, and a cylinder of the resultant mixed glassy sample was recovered by coring. Horizontal equidistant sections (5 mm intervals) of this cylinder were prepared for

56

C. De Campos et al.

electron microprobe (EMP) analyses (Figures 3 and 5). The location of the EMP profile is along the black line in Figure 5. Optical and EMP micro-chemical studies revealed crystal-free filaments of intermediary compositions, changing with depth in complex patterns, with chaotic attractors showing slight variations in vortex strength. Here we present the results for the distribution of CaOand K20-contents (Figures 6 and 7) from a single analytical profile along the black line in section 5 (sample CD-25) at 20 mm depth depicted on Figure 5. This image is a false colour image from a mosaic consisting of over 150 photomicrographs from section 5 on Figure 3. This section crosses several filaments along a striking, stretched vortex structure. In Figure 6 the bottom white line represents the maximum error in CaO-content of the starting materials. This means that all points above this line are due to non-ideal mixing. Two main CaOenriched filaments are visible at about 2.5 mm and 3.5 mm distance. In Figure 7 these points are not correlated with the K20-distribution along the same line. This clearly shows that the mobility of both elements is not correlated. Another way to demonstrate this lack of correlation is shown in Figure 8: ratios of LlC MgO and LlCcao between ideal hybrid concentrations and measured partially mixed concentrations, for MgO and CaO are plotted along the same line. For this plot we assumed an error of 0.02 in the MgO- and CaO-contents due to background signal and sample preparation.

0.6

~

cf

"

()

0.4

8.8 0.0

0.2

0.4

0.6

0.8

1.0

1.2

distance in mm

Figure 6. CaO distribution profile plotted against distance, along a line across slice CD-25 on Figure 5.

~-<

0.0

0.2

~.~~~.~~.-"

0.4

0.6

0.8

1.0

1.2

distance in mm

Figure 7. K20 distribution profiles plotted against distance, along a line across slice CD-25 on Fig. 5

To compare the mixing processes between different elements in the melt, along the analysed profile, we used a coefficient (v E), which we called 'mixing efficiency coefficient': this coeffcient is obtained from

Chaotic Mixing in System Earth

57

the following ratio: VI: = [(Cmean - Chybrid) / (Cend-member- Chybrid)] . Vm where: Cend-member = measured, corrected and normalized concentration of an oxide in the end-member; Chybrid = calculatated concentration of an oxide in the ideal hybrid (100% homogeneous mixing); C mean = measured, corrected and normalized mean concentration of an oxide in the mixed sample, for the most frequent value population; Vm

= Cmean frequency;

The mixing efficiency coefficients calculated for CaO, MgO, Na20, and K20 for section 5, from this experiment, are: CaO = 0.047; MgO = 0.048; K20 = 0.43; Na20 = 0.48. For final experimental VI: computation we will need the mean value obtained from the L of all measured sections. 0.14 ~~~~~~~~I========:=:::'===J • L'.C CaOIL'.C MgO

0.12 0.10 0.08

~

0.06

(]

0.04 0.02 0.00 -0.02

.


. -...... . ...... I...... .. . ,"«,. S./,I.: ........ ....., ... " ·•.. ~ i. ,:...,.If" ..... ,~.. .' •. ~.,

. . .:

•••

e . . . ::_.

•• , .

__ ".. ..... __ •• 1 ........ .

-0.04 L-L-~~~~~L'.~C--";ty1""~g~O~~~~~~~ -0.04 -0.02 0.00 0.02 0.04

Figure 8. MgO and CaO distribution profiles plotted against ideal hybrids, along slice CD-25 on Figure 5.

4. Conclusions Our goal in this work was to present a newly built JBS experimental device for first in-situ chaotic mixing experiments at geologically relevant temperatures (1,300 to 1,700 DC) under experimentally reproducible conditions. Applying this device, a first run, using synthetic basalt (An-Di) inside a granitic starting glass was performed. Due to the very high viscosity contrast and to the predominance of a very high viscosity melt in the crucible this represents the worse experimental scenario.

58

C. De Campos et al.

Previous numerical simulations of mixing between basalt and granite, in the same proportions as in this work (95% granite + 5% basalt) and the same viscosity contrast, point towards mixing probabilities around zero (Jellinek et aI., 1999). The combination of convection and chaotic dynamics, however, greatly enhances the diffusive processes in molten silicate melts over filament or lamellae dimensions. Preliminary analytical results reveal different mobility and mixing efficiency for different elements, which are presented in this work as oxides. This way, despite very low diffusion coefficients and a very short mixing time (154 min), CaO and MgO may reach mixing efficiency coefficients around 5% while K20 and Na20 come closer to 50% (0.43 and 0.48). The application of this device under geologically relevant conditions has demonstrated that chaotic mixing processes are very efficient ways to mix silicate melts. Acknowledgements: Thanks to Philippe Courtialfor REE-rich glasses used in sample preparation and INGV /Unrest / Italy forfunding.

References [1] K.Righter and MJ. Drake. Effect of water ... A high pressure and temperature terrestrial magma ocean and core formation, Earth Planet. Sci. Lett. 171:383-399,1999. [2] C.C.Reese and V.S.Solomatov. Fluid Dynamics of local martian magma ocean. /caras. 154: 102-120, 2006. [3] Y.Abe. Chemical and thermal evolution of the terrestrial magma ocean. Physics of Earth and Plan. Int., 100:29-39, 1997. [4] 1.M.Ottino. The kinematics of mixing: stretching, chaos, and transport. Cambridge Texts in Applied Mathematics, 1989. [5] H. Aref. Stirring by chaotic advection. 1. Fluid Mech., 143: 1, 1984. [6] 1.M.Ottino. Mixing, chaotic advection and turbulkence.Annu.Rev.Fluid.Mech. 22: 207 -53, 1990. [7] M. Funakoshi. Chaotic mixing and mixing efficiency in short time. Fluid Dyn. Res.,40:1-33, 2008. [8] P.D.Swanson and 1.M. Ottino. A comparative computational and experimental study of chaotic mixng of viscous fluids. 1. Fluid Mech., 213 227-249,1990. [9] D. Giordanno, 1.K. Russel and D.B. Dingwell. Viscosity of magmatic liquids: a model. Earth Planet. Sci. Lett. 271: 123-134, 2008) [10] A.M. 1eIlinek, R.C. Kerr and R.W. Griffiths. Mixing and compositional stratification produced by natural convection. 1. Experiments and their application to earth's core and mantle. 1. Geoph. Res. 104 (B4): 7183-7201, 1999.

59

Multiple Equilibria and Endogenous Cycles in a Non-Linear Harrodian Growth Model Pasquale Commendatore 1 , Elisabetta Michetti 2 , and Antonio Pint0 3 1

2

3

Dept. of Economic Theory and Applications University of Naples Federico II Naples, Italy (e-mail: [email protected]) Dept. of Economic and Financial Institutions University of Macerata Macerata, Italy (e-mail: [email protected] t) Dept. of Statistical Sciences University of Naples Federico II Naples, Italy (e-mail: [email protected] )

Abstract: The standard result of Harrod's growth model is that, because investors react more strongly than savers to a change in income, the long run equilibrium of the economy is unstable. We re-interpret the Harrodian instability puzzle as a local instability problem and integrate his model with a nonlinear investment function. Multiple equilibria and different types of complex behaviour emerge. Moreover, even in the presence of locally unstable equilibria, for a large set of initial conditions the time path of the economy is not diverging, providing a solution to the instability puzzle. Keywords: economic growth, Harrodian instability puzzle, capacity utilisation, multiple equilibria, chaos .

1

Introduction

In the Keynesian theory of growth, as developed by Harrod[2] and [3], firms investment decisions can be separated into two components: the first component corresponds to the warranted rate of growth, defined as the rate of capital accumulation that the economy can sustain in the long run with the saving corresponding to the expected or normal level of production or income; the other component reacts to the short-run discrepancies between current and normal degree of capacity utilisation. The standard Harrodian result, following the hypothesis that investors react more intensely than savers to a change in income, is the local instability of the equilibrium rate of growth. Most literature has interpreted this result as the outcome of a dynamic analysis of global stability. However , as argued by Sen [6], Harrod's instability analysis over-stresses a local problem near the equilibrium without carrying the stor'y far enough. Moreover, Harrod himself [2] and [3] has stated that his analysis

60

P. Commendatore. E. Michelti. and A. Pinto

does not give a complete account of the problem, suggesting that he had only tried to underline the existence of some centrifugal forces coming into play as soon as the economy gets out of equilibrium. The reference to these forces does not exclude the existence of other forces producing stabilising effects. In what follows we present a Harrodian growth model integrated by a non linear investment function as in Kaldor's [4] trade cycle model. There can be either one or three equilibrium solutions. The first corresponds to the warranted rate of growth while the other two correspond to under and over utilisation. If the initial capacity utilisation is sufficiently close to normal, two different cases may occur: a) the economy converges towards one of the three equibria; b) the economy fluctuates permanently. Even in this case the long term evolution of the system is bounded, providing a solution to the Harrodian instability puzzle.

2

The model

The model describes a single-good closed economy. Production involves fixed proportions of labour and capital (Leontief technology). Technical progress is excluded. The labour supply is perfectly elastic and capital does not depreciate. Hence, at time t , we have yt = bL t :; y";;P = aKt , being a, b > 0, where y";;P is the potential output, yt the current output, K t the stock of capital, L t the amount of labour employed in production all evaluated at time t (no production lag). Parameters a and b are the reciprocal of the capital and labour coefficients respectively. In each period the capital stock is not fully utilized, so that the potential and current outputs do not coincide. We define the degree of capacity utilisation as the ratio between current and potential output: Ut = yt / Y";;p. Saving corresponds to a constant fraction of the income of the economy:

(1) where St is the ratio saving to capital and s is the (constant) propensity to save out of income. Investment is determined as follows:

(2) The component G in equation (2) is interpreted as the Harrodian warranted rate of growth that corresponds to the saving generated at normal capacity utilisation: G = sail. (3) The value il represents the normal degree of capacity utilisation, that is, the optimal degree of capacity utilization given the existing technology. Capital accumulation G t also depends nonlinearly on the current degree of capacity utilization Ut. On the nonlinear component ¢(Ut) of the investment function we make the following assumption, which assigns to it a sigmoidal shape (see Kaldor[4J; and Bischi et al.[l]):

Multiple Equilibria and Endogenous Cycles

61

Assumption 1. Function ¢ is such that ¢( u) = O. Furthermore ¢ is strictly increasing with respect to Ut (i. e. ¢' (Ut) > 0) and a u E (0, 1] exists such that ¢//(Ut) ~ «)0 iff Ut ::; (> )u.

Investment increases with the current degree of capacity utilisation. However, when the degree of utilisation is substantially below the normal value, the entrepreneurs' incentive for capital accumulation is small; when the degree of utilisation is substantially above the normal value, investment slows down again, due to the increasing installation costs and to the increasing risks of entering other projects. In order to guarantee positive capital accumulation, we also make the following assumption. Assumption 2. We assume G + ¢(O) > 0 , that is the long run expected growth of demand G is always high enough to induce positive investments even in correspondence of a low capacity utiK5ation in the current period.

The dynamics of the system is generated by the variations in the degree of capital utilisation in the face of discrepancies between demand and supply as follows : (4) where Ut+l denotes the one-period forwarded value of the state variable Ut while parameter 8 is the adjustment speed of capacity utilization. By substituting equations (1), (2) and (3) into (4), we obtain the general model for the evolution of the state variable Ut describing the degree of capacity utilisation :

(5)

3

Fixed points and their stability

Concerning the fixed points of (5) the following proposition holds . Proposition 1. Let u* be a steady state of map (5), then the following relat'ion holds: u* = u + ¢(u*)/sa.

Observe that the corresponding equilibrium investment (and saving) is given by G* = S* = G + ¢( U *). It is easy to verify that the following proposition holds: Proposition 2. u* ters.

=u

is a fixed point of (5) for any choice of the parame-

This proposition follows directly from the asumption ¢(u) = 0, hence Proposition 1 is verified for u* = u. As a consequence, the equilibrium . corresponding to the warranted rate of growth always exists. We need to verify the existence of other equilibria. Observe that map (5) can be expressed as the sum of a linear function and a sigmoidal function, that is , Ut+l = f(ut) = g(Ut) + 8¢('ut) = (hUt + m) + 8¢(ut), where h = (1 - 8sa) and m = 8sau. Then we prove the following proposition about the number of fixed points of f.

62

P. Commendatore, E. Michetti, and A. Pinto

Proposition 3. If 1'Cu) :::; 1 then u* = u is the unique fixed point of f , otherwise, if l' (u) > 1 then f has three fixed points (Ul < u < u r ).

Proof. Assume 0 :::; 1'(u) :::; 1 and observe that !"(Ut) 2: «)0 iff Ut :::; (> )u, being f"(ud = eq/'(Ut). Simple geometrical considerations prove that u* = u is the unique fixed point of f. Consider now the case in which l' (u) < O. Observe that 1'(Ut) = h + eq/(ud, \lUt > 0 and that eq/(Ut) :::; eq/(u) , hence 1'(Ut) :::; h + eq/(u) = 1'(u), \:IUt > O. Since 1'(u) < 0 thcn 1'(Ut) < 0, \:IUt > 0, i.e. f is strictly decreasing and consequently is has at most one fixed point. Finally consider the case 1'(u) > 1 and remember that f(O) 2: 0 and that f is convex (concave) for U :::; (> )u. Trivially f must intersect the 45 degree line in two more points, namely Ul and U r such that Ul < u < Ur . Observe that when 1'(u) = 1 the map passes from a fixed point to three fixed points via tangent bifurcation. When the fixed point u* = 'u is unique, we can draw some general conclusions about its stability and the final dynamics of the system, as stated in the following proposition. Proposition 4. Assume that u* = u is the unique fixed point of map (5). (i) If 0 :::; 1'(u) :::; 1 then u is globally stable; (ii) ~f - 1 < 1'(u) < 0 then Ul < u < U2 exist such that u attracts any trajectory starting from an i. c. Uo

E (V'1,U2); (iii) if 1'(u) :::; -1 then the generic trajectory diverges in

modulus. Proof. (i) Trivial. (ii) If 1'(u) < 0 the generic trajectory oscillates. In order to study the asymptotic behaviour of Ut, remember that only simple dynamics can be produced and consider the second iterate of f, namely f2. Then f2 is strictly increasing, it is concave (convex) if Ut :::; (> u) and remember that (j2(u))' = (j'(U))2. Then, - 1 < 1'(u) < 0 =} (j2(u))' E (0,1) and f2 has two more fixed points Ul and U2 such that Ul < U < U2 that are points of an unstable two-period cycle of f. This cycle separates the basin of attraction ofu, given by B(u) = (Ul,U2), from the basin B(oo) = (-oo,ud U (U2'+OO) whose trajectories diverges in modulus. (iii) Otherwise, 1'( u) :::; -1 =} (j2(u))' 2: 1 and, given its shape, j2 cannot have other fixed points, so that every trajectory diverges in modulus. About the case in which 1'(u) > I, and the map admits three equilibria, we can only draw conclusions about the instability of U. Anyway, in order to study the local stability of the other fixed points and to consider the possibility of complex dynamics to be exhibited, we have to specify function 4>. In particular we focus on the case in which 4> is simmetrc around the equilibrium U, i.e. 4>(- (Ut - u)) = -4>( Ut - u). A sigmoidal function enjoying this property is given by: (6)

Assume l' (u) > 1, then two more distinct equilibria are present , as proved in Proposition 3, namely Ul and Ur, which lie symmetrically around u i.c. Ur -

fl

= 'U - Ul.

Multiple Equilibria and Endogenous Cycles

63

Fig.L Two dimensional bifurcation diagram in parameters' plane (Z,8) being (31 = 0.08, (32 = 7.5 and u = 0.5. Curves Gi , i = 1, ... , 5 separate the plane into different regimes as analytically proved.

4

Local and global bifurcation analysis

We now present some local and global bifurcations for the map (5) when ¢(Ut) is given by (6). Define Z = sa, the map (5) becomes:

. F(u.t) = (1- eZ)'Ut

+ gZu + epi arctan(P2(Ut -

n).

(7)

We focus first on the case F ' (u) > 1, which implies P1P2 > Z . In such a case F(ut) may be non monotonic as stated in the following proposition. Proposition 5. Assume

pIfh >

Z and

ez > 1.

Then map F is bimodal.

Proof F is not monotonic iff F' changes sign. In order to have two distinct solutions for the equation [P2(Ut - u)]2(1- eZ) = -(1 + e(p1P2 - Z» being P1P2 > Z, it must be gZ > 1.

If Proposition 5 holds then em and eM are respectively the minimum and the maximum points of F , where em < u < eM and eM-n, = 'tt-e m · Observe that for a given Z , F is bimodal if the parameter () is sufficiently high. Let P1P2 > Z so that F admits three fixed points, then the curve () = l iZ, namely G1 , separates the parameter plane (Z, () into two regions (see Figure 1), For points below such a curve F is strictly increasing so that only simple dynamics can be exhibited, more precisely, every initial condition belonging to B(uI) = (-00, 'II) (B(u r ) = (u, +00» generates trajectories converging to UI (u r ). Otherwise, for points above G1 , F is bimodal and more complex dynamics can be produced, We focus on such a case. In the following discussion we take into account that, by simmetry, the properties of the local dynamics around the fixed point '/II also apply to '11 ' " and, similarly, the results concerning the minimum point also apply to the maximum point. We now distinguish between two cases. If em :s; F(c,n), that is, the minimum point is above the 45 degree line, then the fixed points UI and ttl' are locally stable and no complex dynamics

64

P. Commendatore, E. Michetti, and A.

Pinto

", Fig. 2. Staircase diagram of the map F for

/31 =

(l.OS,

/32 =

7.5, Z

=

0.4 and

u = 0.5. (a) B = 12.9: the attractor Al is chaotic; (b) B = 13.4: the unique attractor A after the contact bifurcation (occurring at B c:::'. 13.379).

can emerge. Condition C m = P(c",) defines a curve C 2 in the parameters' plane (Z, ()) such that for points above such a curve the minimum point is below the 45 degree line (see again Figure 1) . In fact a necessary condition for cycles or complex dynamics to be exhibited is that Cm > P(Cm), i.e. pl(nd < O. According to the previous consideration, in what follows we assume em > F(c,m). If PI(ud > -1, the only way nl may lose stability is via a flip bifurcation, hence the following proposition trivially holds. Proposition 6. Let

Ul

be a fixed point of F. Then for each (Z, ()) in the

region defined as .

J!(Ul)

the point

Ul

=

{

(B, Z) E 1R+ x (O,lh,82 ) : (}Z < 2 +

- B{"h[).,

1+

[,8 ( . ~ . )]2 2 lil -

}

11,

is locally stable.

From the previous result it is easy to determine the flip bifurcation curve C3 in the parameter plane (Z, B) (see Figure 1). Assume Cm > P(cm ), then map F is a non invertible Zl Z3 - Zl map (see Mira et al.[5]) and the set S = [F(cm) ,P(CM)] is trapping, i.e. pt(S) ~ S, 'It E Z+. As a consequence we focus our analysis on the restriction of F to the set S, that is, we consider the structure of the attractor A ~ S owned by P for any initia.l condit ion 110 E 8. The main results are summarized in the following Proposition. Proposition 7. Let Cm > P(cm ). (i) If p 2(c m ) < u then P has two coexist'ing attractors Al <;;;: [F(Cm),p 2(c m )] and AT' <;;;: [F(CM),p 2(cM)]j (ii) at F2(c m ) = 'ii, a contact bifurcation occurSj (iii) if p 2(c m ) E (il, F(cM)) then F has (], nn'ique oJ tractor A ~ 8; ('tv ) at F2(c m ) = F(CJ\.t) a final global

bifnTcation OCcurSj (v) if p 2(Cm) > F( CM) almost all trajectories diverge. Consider Propositon 7. In case (i) map F has two coexisting attractors At and A,. having the same structure. They may both consist of fixed points,

Multiple Equilibria and Endogenous Cycles

65


20

z

0.5

0.6

Fig. 3. (a) One-dimensional bifurcation diagram W.r.t. () for {31 = 0.08, {32 = 7.5, Z = 0.4, U = 0.5 and Uo = Cm. (b) One-dimensional bifurcation diagram w.r.t. Z for {31 = 0.08, (32 = 7.5, () = 15, U = 0.5 and Uo = Cm.

m-period cycles or chaotic sets. Denote by B(AI) and B(Ar) their basins of attraction, i.e. the set of points Uo E 8 which generate trajectories converging to them. Then B(AI) = [F(c m ), u) while B(Ar) = (u, F(CM)], hence the unstable fixed point u separates tha basins of the two attractors. About the structure of Ai, i = l, r, remember that the fixed point Ui, i = r, l, can be locally stable (i.e. Proposition 6 applies) and that the only way it can lose stability is trhough a flip bifurcation, where a stable 2-period cycle is created. After this first bifurcation, a period doubling route to chaos is exhibited (as F restricted to [F(c m ), F2(cm)] is topologically conjugate to the standard logistic map). This case is presented in Figure 2 panel (a). In case (ii) the critical point Cm is pre-periodic, proving the existence of parameter values for which the map is chaotic. In fact no attracting cycles may exist since their basins cannot contain the critical point. Condition F2(C m ) = u define the contact bifurcation curve C4 in the parameters' plane (Z, e) (see again Figure 1).

The contact bifurcation gives rise to a unique connected attractor to which almost alla trajecories converge. This situation occurs in case (iii) and it is depicted in Figure 2 panel (b). When F2(c m ) = F(CM ) (case (iv)) a final global bifurcation occurs (8 is no long invariant). After such a bifurcation, the attractor A disappears and almost all trajectories diverge. Observe that, being F2(C m ) increasing w.r.t. fixing all the other parameters and letting increase the map passes from having two attractors, to one attractor and, finally, to none (as almost all trajectories diverge). This fact is confirmed in Figure 3 (a). Finally the bifurcation diagram w.r.t. Z is presented in Figure 3 (b). Observe that complex dynamics can be exhibited for intermediate values of Z.

e,

e

66

5

P. Commendatore, E. Michetti, and A. Pinto

Conclusions

The analysis presented in this paper suggests a solution to the Harrodian instability puzzle. A possible outcome of our analysis is that, when the warranted growth rate is locally unstable, the economy, moving away from the normal degree of capacity utilisation, tends towards another equilibrium with capacity utilisation above or below normal. The occurrence of a degree of capital utilisation that even in the long run may differ from its normal value may be questioned on the ground that forecasts cannot be systematically wrong. Following Harrod, however, one may assume that, when the capacity utilisation is sufficiently close to the normal value, investors may be unwilling to change their expectations. Moreover, when the economy is subject to fluctuations which are difficult to predict (as in our model in the presence of chaotic patterns and complex basins of attraction) the long-run discrepancy between normal and current degree of capacity utilisation is justified. In this case, long-term statistical properties, as long-run averages, become very important to understand economic processes.

References 1.G. Bischi, R.Dieci, G. Rodano and E. Saltari. Multiple at tractors and global bifurcations in a Kaldor-type business cycle model. Journal of Evolutionary Economics, 11, 527:554, 2001. 2.R.F. Harrod. An essay in dynamic theory. The Economic Journal, 49(193), 14:33, 1939. 3.R.F. Harrod. Towards a Dynamic Economics. London, 1948, Macmillan. 4.N. Kaldor. A Model of Trade Cycle. The Economic Journal, 50(197), 79:82, 1940. 5.C. Mira, L. Gardini, A. Barugola and J.C. Cathala. Chaotic Dynamics in TwoDimensional Noninvertible Maps. Singapore, 1996, World Scientific. 6.A.K. Sen. Growth Economics. Harmondsworth, 1970, Penguin.

67

Hick Samuelson Keynes Dynamic Economic Model with Discrete Time and Consumer Sentiment Loretti 1. Dobrescu,l Mihaela Neamtu2 and Dumitru Opri§3 1

2

3

Australian School of Business, School of Economics University of New South Wales, Sydney Australia (e-mail: dobrescu©unsw.edu.au) Faculty of Economics and Business Administration West University of Timi§oara, Timisoara, Romania (e-mail: mihaela.neamtu©feaa.uvt.ro ) Faculty of Mathematics and Informatics West University of Timi§oara, Timisoara, Romania (e-mail: opris©math.uvt.ro )

Abstract: The paper describes the Hick Samuelson Keynes dynamical economic model with discrete time and consumer sentiment. We seek to demonstrate that consumer sentiment may create fluctuations in the economical activities. The model possesses a flip bifurcation and a Neimark-Sacker bifurcation, after which the stable state is replaced by a (quasi-) periodic motion. Keywords: consumer sentiment, Hick Samuelson Keynes models, Neimark-Sacker, flip bifurcation, Lyapunov exponent.

1

Introduction

The economic empirical evidence ([1], [4], [8]) suggests that consumer sentiment influences household expenditure and thus confirms Keynes' assumption that consumer "attitudes" and" animalic spirit" may cause fluctuations in the economic activity. On the other hand, Dobrescu et all. ([2], [3]) analyzed the bifurcation aspects in a discrete-delay Kaldor model of business cycle, which corresponds to a system of equations with discrete time and delay. Following these studies, we develop a dynamic economic model in which the agents' consumption expenditures depend on their sentiment. As particular cases, the model contains: the Hick-Samulson ([6]), Puu ([7]) and Keynes ([9]) models, as well as the models from [9]. The model possesses a flip and Neimark-Sacker bifurcation, if the autonomous consumption is variable. The paper is organized as follows. In Section 2, we describe the dynamic model with discrete time using investment, consumption, sentiment and saving functions. For different values of the model parameters we obtain well-known dynamic models. In Section 3, we analyze the behavior of the dynamic system in the fixed point's neighborhood for the associated map. We establish asymptotic stability conditions for the flip and Neimark-Sacker bifurcations. In both the cases of flip and NeimarkSacker bifurcations, the normal forms are described in Section 4. Using the QR method, the algorithm for determining the Lyapunov exponents is presented in Section 5. Finally, the numerical simulation is done. The analysis of the present model proves its complexity and allows the description of the different scenarios which depend on autonomous consumption.

Opri~'

68

L. I. Dobrescu, M. Neamlu, and D.

2

The mathematical model with discrete time and consumer sentiment

Let Yt = y(t), tEN be the income at time step t and let: 1. the investment function It = I(t), t = 2 .. N, be given by: It

=

V(Yt-1 - Yt-2) - W(Yt-1 - Yt_2)3,

v

> 0, W :;0, 0;

(1)

2. the consumption function C t = C(t), t = 2.. N, be given by:

Ct = a where St

=

S(t), t

=

+ Yt-I (b + CSt-I),

a:;o, 0, b > 0, c :;0, 0,

(2)

2 .. N, is the sentiment function given by:

St=

1 1 + Eexp(Yt-1 - Yt)

3. the saving function E t

,

EE[O,I];

(3)

= E(t), t = 2.. N, be given by:

E t = d(Yt-2 - Yt-I)

+ mSt-l,

d:;o, 0, m :;0, 0.

(4)

The mathematical model is described by the relation:

(5) From (1), (2), (3), (4) and (5) the mathematical model with discrete time and consumer sentiment is given by: Yt

= a + dYt-2 + (b -

d)Yt-1

+ V(Yt-1

CYt-1 + m + 1 + Eexp (Yt-2 -

Yf-I

)' t

_ N - 2.. .

- Yt-2) - W(Yt-1 - Yt_2):l+

(6)

The parameters from (6) have the following economic interpretations. The parameter a represents the autonomous expenditures. The parameter d is the control, dE [0,1] and it characterizes a part of the difference between the incomes obtained at two time steps t - 2 and t - 1, which is used for consumption or saving in the time step t. The parameter c, c E [0, 1], is the trend towards consumption. The parameter m, m E [0,1]' is the trend towards the saving. The parameter b, b E (0,1), represents the consumer's reaction against the increase or decrease of his income. When the income (strongly) decreases, the consumer becomes pessimistic and consumes < b < 1 of his income. When the income (strongly) increases, the consumer becomes optimistic and consumes b < b + c < 1 of hi" income. Note that Souleles ([7]) finds, in fact, that higher consumer confidence is correlated with less saving and increases in relation to expected future resources. The parameters v and w, 11 > 0, 11J :;0, describe the investment function. If W = the investment function is linear. The parameter E, E E [0, 1], describes a family of the sentiment functions. For different values of the model parameters, we obtain the following classical models: 1. for a = 0, b = 1 - s, d = 0, m = 0, E = 0, s E (0,1) from (6) we obtain the Hick-Samuelson model ([6]):

°

°

°

Yt

=

(1

+v

- S)Yt-1 - VYt-2,

v> O,S

E (0,1);

Hick Samuelson Keynes Dynamic Economic Model

2. for v

=

w

= 0, c = 0, rn = 0, (6) gives us the Keynes model ([6]): Yt

3. for v Yt

=

w

YI

d(YI-2 - Yt-l)

+ a + bYt-l;

= 0, c = 1, (6) leads to the model from [2]:

4. for a = O,b model ([7]):

3

=

d(Yt-2 - YI-l)

=

69

=

CYt-l + rn + a + bYt-l + ( .)' tEN; . 1 + exp Yt-2 - Yt-l

=.5 - I,d = O,Tn = O,W = 1 + V,c = 0, from (6) we get the Puu V(YI-1 - Yt-2) - (1

+ V)(YI-l

- YI_2)3 - (1 - S)Yt-l.

The dynamic behavior of the model (6)

Using the method from Kusnetsov ([5]), we will analyze the system (6), considering the parameter a as bifurcation parameter. The associated map of (6) is F : ]R2 --+ ]R2 given by:

FG) =

(a

+ (b -

d

+ v)y + (d -

+ l+E~!~;(:'-Y»)

v): - w(y - z)3

(7)

Using the methods from [3], [4] and [6], the map (7) has the following properties: Proposition 1. (i). If (1 + c)(I- b) - c > 0, then, for the map (7), the fixed point with the positive components is Eo (Yo, zo), where:

Zo = Yo and

l+c PI

= (1 + c) (1 -

Tn b) - c ,P2

= (1 + c) (1 -

b) - c;

(8)

(ii). The Jacobi matrix of the map F in Eo is given by:

A = (all a 12 ) 1

° '

(9)

where: all = P3 a - P4 + P5, a12 = -P3a + P4, = (1~~)2pl,P4 = d - 11 - (1~1J:)2 - (1~~)2P2,P5 = b + (I~E); (iii). The characteristic equation of matrix A is given by:

P:l

A2 - allA - a12

=

(10)

0;

(iv). If the model parameters d, v, c, b, c, rn satisfy the following inequality:

(1

+d-

v)(l

+ c)((1 + c)(l -

b) - c) - rnc(1 - b)

> 0,

(11)

then, for equation (1 OJ, the roots have the modulus less than 1, if and only if a E (al,a2), where 2P4 - P5 - 1 1 + P4 (12) al = a2 = - 2P3' P3

70

L. I. Dobrescu, M. Neam/u, and D. Opriij

(v). If the model parameters d, v, c, b, c, Tn satisfy the inequality (11) and a = al then, one of equation (10)'s roots 'is -1, while the other one has the modulus less than 1; (vi) If the model parameteTs d, v, c, b, c, Tn satisfy the inequality (11) and a = a2 then, the equation (10) has the TOots fJoI(a) = fJo(a),fJo2(a) = /i(a), wheTe IfJo(a) I = 1.

Using [6] and Proposition 1, with respect to parameter a, the asymptotic stability conditions of the fixed point, the conditions for the existence of the flip and NeimarkSacker bifurcations are presented in: Proposition 2. (i). If (1 + c)(1 - b) - c > 0, the inequality (11) holds and 2P4Ps - 1 > 0, then fOT a E (aI, (2) the fixed point Eo is asymptotically stable. If (1 + c)(1 - b) - c > 0, the ineq'uality (11) holds and 2P4 - P5 - 1 < 0, then for a E (0, (2) the fixed point Eo is asymptotically stable; (ii). If (1 + c)(1 - b) - c > 0, the inequality (11) holds and 2P4 - P5 - 1 > 0, then a = al is a flip bifuTcation and a = a2 is a NeimaTk-SackeT bifuTcation; (iii). If (1 + c)(1 - b) - c > 0, the inequality (11) holds and 2P4 - P5 - 1 < 0, then a = a2 is a Neimark-SackeT bifuTcation,

4

The normal form for flip and Neimark-Sacker bifurcations

In this section, we describe the normal form in the neighborhood of the fixed point Eo, for the cases a = al and a = a2. We consider the transformation: UI

=

(13)

Y - Yo,

where Yo, Zo are given by (8). With respect to (13), the map (7) is where

G:]R2 ---> ]R2,

(14) and gl(UI,1L2)

= - -r1 + (b +c

- d

+ V)UI + (d -

CUI + r I+Cexp(U2- UI)

+-----,------,-

g2(UI, U2)

=

uI, T

=

cYo

+ Tn

The map (14) has 0(0,0) as fixed point. We consider:

V)U2 -W(UI - U2)3+

Hick Samuelson Keynes Dynamic Economic Model

71

We develop the function G(u),u = (U1 , U2)T in the Taylor series until the third order and obtain:

1

G(u) = Au + "2B(u , u)

1

+ "2C(u , u, u) + 0(luI 4 )

where A is given by (9) and B(u , u) = (B1(u , u) ,

of, C(u, u , u) =

(C 1(u , u, u),

of

(15)

For a = aI, given by (12) with the condition (v) from Proposition 1, we have: Proposition 3. (i). The eigenvector q E nents: q1 = 1, q2 (ii) The eigenvector h E

]R2,

=

hI

]R2 ,

=

given by Aq = -q has the compo-

(16)

-l.

given by hT A = _hT has the components :

1

- - - ,h 2 1 + a12

a12

= - ---. 1 + a12

The relation < q, h >= 1 holds. (iii). The normal form of the map (7) on the center manifold in 0(0,0) is given

by:

7) where v =

-1+1 a12

(130 - 31 21

1

--->

-7) + 6v7)3

+ 31 12 -103) +

+ 0(7)4) , 1

3

- all -a1 2

(1 20 - 2111

+ 102)2.

The proof results from straight calculus using the formula (6): v

=< r, C(q , q, q) > +3B(q, (I - A)-l B(q, q)), 1 =

(~ ~) .

For a = a2, given by (12) with the condition (vi) from Proposition 1, we have: Proposition 4. (i). The eigenvector q E ([2, given by Aq = J-i1q, , where J-i1 is the eigenvalue of the matrix A , has the components: q1 = 1, q2 = J-i2 = Ill· (ii) The eigenvector h E ([2, given by hT A = J-i2hT, where J-i2 components: 1 J-i1a12 hI = 2 ' h2 = 2· 1 + J-i1 a12 1 + J-i1 a12 The relation

< q, Ii >= 1

holds.

=

Ill' has the

72

L. 1. Dobrescu, M. Neamfu, and D.

Opri~

Using (15) and (16) we have:

= 120 + 2111/12 + 102/1~, = 120 + 111 (/11 + /12) + 102 /11/12, Bl (q, q) = 120 + 2111/11 + 102/1i.

Bl(q, q)

Bl (q, q)

We denoted by: 920 = VIB1(q,q),911 = VIB1(q,q),902 = VI B1 (q,q),

hil

h~o = (1 - VI - Vl)920,

(1 - VI - Vl)911, h62

=

=

(1- VI -

Vl)902,

h~o

= -(Vl/12 +Vl/11)B 1(q,q),hil = -(Vl/12 +Vl/11)B 1(q,q), h62 = -(Vl/12 + Vl/11)B 1(q, q), w20 = (/112I - A) -1 (h~o) h~o' Wl1 = (I - A) -1 where I

=

(hil) hil' W02 =

(17)

(/122 - A) -1 (h62) h62' (18)

(~ ~), A is given by (9) and

= B 1(q,w2o),rll = B 1(q,Wl1), Co = C 1(q, q, q) = 130 + (/11 + 2/12)121 + /12(2/11 + /12)112 + /11/1~103, 921 = V2(r20 + 2rl1 + Co). r20

(19)

Using the normal form for the Neimark-Sacker bifurcation of the dynamic systems with discrete time ([6]) and (17), (18), (19), we obtain: Proposition 5. (i). The solution of the system (6) in the neighborhood of the fixed point (Yo, zo) E lR n is given by:

Yt

-

1 2

2

2

1

_

2-2

= Yo + /12 Xt + /11 Xt + '2W20Xt + Wl1 XtXt + '2 W02Xt ,

1 1 2 1 _ 1 1-2 Yt-l = Zt = Zo + Xt + Xt + '2W20Xt + Wl l XtXt + '2W02Xt,

where Xt is the solution of the following equation:

Xt+l

1

2

_

1

-2

1

2-

= /1Xt + '2920Xt + 911 Xt X /, + '2902·7;t + '2921XtXt.

(ii) A eomplex variable transformation exists so that the equation (20) becomes:

Wt+l

= /11Wt + C 1W;Wt + O(IWtI4),

where

Cl=920911(tL2-3-2tLIl+ Igl11 2 + 19201 2 +921 2(/1i - /11)(/12 - 1) (1 - /1Il 2(/1i - /12) 2' (ii'i). Iflo = Re(e- W C 1 ) < 0, where B = arg(/1Il,then in the neighborhood of the fixed point (Yo, zo) ther'e is a stable limit cycle.

Hick Samuelson Keynes Dynamic Economic Model

5

73

The Lyapunov exponents

°

If a1 > then for a E (0, ad or a E (a2 ' 1) the system (6) has a complex behavior and it can be established by computing the Lyapunov exponents. We will use the algorithm QR from [6] and the determination of the Lyapunov exponents can be obtained by solving the following system:

Yt+1

3

= a + (b - d + V)Yt + (d - V)Zt - W(Yt - Zt) +

cYt

+m

( ) 1 + exp Zt - Yt

Zt+ 1 = Yt Xt+1 = arctan (

cos Xt . ) 111 cos Xt - h2 sm Xt

= At + In( I(/11 - tanxt+d COSXt COSXt+1 - h2 sinxt cOSXt+11) /-It+1 = /-It + In(1 (/11 - tan Xt+1 + 1) sin Xt cos Xt+1 + h2 cos Xt cos xt+11), At+1

with

c c P f 11 -_ -811( Yt, Zt )-b - - d+ v - 3W(Yt - Zt )2+ + (c +m+CYt)ex (Zt2 - Yt} '

(1

8y

8h(

f 12 = -

8z

Yt, Zt

)

=

d

+ c exp(Zt -

- v + 3W(Yt - Zt )2 - c(m+cYt)exp(Zt - Yt) 2 (l+cexp(Zt - Yt))

Yt))

.

The Lyapunov exponents are:

. At L 1 = hm -,L 2 t~ oo

t

. /-It = t~oo lUll t

(20)

If one of the two exponents is positive, the system has a chaotic behavior .

6

Numerical simulation

The numerical simulation is done using a Maple 13 program. We consider different values for the parameters which are used in the real economic processes. The bifurcation parameter is a. For a = 250, v = 0.1, W = 0, C = 0.1, b = 0.45 , c = 1, m = 0.5 and the control parameter d = we obtain in Figl the evolution of the income in the time domain (t,yd, in Fig 2 the evolution of the income in the phase space (Yt - 1, Yt). In both these cases, the system is not chaotic.

°

Fig. 1. (t, Yt), for d =

°

Fig. 2. (Yt-1 , Yt), for d = 0

74

L. I. Dobrescu, M. Neamju, and D. Opri}

For the control parameter d = 0.6 Fig3 displays the evolution of the income in the time domain (t, yd, Fig 4 the evolution of the income in the phase space (Yt-l, Yt)· The Lyapunov exponent is positive and the system has a chaotic behavior.

,..

Fig. 3. (t, Yt), for d

=

'

0.6 Fig. 4. (Yt-l, Yt), for d

= 0.6

For different values of the parameters, we can obtain a Neimark-Sacker bifurcation point or a flip bifurcation point.

7

Conclusion

Hick Samuelson Keynes dynamic model with discrete time using investment, consumption, sentiment and saving functions is studied. The behavior of the dynamic system in the fixed point's neighborhood for the associated map is analyzed. We establish asymptotic stability conditions for the flip and Neimark-Sacker bifurcations. The QR method is used for determining the Lyapunov exponents and they allows us to decide whether the system has a complex behavior. Using a program in Maple 13, we display the evolution of the income in the time domain and the phase space.

References l.C. Carroll, J. Fuhrer, and D. Wilcox. Does consumer sentiment forecast household spending? If so, why? American Economic Review, S4:1397-140S, 1994. 2.L.1. Dobrescu, and D. Opris. Neimark-Sacker bifurcation for the discrete-delay Kaldor model. Chaos , Solitons fj Fractals, doi:1O.1016/j.chaos.2007.10.044, 2007. 3.L.I. Dobrescu, and D. Opris. Neimark-Sacker bifurcation for the discrete-delay KaldorKalecki model. Chaos, Solitons fj Fractals, doi:1O.1016/j.chaos.200S.09.022, 2008. 4.M. Doms, and N. Morin. Consumer sentiment, the economy, and the news media. Federal Reserve Bank of San Francisco, Working Paper 2004-09. 5.Y.A. Kuznetsov. Elements of Applied Bifurcation Theory. Applied Mathematical Sciences 112, Springer-Verlag, New York, 1995. 6.M. Neamtu and D. Opris. Economical games. Disctrete economical dynamics. Applications. Mirton, 200S (in Romanian). 7.T. Puu and 1. Sushko. A business cycle model with cubic nonlinearity. Chaos, Solitons and Fractals, 19:597-612, 2004. S.N. Souleles. Expectations, heterogeneous forecast errors, and consumption: Micro evidence from the Michigan Consumer Sentiment Survey. Journal of Money, Credit and Banking, 36:39-72, 2004. 9.F.H. Westerhoff. Consumer sentiment and business cycles: A Neimark-Sacker bifurcation scenario. Applied Economics Letters, 14:1-5, 2007.

75

On the Entropy Flows to Disorder c.

T. J. Dodson

School of Mathematics University of Manchester Manchester M13 9PL, UK (e-mail: [email protected] . uk) Abstract: Gamma distributions, which contain the exponential as a special case, have a distinguished place in the representation of near-Poisson randomness for statistical processes; typically, they represent distributions of spacings between events or voids among objects. Here we look at the properties of the Shannon entropy function and calculate its corresponding flow curves, relating them to examples of constrained degeneration from ordered processes. We consider univariate and bivariate gamma, and Wei bull distributions since these also include exponential distributions. Keywords: Shannon entropy, integral curves, gamma distribution, bivariate gamma, McKay distribution, Wei bull distribution, randomness, information geometry. MSC classes: 82-08; 15A52.

1

Introduction

The smooth family of gamma probability density functions is given by

. [0 00) ----) [0 00) . x f . , ,.

----) e-";2 x K -

1

r(K)

(~r

iL,

K

> O.

(1)

Here iL is the mean, and the standard deviation 17, given by K (;;:)2, is proportional to the mean. Hence the coefficient of variation is unity in the case that (1) reduces to the exponential distribution. Thus, K = 1 corresponds to an underlying Poisson random process complementary to the exponential distribution. When K < 1 the random variable X represents spacings between events that are more clustered than for a Poisson process and when K > 1 the spacings X are more uniformly distributed than for Poisson. The case when iL = n is a positive integer and K = 2 gives the ChiSquared distribution with n - 1 degrees of freedom; this is the distribution of ~ for variances 8 2 of samples of size n taken from a Gaussian population (Je

fo

with variance 17~. The gamma distribution has a conveniently tractable information geometry [1,2], and the Riemannian metric in the 2-dimensional manifold of gamma distributions (1) is

(2)

76

C. T. 1. Dodson

So the coordinates (p" K,) yield an orthogonal basis of tangent vectors, which is useful in calculations because then the arc length function is simply

The system of geodesic equations is difficult to solve analytically but numerical solutions using the Mathematica programs of Gray [7] were obtained in [2]. Figure 1 shows a spray of some maximally extended geodesics emanating from the point (p" K,) = (1,1). Geodesic curves have tangent vectors that are parallel along them and yield minimal information arc length distances in the gamma 2-manifold . 2.00 1.75

K, 1.50 1.25 1.00 0.75 0.50 0.25

0.5

1.0

1.5

2.0

2.5

3.0

/1

Fig. 1. Some examples of maximally extended geodesics passing through (/1, K,) = (1,1) in the gamma 2-manifold.

We note the following important uniqueness property: Theorem 1 (Hwang and Hu (8)). For independent positive random variables with a common probability density function f, having independence of the sample mean and the sample coefficient of variation is equivalent to f being the gamma distribution.

This property is one of the main reasons for the large number of applications of gamma distributions: many near-random natural processes have standard deviation approximately proportional to the mean [2].

On the Entropy Flows to Disorder 77

2

Gamma entropy flows

Papoulis [13] Chapter 15 gives an account of the role of the Shannon entropy function in probability theory, stochastic processes and coding theory. The entropy of (1) is shown in Figure 2 using Sf

(tt, r;,)

=

-1

00

f logf

dx : lR 2+

-+

lR

~ r;, -- log (~) + log(r(r;,)) -

(r;,-1)1,b(r;,)

(~3)

with gradient

( ~, J.i

_ (r;, -1) (r;,1,b/(r;,) -1) ) . r;,

(4)

r;:

where 1,b = is the digamma function. At fixed r;" the entropy increases like log J.i. At fixed mean J.i, the maximum entropy is given by r;, = 1, the exponential distribution case of maximal disorder or chaos.

Fig. 2. Shannon entmpy function Sf for the gamma family as a sm:face with respect to mean fJ, and", (left) and as a conto'u r plot with entropy gradient flow and integral curves (right), The asymptote is", = 1, the exponential case of maximum disorder.

Figure 2 on the right shows entropy as a contour plot with superimposed also some examples of integral curves of the entropy gradient flow field, namely curves c satisfying c:

[0,(0)

-+ ]R2:

c(t) =

V'Sf l(c(t)) .

(5)

78

C. T. J. Dodson

By inspection, we can see that the entropy gradient components are each in one variable only and in particular the first component has solution

so the mean increases exponentially with time. Such curves represent typical trajectories for processes subordinate to gamma distributions; the processes become increasingly disordered as K, -+ 1. The entropy gradient curves correspond to systems with ext ernal input- the mean increases as disorder increases. The asymptote is K, = 1, the exponential case of maximum disorder. Conversely, the reverse direction of the curves corresponds to evolution from total disorder to other states (clustered for K, < 1, and smoothed out, ' more crystal-like', for K, > 1) while the mean is allowed to reduce-somewhat like the situation after the Big Bang, see Dodson [4].

3

Constrained degeneration of order

Lucarini [ll] effectively illustrated the degeneration of order in his perturbations of the simple 3D cubic crystal lattices (SC, BCC, FCC) by an increasing spatial Gaussian noise. Physically, the perturbing spatial noise intensity corresponds somewhat to a lattice temperature in the structural symmetry degeneration. With rather moderate levels of noise, quite quickly the three tessellations became indistinguishable. In the presence of intense noise they all converged to the 3D Poisson-Voronoi tessellations, for which exact analytic results are known [6]. Moreover, in all cases the gamma distribution was an excellent model for the observed probability density functions of all metric and topological properties. See also Jarai-Szabo and Neda [9] for some analytic approximations using gamma distributions for two and three dimensional Poisson-Voronoi cell size statistics. Lucarini provided plots showing the evolution of the mean and standard deviation of these properties as they converge asymptotically towards the Poisson-Voronoi case, illustrating the degeneration of crystallinity from K, ~ 00 to lower values. Of course, the constraint of remaining tessellations , albeit highly disordered ones, precludes convergence down to the maximum entropy limit K, = 1. In fact the limiting values are K, ;::,;; 16 for number of vertices and the same for number of edges and K, ;::,;; 22 for the number of faces; actually these are discrete random variables and the gamma is not appropriate. However, for the positive real random variables, polyhedron volume in the limit has K, ;::,;; 5.6 and polygon face area K. ;::,;; 16. Lucarini [10] had reported similar findings for the 2D case of perturbations of the three regular tessellations of the plane: square, hexagonal and triangular. There also the gamma distribution gave a good fit for the distributions during the degeneration of the regular tessellations to the 2D Poisson-Voronoi case; the limiting values were K, ;::,;; 16 for the perimeter of polygons and K, ;::,;; 3.7 for areas.

On the Entropy Flows to Disorder

79

We can give another perspective on the level of constraint persistent in the limits for these disordered tessellations: for infinite random matrices the best fitting gamma distributions for eigenvalue spacings have", = 2.420, 4.247, 9.606 respectively for orthogonal, unitary and symplectic Gaussian ensembles [2]. These values in a sense indicate the increasing statistical constraints of algebraic relations as the ensembles change through orthogonality, unitarity and symplectivity. In the context of analytic number theory, gamma distributions give approximate distributions for the the reported data on spacings between zeros of the Riemann zeta function [12]. The best fit gamma distributions to spacings between the first two million zeros of the Riemann zeta function has "':::::; 5.6 [2].

For the SARS disease epidemic outbreak [3,14] the gamma distribution gave a good model for the infection process with", :::::; 3, cf. [5].

4

Bivariate gamma processes

Next we consider the bivariate case of a pair of coupled gamma processes. The McKay family can be thought of as giving the probability density for the two random variables X and Y = X + Z where X and Z both have gamma distributions. This smooth bivariate family M of density functions is defined in the positive octant of random variables 0 < x < y < (Xl with parameters CYi, C, CY2 E ~+ and probability density functions cCCXl+CX2)XCXl-i(y _

m(x, y) =

x)cx 2- i e -c y

(6)

r(CYdr(CY2)

The marginal density functions, of X and Yare:

(7)

x>o c(CX1 +CX2)y(CX 1+cx 2 )- le- c y

my(y) =

r(CYi

y>

+ CY2)

o.

Note that we cannot have both marginal distributions exponential. covariance and correlation coefficient of X and Yare: 0"12

=

CYi and p(X, Y) 2"" C

(8) The

= { ; ! £ ; i. CYi

+ CY2

Unlike other bivariate gamma families, the McKay information geometry is surprisingly tractable and there are a number of applications discussed in Arwini and Dodson [2]. In fact the parameters CYi, C, CY2 E ~+ are natural coordinates for the 3-dimensional manifold M of this family. The Riemannian

80

C. T. 1. Dodson

information metric is

(9)

It is difficult to present graphics of curves in the McKay 3-manifold M, but it has an interesting 2-dimensional submanifold M1 C M: 001 = 1. The density functions are of form: (10) defined on 0 < x < y < = with parameters c,002 E jR+. The correlation coefficient and marginal functions of X and Yare given by:

(11) (12) (13) In fact 002 = l~t, which in applications would give a measure of the variability not due to the correlation. The matrix of metric components [gij 1 on M1 is

(14) Some geodesics emanating from (002, c) = (1,1) E M1 are shown on the left of Figure 3. The density functions can be presented also in terms of the positive parameters (OO1,CJ12,OO2) where CJ12 is the covariance of X and Y

(15)

x>O

(16) y > O.

(17)

On the Entropy Flows to Disorder

c

81

c

Fig. 3. Geodesics passing through (c, (2) = (1,1) (left) and a contour plot of the entropy wdh some integral gradient CU'f'1JC8 (right), in the McKay submanifold 1111 which has a1 = 1.

T he entropy function is

On the right of F igure 3 is a contour plot of t he entropy showing its gradient field and some integral curves. It may be helpful to express the entropy in terms of a2 and p, which gives

T he McKay entropy in M1 with respect to a2, P is shown in F igure 4 (left) with the gradient flow on a contour plot (right) together with the approximate locus curve of maximum entropy. The two thinner curves show t he loci of a 1 = 1 (upper curve) and a1 + a2 = 1 (lower curve), which correspond, respectively, to Poisson random processes for the X and Y variables.

82

C. T. 1. Dodson

a2

Fig. 4. Surface representation of the Shannon entropy function 8 m f01' the s'U.bmani-

fold !viI of the M cKay family 'with respect to p(L7'ameter' a2 and correlation coefficient p (left), and contour plot (right). Superimposed also are gradient flow arrows and the thick curve is the loc'us of maximum entmpy. The two thinner C1lr'Ves show the loci of al = 1 (upper curve) and al + a2 = 1 (lower curve), which correspond, respectively, to Poisson random processes for the X and Y variables.

Qualitatively, what we may see in Figure 4 is that for correlated random variables X and Y subordinate to the bivariate gamma density (10), the maximum entropy locus is roughly hyperbolic. The maximum entropy curve is rather insensitive to the correlation coefficient p when (X2 > 1 and the difference Y - X is dispersed more evenly than Poisson. When Y - X is actually Poisson random, with 0;2 = 1, the critical value is at p ~ 0.355. However, as 0;2 reduces fnrther- -corresponding to clustering of Y - X values- ~so the locus turns rapidly to increasing correlation. If we take the situation of a bivariate gamma process with constant marginal mean values, then the McKay probability density has constant correlation coefficient p; in this case the gradient flow lies along lines of constant 0;2· Dodson [5] gives an application to an epidemic model in which the latency and infectivity for individuals in a population are jointly distributed properties controlled by a bivariate gamma distribution.

5

Weibull processes

Like the gamma family, the Wei bull family of distributions contains the exponential distribution as a special case; it has wide application in models for reliability and lifetime statistics for random variable t > O. The probability density function can be presented in terms of positive parameters ~,f3 w : jR+ ---; jR : t

h>

f3 ( ~

f)/3-1.e~ (t/O{3 .

(21)

On the Entropy Flows to Disorder

83

Fig. 5. Surface repl-csentaJion of the Sha.nnon cntmpy ftmction Sw for the Weibnll family (left). and conto'Wr plot (T'ight) with gmdient flow and integml carves.

In applications of (21), reliability R(t) is the probability of survival to time t and it is related to the failure rate Z (t) at time t through

i· X

_'t!i.:(J

R(t)=w(t)dt=e (f~) >

t

1)

r.' 1JJ (t) " ( and Z(t) = _ . =[3 -. . R( t) . ~

(I

j

(22)

The INeihull mean, Htandard deviation and entropy are /1",

1

= <,C T(l +. -(1 ) (24)

(25) (26) In (25) 1 is the Euler constant. approximately 0.577. In cFt.'le 3 = positive integer n, then the coefficient of variation is

!!3L /1-11.'

=

("

)I

-(":~')i -n.

*

for

1, and

we sec that tl = 1 reduces (21) to the exponential density with Itw = CJw = ~ and unit eoefIicient of variation. Figure 5 shows 11 surface plot of the Weibull entropy Sw and a corresponding contour plot ,\lith gradient flow and some integral curves. There is a resemblanee to the gamma distribution entropy in Figure 2 a.nd from (26) again here we have /-lw(t) = /-l'/u((J)e t , but in the \Veibull case the asymptotic curve has f3 = r ~ 0.577, which corresponds to it process with coefficient of variation ~ 4.65 compared with unity for the exponential distribution and Poisson randomness.

84

C. T. 1. Dodson

Acknowledgement The author is grateful to V. Lucarini for discussions concerning the interpretation of his simulations.

References 1.S-I. Amari and H. Nagaoka. Methods of Information Geometry. American Mathematical Society, Oxford University Press, Oxford, 2000. 2.Khadiga Arwini and C. T. J. Dodson. Information Geometry: Near Randomness and Near Independence. Springer-Verlag, New York, Berlin, 2008. 3.T. Britton and D. Lindenstrand. Epidemic modelling: aspects where stochasticity matters. http://arxiv.org/abs/0812.3505, pages 1~16, 2008. 4.C. T. J. Dodson. Quantifying galactic clustering and departures from randomness of the inter-galactic void probablity function using information geometry. http://arxiv.org/abs/astro-ph/0608511, 2006. 5.C. T. J. Dodson. Information geometry and entropy in a stochastic epidemic rate process. http://arxiv . org/abs/0903. 2997, 2009. 6.S.R. Finch. Mathematical Constants. Cambridge University Press, Cambridge, 2003. 7.A. Gray. Modem Differential Geometry of Curves and Surfaces 2nd Edition. CRC Press, Boca Raton, 1998. 8.T-Y. Hwang and C-Y. Hu. On a characterization of the gamma distribution: The independence of the sample mean and the sample coefficient of variation. Annals Inst. Statist. Math., 51(4):749~753, 1999. 9.F. Jarai-Szabo and Z. Neda. On the size distribution of poisson voronoi cells. http://arxiv.org/abs/cond-mat/0406116v2, 2008. 1O.Valerio Lucarini. From symmetry breaking to poisson point processes in 2d voronoi tessellations: the generic nature of hexagons. 1. Statist. Phys., 130:1047~1062, 2008. I1.Valerio Lucarini. Three-dimensional random voronoi tessellations: From cubic crystal lattices to poisson point processes. 1. Statist. Phys., page In press, 2008. 12.A. Odlyzko. Tables of zeros of the riemann zeta function http://www.dtc.umn.edu:80/-od1yzko/zeta_tab1es/index.htm1. 2009. 13.A. Papoulis. Probability, Random Variables and Stochastic Processes. McGrawHill, New York. 14.WHO. Cumulative number of reported probable cases of severe acute respiratory syndrome (sars). http://www.who.int/csr/sars/country/en/, 2003.

85

Linear Communication Channel Based on Chaos Synchronization Victor Grigoras 1 and Carmen Grigoras 2

2

Faculty of Electronics, Telecommunications and Information Technology, 'Gh. Asachi' Technical University, Iasi, Romania Faculty of Medical Bioengineering, University of Medicine and Pharmacy "Gr. T. Popa", Ia~i, Romania Romanian Academy - Ia~i Branch, Institute of Computer Science, Romania (e-mail: [email protected])

Abstract: This paper presents the possibility of designing a linear communication channel by modulating chaotic analog systems. After presenting the general setup, conditions for correct demodulation and linear dynamic input-output behavior are demonstrated. For a linear dynamic relation between the modulating and demodulated signals, channel equalization is used to achieve wider bandwidth transmission. The presented case studies, regarding the Lorenz and Chen systems, highlight the applicability of the proposed method for high speed digital communication. The overall performance of the resulting communication system is analyzed in terms of speed, security and occupied frequency bandwidth. The concluding remarks point towards some directions in further research. Keywords: Chaos synchronization, channel equalization, Lorenz system, Chen system.

1. Introduction The present contribution aims at characterizing an analog modulation method to build a wide-band communication channel. The proposed approach is based on chaos synchronization between the emitter and receiver ends in order to achieve a supplementary level of security under the standard digital encryption layer. The general synchronization setup is based on the system partitioning method, first introduced by Carroll and Pecora in [1]. Considering some reasonable restrictions for the chaotic system, we develop a method for the characterization of the input-output relation of the proposed communication channel, namely between the modulating signal applied at the emitter end and the demodulated signal obtained at the output of the receiver. The linear dynamic system that models the studied relation shows the frequency limitations for the modulating signal. In order to increase the transmitted signal bandwidth, a feed forward equalizer is proposed and its efficiency studied. Case studies regarding third order chaotic systems of the Lorenz [2] and Chen [3] types aim at proving the feasibility of the proposed method. The Lorenz system, chosen for implementation advantages [4] and for synchronization simplicity [5-6], proved to be a good prototype for the proposed method, allowing easy synchronization and demodulation. In order to apply the general setup to the Chen system, some coefficient modifications had to be made to allow the proposed demodulation scheme.

86

V. Grigoras and C. Grigoras

The next section concentrates on the demonstration of the proposed method in the general case. The error dynamics method is used, leading to the conclusion that the input - output relation of the proposed channel is linear. Simulation results, confirming the theoretical solution, are presented in the third section, for two case studies.

2. General Results The communication channel based on chaos synchronization is built around a nonlinear chaotic emitter described by the general state equations: x'=f(x) where x denotes the N-dimensional state vector and f : lRN ~ lRN is the nonlinear state transition function. In order to achieve chaos synchronization, using the emitter subsystem approach, we presume that the above state equations can be partitioned in the form:

j

XI ' = ./; I (XI) + ./;? (XR) , -() lXI] ./;1: lR ~ R '/; 2: lR N- I ~ lR; x R =f21(XI)+f22 x R ; X= ;f .m mN-I' f . mN-I mN-I XR 21. "" ~ "" , 22 ' "" ~ "" y=XI

If we choose to transmit the first state variable, receiver results in the form:

Xl,

the synchronizing

XR' =i22 (XR )+f21(Y) where the ';' signals and functions denote the receiver correspondents of the emitter ones. Presuming the receiver has been correctly designed, in the case of parameter matching ((21 (-) = f21(-);£22 (-) = f22 (- p, the error dynamics IS globally asymptotically convergent to zero: E'=f22 (xR)-f22(x R ) ; E(t) ~ 0 where the error E (t ) = x R (t) -

x (t) R

denotes the difference between the emitter

and receiver state vectors. To achieve transmission of the useful information signal, met), a direct modulation approach is used at the emitter end, by modifying the first state equation: XI'='/;I (XI)+./;2 (xR)+m(t) At the receiver end, demodulation is implemented by appending the previous receiver state equations with a similar one, without modulation:

XI'=~I(XI)+~ 2 (XR) The demodulated signal is algebraically obtained giving the receiver output equation:

Linear Communication Channel Based on Chaos Synchronization

87

The dynamic time evolution of the demodulated signal is governed by the resulting nonlinear differential equation, obtained by subtracting the previous state equations: m'(t) = .1;1 (Xl) - ~ 1(j\) + i12 (XR) - 112 (XR) + m(t) As we presumed the error dynamics to be globally asymptotically stable, in the case of parameter matching (~1 (.) = .I; 1 (.); ~2 (.) = i12 (.)), the second difference converges to zero

i12(XR)- 112(X R )-70 After the synchronizing transient, the demodulated signal differential equation is simplified to the form: m'(t) =

111 (X1)- 111 (i\)+m(t)

Assuming the average value of the modulating signal is zero, we can decompose the algebraic nonlinear function, ill (.) in a power series around this value, resulting:

.1;1 (X) =

f ~n. d"!:1 (O)·x" dx n=O

If the nonlinear function,!II(.), is smooth enough and the modulating signal

small, we may neglect the higher order terms, to obtain: d.l; I (0)· met) + met); met) = XI (t) - iJt) dx The resulting linear dynamic system, characterizing the proposed communication channel based on chaos modulation, is stable if the structural constant, a, is negative: m '(t) =

a=d.l;l(O)
The obtained result shows that the modulating signal cannot be perfectly recovered unless the maximum frequency in its spectrum
s+a

l/mc' s+ 1 The overall communication system is linear, characterized by the transfer function:

88

V. Grigoras and C. Grigoras

3. Case Studies 3.1. The Lorenz system In this particular case, the emitter subsystem is a third order, analogue, Lorenz type system:

j

X'

=

a· (y - x)

y' = p·x- y-x· Z z'=x-f3·z+x·y

For a large enough parameter range, this system exhibits chaotic behavior, useful in ensuring the secrecy the modulating signal. To achieve synchronization, the y state variable is transmitted to the receiver subsystem, designed by using the system partitioning method: i' = a·(y-i) { z'=i-f3·z+i· y The second equation, for y' , will be introduced later on in the state description of the receiver for further use in the modulation/demodulation process. Taking the autonomous case, y = 0, its state equations become linear, with a state transition matrix of the form: A =(-a 1

0)

-13

The resulting eigenvalues are real and negative, for positive a and p, ensuring global asymptotic stability of the receiver. The error dynamics can be analyzed by subtracting the receiver equations from their emitter counterparts. The state errors, EJ = x - x, E3 = Z - Z, are described by the dynamic equations: {

EJ ' =

-a . EJ

E3 ' = EJ

-

13 .E3 + E

J •

Y

Due to the fact that all state variables are bounded, it is easy to demonstrate iteratively that all errors decay exponentially to zero. Thus the proposed receiver synchronizes with the chaotic emitter. In order to use this synchronizing setup to transmit an information signal, met), by the direct modulation technique, we add the modulating signal to the second equation of the emitter: X' = a·(y-x)

j

y:= p·x- y-x· z+m(t) =x-p·z+x·y

Z

At the receiving end of the communication channel, we demodulate the received signal, y(t), by subtracting from it the locally recovered second state variable, y:

y'= p·x- y-x· z

Linear Communication Channel Based Oil Chaos Synchronization

Figure 1.a The PSD of the modulating signal

Figure l.h The PSD of the chaotic signal for amplitude 10

89

Figure l.c 'nlC PSD of the chaotic signal for amplltude 20

The error dynamics for the second state variable: £2 • =

P . £J -

£2 -

(x· z -

i . z) + m

leads to the differential eqnation for thc demodulated signal: !n' == -/iI-(z - P )'£1 -.x·e, + m After the synchronization transient died out the demodulator equation can be approximated as follows: iii' "" -J'iI+m This highlights the linear memory character of the resulting communication channel, which can be characterized by the transfer function :

H(s)=!Yn~) ",_1__ M(s) s+1 The dynamic range of the amplitude for the modulating signal must ensurc both linearity of the communication system and the lack of visibility of the modulating signal in the transmitted chaotic one. In order to track the visibility of the modulating signal in the transmitted chaotic one, we studied the power spectral density (PSD) of the transmitted signal, for different amplitudes of the modulating signal, depicted in figure 1. As seen from the simulation results, if the amplitude of the modulating signal is as high as 10, its PSD is not visible in the one of the transmitted signal. Beginning with amplitudes of the order of magnitude 20, the modulator line becomes visible in the PSD of the chaotic signal. This gives a dynamic range of modulating sigmtl amplitude comparable to the amplitudes of the state variables and transmitted signal, thus large enough for practical applications.

Figure 2.a The modulating signal

Figure 2.b The demodulateD signal without equalizer

Figme 2.e The demodulated signal with equalizer

90

V. Grigoras and C. Grigoras

To increase the bandwidth of the transmitted signal, we use for our equalizer a transfer function of the form: s+1 s+l l/mc ·s+1 O.OI·s+1 In figure 2, we give example results showing the demodulated signal without and with the feed forward equalizer. H E(s)=

3.2. The modified Chen system The Chen system is described by the state equations: x,=-a.x+a. y

!<=

(c-a )·x+c· y-x· z

z =-b· z+x· y where the standard values for the coefficients are a = 35, b = 3 and c = 28. To achieve synchronization, the second state variable must be transmitted and the synchronizing receiver is built around the first and third state equations. Due to the fact that the y coefficient in the second state equation, c, is positive, the proposed demodulation method cannot function since it will lead to demodulator instability. In order to use our approach on the Chen system, some coefficient modifications need to be made: x,=-a . x+a. y

!

y.'=d.x-c.y-x.z z =-b·z+x· y

As a consequence of the sign change of the c coefficient, the values for the other ones have to be modified to achieve chaotic behavior. For instance, a = 15, b = 3, c = 1 and d = 32 is a set of values leading to a chaotic attractor, as seen from the simulation results given in figure 3, where the 3D system trajectory suggests chaos. The resulting receiver is described by:

{ x'=-~ .x +a.y

z' = -b . z+ X· Y In the autonomous case, the receiver state equations are linear and the state transition matrix is: A =( -a

J

0 -b Obviously, the synchronizing receive is globally asymptotically stable, with negative eigenvalues -a and -b. The error dynamics, in the case of parameter match, is given by:

lo

{

C]

'=-a·c]

c3 ' =-b . cJ + c] . Y

Linear Communication Channel Based on Chaos Synchronization ,-

,

91

--,-.

"?]I ~ .•;.•/~) j~~~;o~~20 ~...• • i.

.•.•..

............... .. ...............

o ~<.~___ ~ O y

-40

-20

o

Figure 3 The modified Chen attractor

200

400

600

800

1000

Sample no.

Figure 4 Time evolution of the state synchronization errors

The state errors, EI = X - i , E3 = Z - Z, exponentially converge to zero, with the time constants -1 / a and -1 / b. The demodulator is based on the second state variable: y'=d·i-c·y-i·z Subtracting from second equation of the emitter, the demodulated signal dynamics results in the form: m'=-c·m-( z -d)·E1 -i·E3 +m After the synchronization transient, of time length 'rTr = Max(-1 / a, -J / b), the state errors become negligible, and the time evolution of the recovered signal can be approximated by: m'''''-c 'm+m The corresponding transfer function results in the form: H(s)=M(s) ",,_1_ M(s) s+c

Following the general method, the feed forward equalizer has the transfer function: s+c

HE(s)=

I/wc ·s+l

s+l O.Ol·s+l

Simulation results confirm the design calculations previously presented. For the same parameter values as above, the synchronization errors decay exponentially with the predicted time constants, as presented in figure 4. The efficiency of the proposed equalizer is highlighted in figure 5, where the modulating signal and the demodulated one are compared.

. 1 50~-iOO- ~",,~", ;oo '----';'- ~ooo

Sample

no.

Figure 5.a. The Modulating signal

'o!ll-~-",""'-- -660- ~DO Sample

no.

Figure 5.b. The demodulated signal, using the equalizer

92

V. Grigoras and C. Grigoras

4. Conclusions We proposed an analog wide-band communication channel based on the chaos synchronization and direct modulation principles. Using the system partitioning synchronization and error dynamics methods, we demonstrated, in a general enough setup that a linear dynamic relation exists between the modulating signal and its demodulated counterpart. A feed-forward channel equalization technique ensures that the modulating signals can have large enough bandwidth, leading to the possibility of high speed digital communication. The presented case studies highlight the feasibility of the general method. Further research is needed to approach noise and parameter mismatch problems, specific to analog implementation of synchronizing transmissions. A possible approach can be to extend the use of the communication setup to digital modulation.

References [1] T. L. Carrol, L. M. Pecorra. 'Synchronizing Chaotic Circuits', IEEE Transactions on Circuits and Systems, Volume 38, No.4, April 1991, Page(s): 453-456. [2] Sparrow, C. 'An introduction to the Lorenz equations', IEEE Transactions on Circuits and Systems, Volume 30, Issue 8, Aug 1983 Page(s):533 - 542 [3] T. Veta and G. Chen. "Bifurcation analysis of Chen's equation," International Journal of Bifurcation and Chaos, Vol. 10, Aug. 2000, Page(s): 1917-1931. [4] Gonzales, O.A.; Han, G.; de Gyvez, J.P.; Sanchez-Sinencio, E.; 'Lorenz-based chaotic cryptosystem: a monolithic implementation' IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Volume 47, Issue 8, Aug. 2000Page(s):1243 -1247 [5] Cuomo, K.M. ; Oppenheim, A.V.; Strogatz, S.H.; 'Synchronization of Lorenzbased chaotic circuits with applications to communications' IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing. Volume 40, Issue 10, Oct. 1993 Page(s):626 - 633 [6] Corron, N.J.; Hahs, D.W.; 'A new approach to communications using chaotic signals', IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Volume 44, Issue 5, May 1997 Page(s):373 - 382

93

Maxwell-Bloch Equations as Predator-Prey System A. S. Hacinliyan 1 ,2, O.

1

2

O.

Aybar 1 ,2, i. Ku:;;beyzi 1 ,2 , E. E. Akkaya 1

i.

Temizer 1 , and

Yeditepe University 26 Agustos Yerle:;;kesi Kay:;;dagl , istanbul, Turkey (e-mail: [email protected] ) Gebze Instute of Technology Kocaeli, Turkey

Abstract: The dynamics of laser models based on the Maxwell Bloch equation is studied. Instances of stability and chaotic behavior are investigated . Special solutions of the system one of which reduces to the Lotka Volterra system are derived. Absence of oscillating solutions in the reduced systems is studied. Keywords: Maxwell Bloch, Lotka Volterra, bifurcation.

1

Introduction

The pure Bloch model is linear so that chaotic behavior is not ordinarily expected, although for strongly coupled AC components, trajectories that are dense in the phase space can result in seemingly chaotic like regions. When the Bloch system is coupled to the Maxwell equations in the presence of material media, chaotic behavior as evidenced by positive Lyapunov exponents can be seen. These systems are frequently quoted as laser models. Transition to chaos is based on considering the resonance between the laser cavity frequency and atomic cavity TEM modes. The Maxwell Bloch equations (to be referred to as ME henceforth) as given by and involve the coupling of the fundamental cavity mode, E with the collective variables P and L1, representing the atomic polarization and the population inversion. The equations are:

E = -kE+gP P= Li =

-'l.P + gEL1

- 'II(L1- L1o) - 4gPE

For k = (1, Il. = g2/k = 1, g2L1o / k = r , III = b, the system can be transformed into the Lorenz system about the equilibrium point L1 = L10 by setting x = E, y = gP/k,z = L10 - L1. The meaning of the parameters

94

A. S. Hacinliyan et al. The basic operating point given by the MINUIT Analysis showing "Throw and Catch" chaotic behaviour similiar to LORENZ System. p

7 .5

5

3

-5 - 7.5

k=11.7S, g=6.06, gper=2.66, gpar=2.7S, d10=28

Fig. 1. The Strange Attractor for basic operating point with given parameters

in the original equations are given by Arecchi, while (T, r, b are the Lorenz parameters. A basic operating point given in Figure (1) was found by minimizing the negative of the maximal Lyapunov exponent using the CERN MINUIT package. The dominant chaotic behavior exhibited by the system is the so-called Throw and Catch mechanism , in which two linearized equilibrium points with a pair of complex conjugate eigenvalues with slightly positive real parts surround an unstable third equilibrium point with eigenvalues (+ - - ). The chaotic behavior is predictably similar to the LORENZ System in Figure 1.

Ranges of parameters about the operating point for which a positive maximal Lyapunov exponent has been observed is given in the following Table 1. Parameter Lower Upper Hopf Point k 6.75 11.5 2.86 3 g 0 60 ')'.1 0 60 5 ')'11

Table 1. Parameters about the operating point for which a positive maximal Lyapunov exponent has been observed

Results of a numerical study of the Hopf bifurcation point is also given in the same Table 1.

Maxwell-Bloch Equations as Predator-Prey System

95

This range of parameters is true for Class C lasers (for infrared). Class A lasers require 1'.1.. CC:' I'll » k. Class B lasers require 1'.1.. » I'll CC:' k.

2

Special solutions of the MB equations and its relation to the LV equations

It is possible to seek special solutions to the MB equations using subsidiary conditions that replace either of the two nonlinear terms P E and E.6. by the remaining variable which yields a two by two system and an exponential solution for the third variable. We investigate these solutions and their possible relation to the Lotka Volterra (Henceforth referred to as LV) equations. One of these has been studied. 2.1

Trivial subsidary condition decoupling Ll

By the change of variables PE = .:\.6.

The MB equations can be reduced to the following system

Li =

- I'll (.6. - .6. 0 )

EE = -kE2

P= The change of variables I

=

-

4gP E

+ gP

-l'l..P + gPE 2 f.:\

E2, j 1 .

= 2EE gives

"21 = P=

-k1 + gP

-l'l.. P

+ gP1f.:\

.6. is thus decoupled to give:

This gives the following solution for .6. :

.6. =

I'll

1'11.:\

+ 4g

+ Aexp[-bll + 4g.:\)t]

The remaining system consists of the two equations given by the following two by two matrix system:

96

A. S. Hacinliyan et al.

We seek solutions of the form :

Here

fJ,

are the eigenvalues of the system given by: 2 fJ,

-

('II

+ k) fJ, + k'il

-

92'II 4 .A = 0 'II + 9

To have complex conjugate eigenvalues, that can give oscillatory behavior, the discriminant must be negative.

This is impossible. So the approximate solution for the MB equations has been derived, and shown to reach steady state without oscillation. This special solution is related to the Type II Lasers. 2.2

Trivial subsidiary condition decoupling P

The MB equations can be transformed into a system proposed by, that resembles the Lotka Volterra problem with I representing the intensity of the laser field and N denoting population density, the dimensionless rate equations for I and N are obtained by assuming I = E2 and the subsidiary condition gP = kE11. Multiplying the first MB equation by E gives EE = -kE2 + gP E, P = - '.l.P + gE11 = - '.l. P + g2p/k The subsidiary condition enables one to decouple and solve for the polarization as P = Po exp [- b.l.P - g2 / k )t] and the third equation becomes Li = -'II (11 - 11 0 ) - 4kE211. Finally, the following equations for the intensity an~ population density are obtained by defining 11 = N, 2kt = T, A = 110 and 1= 2I,.

I

=

(-1

+ N)I

N= (A-N -NIh If A = 0 then these equations can be reduced into the LV equations. For negative values of " we get a pair of purely imaginary eigenvalues namely ±v(hl)/2 The normal form of these equations are in resonance and give the limit cycle N I = constant for, = -l. For values of A =I 0, there also is a way in which the equations can be shown to be compatible with the LV system. Augmenting the system by the equation A = 0, we have a 3-dimensional system which can be studied with the method of invariant manifolds. We construct the augmented system

: i: = x( - l+a+y)

Maxwell-Bloch Equations as Predator-Prey System

97

iJ = ,( -a:r - y - xy)

°

Q, =

The linearized eigenvalues are the original eigenvalues and the additional zero eigenvalue for this system, (-1, -,,0). If, > 0, this gives a stable invariant manifold on which the flow is LV. If, < 0, the manifold becomes unstable and we need to study the general case.

3

The general case for arbitrary A I=(-l+N)I IV=(A-N-NI)[

These equations possess two equilibrium points, I = 0, N = A or I = = 1. The first equilibrium point when linearized has eigenvalues A-I, - , so that no resonance or oscillation is expected. At the second equilibrium point:

,(A - 1), N

:...-

I=(-l+A+N)I

IV = (-I - AN - N 1)[ An integer ratio between A - I and , can indicate resonances in the normal form expansion. The latter point when linearized has eigenvalues -A-"y±y!4-4A+A2, . . f 2 so th at w h en 4 - 4A + A2 , ch anges· SIgn, a transItIOn 0 domain indicating a Hopf bifurcation can occur. The normal form expansion about this equilibrium point is not expected to yield a resonance condition.

4

Possibility of oscillatory solutions

It has been shown that oscillatory solutions for the special solutions of the MB equations using subsidary conditions require unphysical parameter values. The possibility for oscillatory solutions can be investigated, assuming dominance of the nonlinear terms, and dropping the exponential decay causing term in the E equations. In effect this is a small k, g dominated regime.

.

iJ.

jj; =

-kE + g2 EiJ.

Li =

'1liJ. o

jj;

-kE + l

=

=

-

4gPE EiJ. .

'1liJ. o

-

2

4(EE + kE )

98

A. S. Hacinliyan et 01.

E=gP

P = gE11 Li = -4gPE

£ = g2E11 Li = 11

E= E 2/2

-2E2

+C

lE(-2E2

+ 11 0)

= g2( E 4/2 + 11oE2 / 2) + cat E = EaE = 0 EE

5

=

-4EE

=

)(g2)E2(110 - E2)

Periodic Analysis of Maxwell Bloch Model

When we separate exponentially decaying part, we have steady state solution from E=gP

P=

gE11

Li = - 4gPE P = g- l £ Li =

-4gEg - 1E

Li +4EE = 0 11 + 2E2 = C1 .

2··

P=gE(Cl-2 E ) = E/g

£

=

lE( Cl - 2E2)

If E = p then p(dp/dE) = l EC1 - 2g2 E3 p2/2 If C2

= 0 then E

If C2

= g2 C1 E 2/2 - g2 E4 /2+ C2 / 2

E=

gE(Cl - E2)1 /2

= 4 Fr exp( Frgt + Frc2)/(1 + 4 exp(2Fr(gt + C2))

= 0 t hen the equation is elliptic and approximately periodic.

Maxwell-Bloch Equations as Predator-Prey System

The equilibrium points are E runaway solution.

= 0 and E = ) cl/2. For E = 0 we have ..

E E For large E

E=

_2g2 E3 for E

99

=

9

2

C1 E

= A exp(gylClt)

=

)cl/2

+~

Near the other equilibrium point after qualitative analysis, we see that the system has periodic solutions with 2g)cl/2

6

Impossibility of trivial resonance k = -b.L

k =

0: ±

+ 'II + 1'1 + 21'2)

- ('.L + 'II) is the requirement for resonance. pi, TIf -0: = 2, then characteristic polynomial is

(0:

+ pi - ,)(0: + pi - ,)( -2, -

Since the sum of the roots is 0, we have -k - '.L - 'II one parameter is unphysical.

7

The eigenvalues are

A)

= 0 which implies that

Conclusion

In conclusion, the Maxwell-Bloch system that is usually used as a model for the Helium-Neon L&<;er has been studied. This system has a rich structure as a function of its parameters. We have shown that this system exhibits several types of instability. The Lyapunov spectrum is compatible with the eigenvalues of the linearized system, which consist of a negative value plus a pair of complex conjugate eigenvalues with slightly negative real part. These parameters correspond to far infrared lasers where Lorenz like chaos has been observed. The bifurcation mechanism that characterizes the possible transition to chaos from this operating point is studied by the MATCONT package and Hopf bifurcation is identified in several instances. Emphasis has been placed on one operating point where chaotic nature as evidenced by a positive maximal Lyapunov exponent is shown while the change of the varying parameters are plotted for each item. However, this behavior is sensitive to variations in the parameters and rapidly changes to steady state. As each parameter is varied, transition to/from chaotic behavior can be seen. We have identified several instances of Hopf bifurcation at this operating point, the operating point is within limits of the parameter range for which chaotic behavior has been studied in the He-N e laser.

100

A. S. Hacinliyan et al.

References • A. Hacinliyan, N. Z. Perdah<;l, G. ~ahin , H. A. Yddmm. Transforming to Chaos by Normal Forms. CMDS 10, Shoresh, Israel, 15-20, 2003. • F. T. Arecchi. Chaos and Generalized Multistability in Quantum Optics. In A.F. Round, editor, Physica Scripta, 85-92, 1985. • H.Haken. Light, Volume 2, 1985. • A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano. Routes to chaotic emission in a cw He- Ne Laser. Physics Letters A, Vol. 28,2, 1983. • P. Grassberger, editor. Non-linear time sequence Analysis. Int. J. Bifurcation and Chaos, pages 521, 1991. • J. P. Eckmann, S. O. Kamphorst, D. Ruelle. Ergodic Theory of Chaos and Strange Attractors. Reviews of Modern Physics , Vol. 57, pages 617-656, 1985. • H. Kantz, editor. A robust method to estimate the maximal Lyapunov exponent of a time series. Physics Letters A, Vol. 185, pages 77-87, 1994. • J. Chrostowski. Noisy Bifurcation in acousto-optic bistability. Physics Letters A, 26, 1982. • Thomas Erneux,Kozyreff G, editor. Nearly Vertical Hopf Bifur-cation for a Passively Q-Switched Microchip Laser. Vol. 101 , 2000. • Erneux, T., Kozyreff, G. Nearly Vertical Hopf Bifurcation for a Passively Q-Switched Microchip Laser Journal of Statistical Physics, Vol. 101, Nos. 1/2, 2000. • Krise, S., Choudhury, S. R. Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations Chaos, Sol'itons and Fractals 16 59-77, 2003. • Clarici, G., Griese, E. Analytical Laser Model for Optical Signal Integrity Analysis International Students and Young Scientists Workshop "Photonics and Microsystems", 2005. • Siegman, A. E. Lasers Univer-sity Science Books ISBN 0-9357022-11-5, 1986. • A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano. Deter-mining Lyapunov exponents from a time ser-ies. Physica D, pages 285- 317, 1985.

101

Identifying Chaotic and Quasiperiodic Time-Series Candidates for Efficient Nonlinear Projective Noise Reduction Nada Jevtic 1 ,2 and Jeffrey S. Schweitzer2 1

2

Physics and Engineering Technology, Bloomsburg University 400 East Second St, Bloomsburg, PA 17815, USA (e-mail: nj evtic©bloomu. edu) Physics Department U-3046, University of Connecticut Storrs, CT 06269 , USA (e-mail: schwei tz©phys. uconn. edu)

Abstract: One of t.he principal problems in identifying chaos using nonlinear timeseries analysis of real-world data is additive noise. A straightforward procedure in traditional phase-space is introduced that can be used to identify data sets amenable to nonlinear projective noise reduction. This methodology can be used both as a tool for identifying candidates for noise reduction and, with some adapt.ation, for zeroth order noise quantification. Results for simulation data are compared with that for a quasiperiodic measured time series to illustrate when series can benefit most from nonlinear projective noise reduction. Keywords: nonlinear time-series analysis, Poincare section, Poincare section return time, local projective nonlinear noise reduction, PG 1351+489.

1

Introd uction

During the nineteen eighties the study of chaotic deterministic systems resulted in the first nonlinear tools for the analysis of time series generated by these systems. It , however, soon became clear that chaotic systems were very hard to find in the real world. Since the theoretical basis for these methods was initially limited to dynamics that can be described by a system of autonomous differential equations (Takens [5]), enthusiasm flagged. Since then, the validity of the embedding theorems has been extended to deterministically driven (Stark [4]) and stochastically driven systems (Stark et al.[I]). Moreover, now the computing power needed to do these analyses is available on our desktops. One of the principal problems in identifying chaos using nonlinear time-series analysis of real-world data is additive noise. Additive noise has also limited our capability to study both nonlinear and linear systems. We introduce a straightforward procedure in traditional reconstructed phase-space that can be used to identify data amenable to nonlinear local projective noise reduction. Simulated solar data are compared with a quasi-periodic measured time series of a variable star light curve in classical reconstructed time-delay phase space to illustrate when series can benefit most from nonlinear projective noise reduction . We also discuss how this methodology might be used for zeroth order noise quantification.

102

2

IV levtic and 1. S. Schweitzer

Time-Delay Reconstructed Phase Space

Of the tools developed for time-series data analysis in time-delay reC011:,;tructed phase space of interest to this discussion are Poincare sections and locally projective nonlinear noise reduction. 2.1

Poincare sections

Flow data such as in the example of variable white dwarf PG 1351 +489 which will be used here are characterized by the fact that errors in the direction tangent to the t.rajectory neither shrink nor increase exponentially. This is a so called marginally stable direction and thus possesses one Lyapunov exponent which is zero. Any perturbation in this direction can be compensated by a simple shift of the time. In many data. analysis ta.sks this direction is of marginal interest, a.nd it can be eliminated. The theoretical concept to do so is called the Poincare section Thus, a Poincare section consists of points in phase space generated by the sequence of intersections of a trajectory with a generalized surface. In three dimensions, the generalized surface is a plane. Points A and B in Fig. 1 at which the trajectory intersects this plane in the same direction constitute the Poincare Section. This is the 'space' aspect of 11 Poincare section. It is importa.nt to note here time is an irnplicit variable.

Fig. 1. A phase-space trajectory intersects the Poincare plane at points A and B.

In order to recoup time information, time intervals beb,veen successive crossings through the Poincare surface (between points A and B) may be used. These times meaningful observables, too. They will be referred to as "Poincare section return times" or PSRT. The inverse of a Poincare section return time (PSHT) will be introduced as the Poincare section return frequency or PSRF. This is the 'time' aspect of a Poincare section. 2.2

Locally projective nonlinear

nois~~

reduction

It is assumed that the measured data are the output of a low-dimensional dynamical sy:,;tem and random high-dimensional noise spreads the data off

Identifying Chaotic and Quasiperiodic Time-Series Candidates

103

Fig. 2 . In locally projective nonlinear noise reduction, if the pha.5e-space trajectory lies on a surface, points that have been forced off the surface by noise are orthogonally projected back onto the surface.

the manifold that the phase-space trajectory lies on. FbI' small amplitude noise, the data points are distributed closely around the manifold (Fig. 2). The goal in local projective nonlinear noise reduction is to identify the manifold and orthogonally project the data points onto it. In this procedure we assume that if the dynamical system with noise forms a q-dimensional manifold containing the tmjectory, and the delay vectors lie inside another lower m-dilnensional manifold with q > m. As a result overembedding is used in an iterative process t.o obtain the highest degree of noise reduction.

3

Analysis of Measured Data

0.15

~,1(l

.

+-c---r-.,......,.~.,.~.--~ .....,.......-,~ !J

5001000

1500

2000

2500

;>Q!lC

3WQ

400U

.tSQO

soon

t(sj

Fig. 3. PG 1351+489 relative light intensity curve of 10 s binned data.

Fig. 4. Power spectrum o[PG 1351 +489 of 30 s integratpd data.

The measured data is the light curve of DB variable white dwarf PG 1351 +489 that was observed by the vVhole Earth Telescope. A section of the light. intensity curve is shown in Fig. 3. The light curve data are available a.t two integration t.imes: 10 sand ::\0 s. The lOs binned light curve is over 40 periods long. The PG 1351 +489 power spectrum consists of the fundament.al frequency fo = 2.041 mHz, (period of 489 s) , its harmonics, subharmonies and other nonlinearly generated lines (Fig. 4)

]04

3.1

N. Jevtic and J. S. Sch1,>'eitzer

Phase-space analysis of PG 1351+489 light curve

The :3-D time-delay phase space portrait of the 10 H integrated data is shown in Fig. 5.

Fig. 5. ;j-D time-delay phase space port.rait of the PG 1:551+4S9 lOs integrated data.

As part of a comprehensive nonlinear analysis of PG 1351+489 data Poincare sections were obtained at the average of the data for two direct.ions. The Poinca.re section 'space' aspects in the two directions are shown in Fig. 6. The effect of noise is evident in the form of 'upward' point.s appearing in the 'downward' bundle of trajectories and vice versa. c· ...

IJpw.ind d ('<W fil,!'l,l "ff d

-1:!lD·

, HUl

«I)

Fig. 6 . 'Space' Poincare section of the PC 1:351+489 lOs binned data at average of data.

The Poincare Section Return Times (PSRT) are shown in Fig. 7 (left) 311dFig. 7 (right) for the 30 sand 10 s binned data, respectively. nIH period points at 48D s and some noise at short P SRT are present in Fig. 7 (left). In Fig. 7 (right) however, in a.ddition to the full period and, at short PSRT, noise, there is an excess of points at about half the fundamental time. In theF'ourier power spectrum, the fundamental and the first harmonic are in a, 10:1 ratio. Here the ratio is about 1:1.

ldenti/ving Chaotic and Quasiperiodic Time-Series Candidates

J05

b.

=

~-'-'-'T-"'--""'---''''''''''''-I II :;rl .0(;' CI ;n

Cifl)"urt/>t$rlb<:t

4

Fig. 7. Poincare Section Return Times (PSHT)of the PG 1:351+489 at average of the data: left - :~o s binned data and right - 10 s binned data.

Analysis of Model Data: Flux Transport Solar Dynamo with Meridional Circulation

At the time ·work was progressing on the analysis of model data f()l' the magnetic field energy of a flux transport solar dynamo with meridional circulation generated by Iv1. Dikpati at the HAO, NCAR where stochastiC'fiuctuatiolls were added to the meridional velocity.

Fig. 8. Flux transport dynamo model with stochastic fluctuations added to the meridional circulation velocity: Phase space portrait (left) and PSRT (right).

The phase-space portrait for the model data for the magnetic fieJd energy of a. flux-transport solar dynamo with stochastic fluctuations added to the meridional circulation velocity is shown in Fig. 8 (left). The Poincare section return times are shown in Fig. 8 (right). These exhibit the sa.me kind of increase in the number of PSRT at the times corresponding to about the first harmonic as the data for PC 13.51+489 with the advantage that it is clear in Fig. 8 (right) that the increase of the first ha.rmonic times is at the expense of the fundamental period points. The sum of t.wo adjacent ha.lf-times were shown to be equal to the fundamental PSRT.

5

Comparison of Model and Measured Data

Insights gained for the dynamo model w(;)re then applied to the PG 1351 +489 data. There, as well, it was shown that adjacent fractional PSRT times or the 'first harmonic times' add to the fundamental period. Fig. 8.h. In an attempt to better understand the observations, a series of Poincare pla.nes were positioned along the phase-space portra.it in one specific direction

] 06

N. levtic and 1. S. Schweitzer

a) §

b) y"

' i~-_-·1· "·.,.., :,f&.··· &

•

••

. . .. . .

ll

~ ~j

"

i

i~j :'" i ~ ·1 ,; :;

,,

_if

" -,--.-···r--·····. --.. .--,-·-·r -···-'100 O,2()401I080

Intersection number

Intersection number

Fig. 9. Comparison of the PSRT for the solar dynamo with resetting (left) and PG 1351+489 (10 s binned data)(right).

and histograms were (Fi?;. 10) obtained for the PSRT at each position.

Fig. 10. Histograms of the PSRT of PG 1351+489 PSRT as the Poincare plane sweeps the portrait.

At the average of the data the number of full period PSRTs was smaller (Fig. 10) than the data requires and there were too many almost equaltime PSRTs of 200-300 s. As the plane swept the representation towards the apex, two return-time groups differentiate whose PSRT times add to the fundamental period. These diverge, one group of PSHT times becomes shorter and the other becomes longer. They revert to two groups of times, one corresponding to the fundamental period and the other to noise at the very apex of the figure .

6

The Effect of "Resetting" on the PSRT

The observed PSHT diagrams and histograms are best explained in the following manner. If a phase-space point in Fig. 11 (left) at C spikes off the trajectory due to noise and then falls back through the plane towards D , it returns to the trajectory at A. Two Poincare return times result or time resetting occurs and as a result the process has been dubbed resetting. For positions of the Poincare plane near the average of the data, these return times are about half the fundamental period.

Identifi;ing Chaotic and Quasiperiodic Time-Series Candidates

107

Fig. 11. The Effect, of "R.esetting" on the PSHT. Poincare plane at. the average of the data (left.) and Poincare plane towards the apex (right).

As the Poincare plane is taken toward the apex of the phase space portrait (Fig. 10 (right)), the difference in the two times grows: one time grows longer (B to C) while the other grows shorter (A to B). The single branch centered around the first harmonic at the average of the data, as the plane is moved, spreads out to become two branches. These diverge towards lower and higher frequencies.

7

The Three-Dimensional PSR-Frequel1cy Spectrum

The previous discussion justifies the introduction of the Three-Dimensional PSR-F'reqllency Spectrum shown in Fig. 12. This spectrum can be obtained using rela.tively simple softwa.re and integrates the above procedure. A 3-D spect.rulll obtained in the following manner:

Fig. 12. ;3-D Poincare Section Return Frequency Spw;trum.

a. A series of Poincare planes is taken along a selected direction and PSRT obtained on a fine grid of positions. b , In order to enhance the visibility of the noise-induced branches, a. number of these histograms are added (binned). c. R.eturn times are converted into 'frequencies' at each position.

108

N. Jevtic and J. S. Schweitzer

d. Histograms of these PSRF are obtained at eaeh position. c. The histograms are assembled according to position along the seleeted direction.

8

An Example of Efficient Nonlinear Local Projective Noise Reduction

0.002

0.004

0.008

0.000

0.0"10

0,012

(1.014

0.016

O.O'j8

Frequency (mHz)

F ig. 13. Comparison of pmvor spectra of PC 1351+489 data prior to (gray) and after loeal projective nonlinear noise reduction (black). (Inset: Phase space portrait after nOiSf) n,duction.)

The importance of reliable identification of candidates for local nonlinear noise reduction can best be seen on the example of the power spectrum of the light; curve of DB variable white dwarf PG 1i351 +489 in Fig. 1:5 (Jevtic et a1.[2]). In the case of PG 13C)1 +489 noise reduction was most effective when working in a 7-dimensional phase space and 6 iterations were used. The 'white-noise tail was lowered by a factor of 50, in efIt~ct shortening the time needed to collect data by a factor of >7.

9 9.1

Discussion Poincare plane orientation

Theoretically a Poincare section is taken in the direction perpendicular to the trajectory. However, since "resetting" has been defined for stochastic fluctuations that are perpendicular to the trajectory, no PSHT resetting would be observed for these Poincare plane orientations. For the identification of candidates for efficient noise reduction the orientation has to be chosen in a.n

Identifying Chaotic and Quasiperiodic Time-Series Candidates

109

arbitrary direction in respect to the trajectory. This 'fuzziness' can, however, work for us. If in addition to the position of the Poincare surface, its orientation is varied through 27r, then for resetting which is normal to the trajectory, the 'theoretical' direction would yield a minimum in the the number of fractional PSRT. As a result, this orientation can be used to select a reference level that has minimal resetting, with additive noise being responsible for the observed noise. This has the potential to allow us to quantify the relative contributions of resetting and additive noise. Further work will address the question of the optimal orientation of Poincare surfaces for the purpose of noise reduction and quantification in other types of systems.

9.2

Noise quantification

Many questions remain to be answered before a noise quantification measure can be defined. The first that come to mind are: If a 'clean' traj ectory can be defined for comparison, a measure of resetting are both the amplitude of the displacement off the trajectory and the length of trajectory missing before reset. Would a product or a ratio of the two be a 'good' measure of the effect of resetting stochastic fluctuations? This in turn raises the question of the definition of a 'clean' trajectory. In an iterative process what is the standard number of iterations? And finally, whether we decide that the product or the ratio of the excursions off the trajectory and the length of the trajectory involved in the excursion are a 'good' measure, should an average of this measure or sum along the trajectory be used? Or should it be normalized on the whole trajectory?

10

Closing Comments

At this time, many issues remain open as can be seen in the previous section. In addition to the procedural issues addressed above also of importance is the more general question of the relationship of the traditional Fourier power spectrum and the 3-D PSRF spectrum proposed here. In the Fourier spectrum of PC 1351+489 the ratio between the power in the fundament al and the first harmonic is 10:1. In the 3-D PSRF spectrum at the average of the data this ratio is 1:1. Could this also be used as a measure of noise? Moreover , digital data acquisition, with the possibility of variable binning as in the example of PC 1351+489, can yield an estimate of the time scale of the turbulence. The answer to this question can only be given after the effect of the delay is identified and provision is made to eliminate this dependence. And last but not least, could this lead to an estimate of time-local parameters of the medium such as thermal response TC timescales (Montgomery [3])?

11

Acknowledgements Dr. Mausumi Dikpati of HAO, NCAR for the dynamo model data.

110

N. Jevtic and J. S. Schweitzer

The WET for the variable white dwarf data: the xCov12 campaign team, particularly Antonio Kanaan and Steve Kawaler. The University of Connecticut Research Foundation and the Bloomsburg University Faculty Travel Fund The Max Planck Institute for Complex Systems for the TISEAN software: R. Hegger, H. Kantz, and T. Schreiber, Practical implementation of nonlinear time series methods: The TISEAN package , CHAOS, 9,413 (1999)

References l.J. Stark et al. Embeddings for forced systems ii: Stochastic forcing. JNonlinSci , 13:519, 2003. 2.N. Jevtic et al. Nonlinear time series analysis of pg1351 +489 light intensity curves. ApJ, 635:527- 539, 2005. 3.M. Montgomery. A renorma lization group with periodic behavior. CoAst, 154:38, 2008. 4.J. Stark. Embeddings for forced systems i: Deterministic forcing. JNonlinSci, 9:255- 332, 1999. 5.F. Takens. Detecting strange attractors in turbulence. In D. A. Rand and L.S. Young, editors, Dynamical Systems and Turbulence, page 366381, London, 1981. Springer-Verlag.

III

Chaos from Observer's Mathematics Point of View Dmitriy Khots 1 and Boris Khots 2 1

2

3710 S. 202nd Avenue Omaha, Nebraska, USA (e-mail: [email protected] ) Compressor Controls Corp 4725 121st Street Des Moines, Iowa, USA (e-mail: [email protected])

Abstract: This work considers Chaos aspects in a setting of arithmetic provided by Observer's Mathematics (see www.mathrelativity.com). We prove that the physical speed is a random variable, cannot exceed some constant, and this constant does not depend on an inertial coordinate system. Certain results and communications pertaining to these theorems are provided. Keywords: observer, chaos, arithmetic, derivative, Lorentz.

1

Introduction

The following discussion is based on the work introduced in Khots et al.[l]. Further information can also be found in Khots et al. [2] and Khots et al. [3]. We consider a finite well-ordered system of observers, where each observer sees the real numbers as the set of all infinite decimal fractions. The observers are ordered by their level of "depth", i.e. each observer has a depth number (hence, we have the regular integer ordering), such that an observer with depth k sees that an observer with depth n < k sees and deals (to be defined below) not with an infinite set of infinite decimal fractions, but, actually, with a finite set of finite decimal fractions. We call this set W n , i.e. it is the set of all decimal fractions, such that there are at most than n digits in the integer part and n digits in the decimal part of the fraction. Visually, an element in Wn looks like _ ... _ . _ ... _. Moreover, an observer with a given '-v-" '-v-" n

n

depth is unaware (or can only assume the existence) of observers with larger depth values and for his purposes, he deals with "infinity". These observers are called naive, with the observer with the lowest depth number - the most naive. However, if there is an observer with a higher depth number, he sees that a given observer actually deals with a finite set of finite decimal fractions, and so on. Therefore, if we fix an observer, then this observer sees the sets W n1 , . . . , W nk with nl < .. , < nk indicating the depth level, and realizes that the corresponding observers see and deal with infinity. When we talk

112

D. Khots and B. Khots

about observers, we shall always have some fixed observer (called 'us') who oversees all others and realizes that they are naive. The "Wn-observer" is the abbreviation for somebody who deals with Wn while thinking that he deals with infinity. The following sections describe application of the idea of relativity in mathematics to various mathematical fields.

2

Arithmetic

We begin by defining sets Wn which consist of all finite decimal fractions such that there are at most n digits in the integer part and at most n digits in the decimal part. That is, the set Wn contains all elements of the form a = aO.al ... a n where the integer part can be written as ao = bn-1 ... bo, where bn - l , ... , bo, aI, .... , an E {O, 1, ... , 9}. If n < m, then Wn naturally embeds into Wm by placing O's in the n + pt through mth decimal places. We call the embedding ipn,m : Wn --> W m . Here are some examples: let 2.34 E W 2 and then ip2,4 (2.34) = 2.3400 E W 4 . Similarly, Wm projects onto Wn by cutting off the superfluous digits on the right of the decimal point. Let cPm.n : Wm --> Wn be the projection, then, for example, if 45.4301 E W 4 , then cP4,2 (45.4301) = 45.43 E W 2. If the integer part of a fraction contains more than n digits, then cPm,n is not defined. Now, given C = CO,Cl"'C n , d = do.dl ... d n E Wn we endow Wn with the following arithmetic (+n, -n, X n , -;-n):

Definition 1. Addition and subtraction C

±n d = { C ± d, if C ± dE Wn not defined, if C ± d ¢:. Wn N

and we write (( ...

(Cl

2.= nCi

+n C2) ... ) +n CN) =

for

Cl, ... ,

CN iff the contents

i=l

of any parenthesis are in W n

.

Definition 2. Multiplication n

cXnd=

2.=

n-k n

k=O

where c, d > 0, -

Co'

2.=

nO.O ... Ock·O.O ... Odm

m=O

'-v-"

'-v-"

k-l

m-l

do E W n , 0.0 ... 0 Ck' 0.0 ... 0 d m is the standard product, and '-v-" '-v-" k-l

k = m = 0 means that

O.~Ck k-l

=

m-l Co

and

O.~dm

= do. If either C < 0 or

m-l

d < 0, then we compute iel Xn Idl and define C Xn d = ± Ici Xn Idl, where the sign ± is defined as usual. Note, if the content of at least one parentheses (in previous formula) is not in W n , then C Xn d is not defined.

Chaos from the Observer's Mathematics Point of View

113

Definition 3. Division

-'-- d_{"if3!IEWn Ixnd=c not defined , if no such 1 exists or it is not unique

C • n

Let n = 2, so we are in W 2 . Here are some examples of elements of W 2 : 3.14, -99,0.1 E W 2 and 0.115, 123.9, - 100000 1- W 2 . Now, the examples of arithmetic: 2.08 +2 11.9 = 13.98; (-2.08) +2 11.9 = 9.82; 80 +2 24 = not defined; 21.36- 20.87 = 20.49; 1.36- 2 16.95 = -15.59; 1.36 -2( -99 .95) = not defined; 11 X2 8 = 88; (-5) X2 19 = -95; 11 X2 12 = not defined; 3.41 X2 2.64 = 8.98; 3.41 X2 (-2.64) = -8.98; 3.41 X2 42.64 = not defined; 99.41 X2 1.64 = not defined; 0.85 X2 0.02 = 0; 80724 = 20 ; 1 7n 0.5 = not defined (since we get 10 different I'S); 1 7n 3 = not defined (since no 1 exists) .

3

Derivatives

From the point of view of Wn-observer (we will call such observers " naive", since they "think" that they " live" in Wand deal with W) a real function Y of a real variable x, y = y(x), is called differentiable at x = Xo (see Khots et al. [4]) if there is a derivative '() y Xo

=

l' 1m

x->xo.x#xo

y(x) - y(xo) x - Xo

What does the above statement mean from point of view of lVm-observer with m > n? It means that I(y(x) - n y(xo)) - n (y'(xo) Xn (x - n xo))1 ::; 0.0 ... 01 whenever Iy(x) -n y(xo)1 = 0.0 . .. OYI YI +1 ... Yn and I(x _n xo)1 = ~ I

'-v--' n

0. 0 ... OXk Xk+1 ... Xn for 1 ::; k, I ::; n, and Xk - non-zero digit . '-.;---' k

We now state the main theorems.

Theorem 1. From the point of view of a W m -observer a derivative calculated by a Wn -observer (m > n) is not defined uniquely. Proof. Put y'(xo) = ±ao·a1 ... a p a p +l . ,. an with aO ·al ... a p a p +l ... an ;:::: Then 0.0 ... OYI Yl+l .. . Yn =

o and p ::; n.

~

I

aO.a1 ... a p ap +1 ... an

Xn

0.0 ... OXk Xk+1 ... Xn = '-.;---'

k

aO.a1 ... a p bp +1 .. , bn

Xn

0.0 ... OXk Xk+1 ... Xn '-.;---'

k

for any digits bp + 1 , ... ,bn and p = n - k. Hence

and

y'(xo) E V

= {±ao.a1 ., . ap a p +1 ... an la p +1,"" an E {O, 1, ... , 9}}

IVI = 10 k .

QED.

114

D. Khots and B. Khots

Theorem 2. From the point of view of a Wm -observer with m > n, ly'(xo)1 ::; C~k, where C~k E W n is a constant defined only by n, I, k and not dependent on y(x).

Proof. We have ± O. 0 . .. OYI Yl+1 . .. Yn = (±aO·a1 ... an) Xn (±O.O ... OXk Xk+1 . .. xn) '--v--'

'--v--"'

I

k

with Xk - non-zero digit and aO.a1 ... ap a p +1 .. . an 2' O. Now , if I > k then ao = 0; if 1 = k then ao::; 9 and if 1 < k then ao < 9 x 10k-1. Hence

Cl,k n =

I, if I > k { 10 , if I = k 9 X 10k- 1, if I < k

QED. Theorem 3. From the point of view of a W m -observer, when a Wn -observer (with m > n 2' 3) calculates the second derivative:

Y"(Xo) =

Y(X3)-Y(XJ) (x3-xd

lim

Y(X2)-Y(XO) X2 - XO

Xl -+XO , X l¥=- XO , X2---+XO , X2¥:XO,X3---+X l ,X:3# X l

we get the following inequality:

n

provided that y" (xo)

i- O.

Proof. For the Wm-observer existence of y"(xo) means that 1((y(X3) -n y(xd) Xn (X2 -n xo) -n ((Y(X2) -n y(:ro)) Xn (X2 -n xo))) - n y"(xo) Xn ((I X2 -n xol Xn IX3 -n :];11) Xn IX1 -n xol)1 ::; O.~, whenever I(X2 -n n

n

n

~

~

'-v-"

'-v-"

k

I

xo)1 ::; 0.0 ... Op * ... * and I(X3 -n xdl ::; 0.0 . . . Oq * ... * and I( X1 -n xo)1 ::; n

~

0.0 .. . Or * ... * where p, q, r are non-zero digits, asterisks are any digits and '-v-" s

3 ::; k + I + s ::; n. Then given y"(xo) Xli) Xn IX1-nxol2' O.~. QED. n

i-

0 we have (IX2 - n xol Xn IX3

- n

Chaos from the Observer's Mathematics Point of View

4

lIS

Chaos theory interpretation

The following hypotheses illustrate possible Chaos theory interpretation of previous theorems. • Hypotheses 1. Theorem 1 could offer an explanation of why physical speed (or acceleration) is not uniquely defined and, from the point of view of a measurement system (observer), it is possible to consider speed (or acceleration) as a random variable with distribution dependend on the measurement system. Let v be the speed with v = Vo.1il ... Vn - k + ~;;-,:k where ~;;-,:k E {a. 0 . . . Vn-k+1 ... Vn } - random va riable, m > n, and the

°

'-v--"

n-k

distribution function is F::.,k(x) = p(~;;-,:k < x). • Hypotheses 2. Theorem 2 could offer an explanation of why the speed of any physical body cannot exceed some constant, (the speed of light, for example). Independence of this constant on explicit expression of spacetime function could offer an explanation of why the speed of light does not depend on an inertial coordinate system. • Hypot heses 3. Theorem 3 could offer an explantion of the various uncertainty principles, when a product of a finite number of physical variables has to be not less than a certain constant. This can be seen not just from consideration of second derivatives, but of any derivative. • Hypotheses 4. Theorems 1, 2, and 3 combined may provide an insight into the connection between classical and quantum mechanics.

5

Newton equation with chaos theory interpretation

Let F(x , t)

= m Xn X. Then we have the following

Theorem 4. If the body with mass m = mO.m1 ... mkmk+1 ... m n , with mE W n , moves with acceleration X, Ixl = XO.X1 ... XI XI+1 .. , Xn, with X E W n , and mo = TTL1 = ... = mk = 0, mk+1 01 0, k < n, Xo = Xl = ... = Xl = 0, l < n, k + l + 2 E W n, n < k + I + 2 ::::: q, then F (x, t) = 0.

Proof. In t his case m Xn

Ixi

=

Corollary 1. If 1= n - 1 and k Theorem 5. If 1= n - 1 and xn

Proof. IF(x , t)1 = m Xn

0 . QED.

= 0, i.e., m < 1, then F(x, t).

01

°

then IF(x , t)1 < 9.

Ixl <- '-v--" 9 ... 9.9 . .. 9 xn O. '-v--" O... 09 = 8.9 .. . 91 < 9. '-v--" '-v--" n

n-1

n- 1

n

QED.

Theorem 6. Ifmo ~~, p

°< p ::::: n, xo

~~,

°< r::::: n, n < p+r:::::

r

q, then there is no force F(x, t), such that F(x , t)

=

m

Xn

X.

Proof. m Xn x> mo Xn Xo does not exist in Wn . QED.

116

6

D. Khots and B. Khots

Schrodinger equation with chaos theory interpretation

Consider the following: ~Ch Xn h) Xn wxx +n ((2 Xn m) Xn V) Xn W = i((2 xn m) xn h)Wt , where W = w(x, t), h is the Planck's Constant, h = 1.054571628(53) X 10- 34 m 2 kgjs. Then we have the following Theorem 7. Let 36 < n < 68, m = mO.m1 ... mkmk+1 ... m n , with m E W n , mo = m1 = ... = mk = 0, mk+1 # 0, k + 35 < n, V = 0, then ,T, _ ,T,O ,T,l ,T,I,T,I+l ... '¥t ,hn an d '¥t ,T,O -- ... '¥t ,T,l -- ,'¥t ,T,I+l , ... , ,T,n f d '¥t - '¥t ·'¥t ... '¥t'¥t '¥t are ree an

°

in {O, 1, ... , 9}, where l = n

~

k

~

36, i.e., Wt is a random variable, with

n

Wt

~

E {(O.~*

... *)}, where * E {O, 1, ... , 9}.

Proof. We have ~(hxnh) = 0, (2x n m)x n V = 0, i.e., i((2x n m)x n h)wt and l = n ~ k ~ 36. QED.

°

=

Corollary 2. Let 36 < n < 68, m = mO.m1 ... mkmk+l ... m n , with m E W n , rno = rn1 = ... = mk = 0, mk+1 # 0. Also, let V = VO'V1 ... VsVs+1 ... Vn ,

with V E W n , Vo =

1JI = ... =

wp

Vs = 0, V s+1

Wt = .wl . . . wiw;+l .. . w;: and in {O, 1, ... , 9}, where l = n ~ k

#

wp = ~

+ 35 < n , then +s+ 2 >n

0, with {kk

.. . wi = 0, W;+l, . .. ,w;: are free and 36, i.e., Wt is a random variable, with

n ~

Wt E {(O.~* ... *)}, where * E {O, 1, ... , 9}. I

7

Lorentz transform with chaos theory interpretation

Let K and K' be two inertial coordinate systems with x~axis and x' ~axis permanently coinciding. We consider only events which are localized on the x(x')~axes. Any such event is represented with respect to the coordinate system K by the abscissa x and the time t, and with respect to the system k' by the abscissa x' and the time t' when x and t are given. A light signal, which is proceeding along the positive x~axis is transmitted according to the equation x = C Xn t or x ~n C Xn t = 0. Since the same light-signal has to be transmitted relative to k' with the velocity c, the propagation relative to the system k' will be represented by the analogous formula x' ~n C Xn t' = Those space-time points (events) which satisfy the first equation must also satisfy the second equation. Obviously will be the case when the relation Al xn (x' ~n C xn t') = ILl Xn (.1: ~n C Xn t) is fulfilled in general, where A1,IL1 E Wn , IA11, IIL11 ?: 1 are constants; for, according to the last equation disappearance of (x ~n C Xn t) involves the disappearance of (x' ~n C Xn t').

°

Chaos from the Observer's Mathematics Point of View

117

Note, that classical equation x' - ct' = ).,(x - ct) is not valid since if)., < 1, x - ct = 0.0 ... 01, then)., Xn (x -n C x t) = x' -n C Xn t' = O. If we apply '-v--' n-l

quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain the condition ).,2 Xn

(x'

+n

c

Xn

t')

=

IL2

Xn

(x +n c

Xn

t)

with )"2,IL2 E W n , 1).,21, IIL21 ~ 1. For the origin of K' we have permanently x' = 0 and x = v Xn t, where v is the velocity with which the origin of K' is moving relative to K. Then we have { ).,l

Xn

).,2 Xn

(v (v

Xn Xn

t t

- n C Xn t') = +n C Xn t') =

ILl Xn (-C Xn t') (Ad IL2 Xn (C Xn t') (A 2 )

These two equations (Ad and (A 2 ) have to be identical, i.e., for each of equation, the sets (t', t) have to be the same, or there is a dependence A,

t=f(t') Furthermore, the principle of relativity teaches us that, as judged from K, the length of a unit measuring-rod which is at rest with reference to k' must be exactly the same as the length, as judged from K', of a unit measuringrod which is at rest relative to K. Let x' = 1 and t = O. Then we have the following system (Ed: {

).,l

Xn

).,2 Xn

Also, let x

x = ILl x = IL2

Xn Xn

(1 (1

-n C Xn

t')

+n C Xn

t')

= 1 and t' = O. Then we have the following system (E2):

And now we have the equation (E): x = x'

where x is the solution of (Ed and x' is the solution of (E2)' Now we have a system

{

(A)

(E)

Solution of this system is the set S = {().,l, ILl, ).,2, IL2)}, where ).,1, ILl, ).,2, IL2 E W n , 1).,11, lILli, 1).,21, IIL21 ~ 1. We call the system of equations {

).,l

Xn

).,2 Xn

(Xl (x'

tl) = ILl Xn t') = IL2

-n C Xn

Xn

+n C

Xn

(x -n (x +n

C Xn C Xn

t) t)

with ().,l, ILl, ).,2, IL2) E S the Lorentz Transform in Observer's Mathematics.

118

D. Khots and B. Khots

8

Geodesic equation with chaos theory interpretation

Consider the following: ··i+ n '\""' L...j

X

n,\"", n r i L...k jk Xn

(.j X

·k) = 0

Xn X

with j, kEG. Then we have the following .P .P •P . P .P ·th P E G· P .P Theorem. 8 If x· P -- XO.X1 ... Xl X I +1 ... Xn' WZ ,XO = Xl = ... = = 0, 0::;; I ::;; n, n < 21 ::;; q, then we have Xi = 0, i.e., the geodesic curve is a line.

xf

P roo. 1

·j ·k X Xn X

(0.0 ... OxI+1

9

... x~

Xn

-

.J.J .j.j .j ·k·k ·k·k ·k X O·X 1 ··· Xl X I +1 ... Xn Xn Xo .X1 ... Xl X I + 1 ... Xn

0.0 ... 0#+1 ... x~) = O. QED.

Acknowledgments

The authors thank Professor Christos H. Skiadas for his invitation to participate at the Chaotic Modeling and Simulation International Conference (CHAOS2009).

References l.B.Khots and D.Khots. Mathematics of relativity webbook, urI: www.mathrelativity.com. 2004. 2.B.Khots and D.Khots. Extraction of dynamical equations from chaotic data. Lecture Notes in Theoretical and Mathematical Physics, 7:269-306, 2006. 3.B.Khots and D.Khots. Observer's mathematics - mathematics of relativity. Applied Mathematics and Computations, 187:228-238, 2007. 4.D.Khots and B.Khots. Quantum theory and observer's mathematics. American Institute of Physics (AlP), 965:261-264, 2007.

119

New Models of Nonlinear Oscillation Generators Alexander A. Kolesnikov Technological Institute of Southern Federal University, Synergetics and Control Processes Department 44, Nekrasovky str., Taganrog, 347928, Russia (e-mail: [email protected])

Introduction One of the fundamental problems of the modern nonlinear dynamics is a complex problem of oscillations control in various nature systems. Numerous foreign and Russian scientific articles and studies are published concerning regular and chaotic oscillations control. Also wide international conferences and symposia are conducted concerning the same problem which is one of causes of nonlinear dynamics and synergetics (a science of self-organisation in complex systems) evolution. That important control problem has not properly resolved yet. In our opinion it requires to be further developed. Hereof the synergetic approach for auto-oscillation system synthesis is considered in this report. The approach is based on the known Analytic Design of Aggregated Regulators (ADAR). According to ADAR, desired invariant manifolds (energetic integrals of motion) are added to state space of synthesized system [1-3]. Such approach provides analytical calculation of feedbacks forming required behavior of the nonlinear oscillating processes. Three basic models of oscillators (self-oscillating 2-order systems) studied all over the world for many years are the following: Van-der-Paul Equation

x(t)-,ul{1-alx 2 )x(t)+w 2 x=O

(1)

Rayleigh Equation (2)

Poincare Equation

(t) = -WX2 + XI (A 2 X2 (t) = +WXl + X2 (A 2 XI

X? XI2 -

xi), xi).

(3)

Dozens of studies and scientific papers are indented to analysis and application of these Equations showing behavior of wide group of engineering systems and devices. Now only estimated periodic

120

A. A. Kolesnikov

solutions of Equations (1) and (2) are known. Amplitude and frequency can be estimated depending on values of parameters JLi and a i from these solutions. Here it's obviously that the lesser JLi ' the closer oscillations to harmonic and vice versa - the more JLi' the more rapid relaxation oscillations. In regard to Poincare Equation (3), one has the exact analytical solutions [23]. In fact, if polar coordinates XI = rcos¢ and x 2

= -rsin¢

are input to (3) then:

r(t)=r(A 2 -r 2 ) and i>(t)=w.

(4)

Equation (4) solutions are the following: 2

r(t)= A

ro . ¢(t) ={J)t +¢o . 2 + (A 2) _2A 2( ' ro -ro e

(5)

(4) and (5) imply that stable harmonic oscillations with amplitude A and frequency (J) are formed in system (3) in time: Xis (t) = Acos(wt + ¢o); X2 s (t) = -Asin({J)t + ¢o). (6)

Note the following one feature of equations (3) rarely mentioned in scientific literature. For that take 2 2 2 (7) If/ = A - XI - X2 , as some energetic macro variable, where A is analog of oscillations energy. If allow If/e = 0 (7) then from system (3) we get the following equations:

or

(7) implies that motion on manifold If/e

=0

is also shown

10

the

following folded differential I-order equation:

XJ/f/(t)=~2E-W2XJ2/f/ ' Where E

A 2 {J)2

=- -

IS energy of steady oscillations. It means that 2 Ife =0 system (3) behavior on manifold is shown in conservative

New Models of Nonlinear Oscillations Generators

121

model (8) which has harmonic solution of type (6). Meanwhile prior to entry on 'fie = 0 the system (3) is dissipative one. It's easy to show with its divergence:

. .()t =aX-I+aX - = 2 (2 A -Xl2-X 22) -2XI2-2X22<0. 2

dlVX

aX l

In fact on manifold

aX 2

'fie

=0

(7) input energy and dissipation

energy is balanced. So all trajectories of system (3) are attracted to energetic invariant manifold 'fie = 0 (7) (harmonic attractor, i.e. stable boundary cycle on its phase plane). This fact is important for development of new methods of oscillation generators synthesis on base of synergetic approach [1-3]. Unfortunately such analytical expressions of Van-der-Paul (1) and Rayleigh (2) Equations boundary cycles are unknown yet. Now other forms of self-oscillating systems significantly of lesser popUlarity than Van-der-Paul, Rayleigh, and Poincare Equations are described in literature. It must be emphasized that there are no regular methods of analytical synthesis of non-linear oscillation generators in literature. Below we describe analytical efficiency of ADAR [1-3] method for non-linear problem of synthesis of various self-oscillating systems with specified amplitude and frequency of harmonic oscillations.

1. Unified Van-der-PauVRayleighIDuffing Generator Consider a system:

(9)

xJ

It's required to synthesize a feedback law u{xI' providing appropriate harmonic attractors (stable boundary cycles with desired amplitude A and frequency OJ of self-oscillations) in phase space of closed system. According to ADAR [27-29] introduce basic energetic macro variable: (10)

and invariant expression (11)

where function I/fl (t) can be interpreted as power of synthesized system. According to (11) and equations (9) and (10) form feedback law

122

A. A_ Kolesnikov

U]

aF x 2 =--+-'1/]ax]

(12)

~

Obviously that equation (11) is asymptotically stable in large relative to condition '1/] = 0 (10). It's shown with Lyapunov function V = 0,5'1/]2 . Time derivative of that function (13) at T] > 0 is always negative. It's suggesting that (11) is stable. Subject to (10) and (12) closed system equation (9) is:

.. ()t - [2 x] A - F () x] - 05. , x]2].x] () t + aF{xJ =0.

(14)

ax]

According to (13) the system (14) has energetic attractor '1/1

=0

(10).

Subject to potential function F{x]) selected in (14), various types of self-oscillating systems can be get. Motions of these systems on manifold '1/] = 0 (10) are of the following types, for example: (a) harmonic (15) (b) Duffing (16) Substituting potential functions (15) and (16) in (14) we get harmonic oscillation generator or Duffing generator respectively. For instance, using Fd (Xl) we get unified Van-der-PaullRayleigh/Duffing oscillator according to (14):

x J(t)- ~ [A2 - 0,501 x~ - fJ x~ ~

On manifold 'l/J equation:

=0

0,5xJ2 ~] (t)+ 0/ x J+ 4fJx~ = O.

(17)

(10) the system (17) is described with Duffing (18)

This equation has different dynamic characteristics depending on relation of OJ and fJ. For example, at fJ = 0 from (18) we get

New Models of Nonlinear Oscillations Generators

123

conservative equation of type (S) with potential function Fg (15). Then equation (17) become an unified Van-der-PaullRayleigh/Duffing generator:

with asymptotically stable harmonic attractor (boundary cycle If/l = 0 (10) to which all phase plane trajectories are attracted). In other words, equation (19) is generator of harmonic oscillations:

x(t)=asin(mt+¢o)

(20)

with specified amplitude a and frequency (f). Generator (19) combines Van-der-Paul (1) and Rayleigh (2) generators into unified non-linear oscillator of new class. It is interesting to note that system (19) has a property of self-similarity as after affix enters on attractor If/] = 0 (10) at fJ = 0 the system continues to be described with the same equation (IS). In other words, attractors are "embedded" one into

oi

another [1-3]. Obviously that if

»1, then member 0,5alx~ will

dominate in comparison with member 0,5x~. In this case properties of equation (19) are closer to properties of Van-der-Paul Equation. On the contrary if m2 « 1, then member 0,5x~ will dominate and equation (19) will be similar to Rayleigh Equation (2). Results of generator (19) simulation with parameters A = 1; m = 1; T = 0,5 and XlO = 1; x 20 = 0 are shown in fig. 1-3. Results of generator (19) simulation with parameters A = I; m = 10; T = 0,5 are shown in fig. 4-6. These results fully verify theoretical statements of ADAR [1-3].

x<~ ,

1

-

o

-

,

,, ,, _ J _ , , , J

L_

-1

,,- ,,- -

, , , , , ,

-;

V,

-

, , ,- , ,- , , , - ,,, , , , -

, \I

o

,

'V

10

20

Fig. 1

Xl

,II , /\ , , , , , , - , - , ,, , ,,, , , - , ,, , , , , ~ ,, - -,, ,,, , _ 1~ , ,- , , , , L , ,- - ,

v'

30

(t)

,

X2

-

-1 t,

time graph

,, , l, ,, , , l _

o

,-

50

,' /\ , , , -, , , -, , ,

1J

C

0

L

, ,

~-

v:

10

,

,

,,, - ~,, -

,

,II ,

, , , ,,,- - I,,- - ,,, ,- - , -' '- , ,,, , ,, ,,, ,,, - - , - , -', ,, , , , , - - ' - , ~, ,, , , y

-

,

, , _ , , , _L , ,

40

(t) ,

_L ,,, -

1

',

'v

20

-

, , , ,, ~ ,

-,

;-

, ,,

_L_

-

II:

30

40

Fig. 2 x 2 (t) time graph

-

Y

t, c 50

124

A. A. Kolesnikov

-- - - - - ~- -- -- - -~- - -- --

, ,

1

, ,

0,8 -- - - - - ,- --- - -- r - - -- --- - --- ~- -- - - - -~- - -- --

, , 0,4 --- - - -, - - - - - -- r- - - - --- - --- ~- -- - - - -~- - - - --

-1

-1

0

1

Fig. 3 Phase-plane portrait

I

I

--- -,- - - --- - r - -- -- 1

o -1 -- - - -I -- - - - - - ,- - - - - -_ _ _ _ _

-2

~

,

I

5

I

I

I

I

I

I

I

L -_ __ __ _~ _ __ _ _ __ L_ __ __ _~ t,c

o

I

-4 - T - .., - - - - ,- - T - .., - -, - - ,- - ,- Xl-(t)

__ __ _ __ L __ _ _ _ _

10

-0,4

15

Fig. 5 x 2 (t) time graph

-0,2

0

0,2

0,4

Fig. 6 Phase-plane portrait

2. Unified Poincare Generator Basic equations of Van-der-Paul (1) and Rayleigh (2) generators as well as above-mentioned synthesized model of unified oscillator are differential 2-order equations:

x(t)- 1(;1, x, x(t))+ oJ x = o.

(21)

However in addition to form (21), in applications it's more convenient to form generators as system of two symmetrical non-linear differential equations of Poincare type generator (3). Therefore use ADAR to synthesize such symmetrical oscillators generating selfoscillations with prescribed amplitude and frequency. Suppose that there is a system of differential equations: Xl (t) = x 2

+ klu;

x (t) = -oJ Xl + k u, 2

2

(22)

New Models of Nonlinear Oscillations Generators

with some coordinate function

U(X j

,

125

x 2 ) in the right part. It is required

to synthesize u(x l , x 2 ) so as self-oscillations with desired amplitude and frequency are always formed in system (22). According to ADAR introduce the following energetic macro variable: If/ = A 2 - o,Sui x~ - O,Sx~ .

(23)

Then substituting If/ (23) into invariant expression

Tvi(t) + F(x p x 2)1f/ = 0; T > 0, subject to system (22) we get feedback law U

(XI ,X2 )(k JOJ2 Xl +k2X2 ) = F(x j ,x2 ) If/. T

(24) Compose equations of closed system (22), (24): . Xl

( t ) =X2 +kl (klOJ 2 Xl +k2X2 ) ( A 2 -O,SOJ 2 Xl2 -0,SX22 ) ; T (2S) . ( t ) = -OJ 2 Xl + T k2 (klOJ 2 Xl + k X2) ( A 2 - O,SOJ 2 Xl2 - 0,SX22 ) . X2 2

Given symmetrical system (2S) is unified Poincare type generator (3). On manifold IjI = (23), function U = (24) and system (2S) is formed with confluent equations

°

°

Xl \If

(t) = X2\1f; X2\1f (t) =- OJ2 Xl \If

fully coinciding with (8). It means that stable harmonic oscillations with prescribed amplitude and frequency are always appeared in the system (2S) by some time depending on parameter T. Results of system (2S) simulation with parameters A = 1 ; (j) = 1; kl = k2 = 1; T = O,S are shown in figures 7 and 8. Results of system (2S) simulation with parameters A = I; OJ = 10; kj = 10; k2 = I; T = 0,5 are shown III figures 9 and 10. These results also verify theoretical statements of ADAR [1-3].

126

A. A. Kolesnikov

-1 -1 -1 5 L---'L""-_c-=--=---L-=----=--L----'-L------.J

, 0

Fig. 7

10 Xl

20

(t) and

X2

t, C

30

x 1 (t)

_2L-~L-~---L---L--~--~~

-1

(t) time graphs

1,5~~~--!c.--~-~-~~

0

1

2

Fig. 8 Phase-plane portrait

11---':"-~

1

0,5 -

,

0--

-1 -1 5 , 0

Fig. 9

1 Xl

2

(t) and

3

--:--

, , --,--

t, C

x 2 (t) time graphs

-0,1

0

0,1

Fig. lO Phase-plane portrait

Obviously that depending on kl and k2 coefficients relation the system (25) is similar to either Poincare generator (3) (when kl - k 2 ) or Vander-Paul and Rayleigh oscillator (19) (when kl »k2 or k2 »kl ). For example, if k2 »kl or kl == 0 then from (25) we get

Xl (t)- ki T

(A 2 -

O,5aixj2 -

O,5x~ )x (t)+ alxl = 0 . j

(26)

On the contrary, if kl »k2 or k2 == 0 then we similarly get X.. 2 () t -

2 -kl T

(A2

(j)2 -

2 05 , (j)2 X 22 - 05. , X 2 ). X 2 (t) + (J) 2 X 2 = 0 .

(27)

Both systems (26) and (27) are models of generator type (19). On the whole it means that synthesized unified Poincare type generator (25) particularly includes or Van-der-Paul and Rayleigh oscillator (19).

New Models of Nonlinear Oscillations Generators

127

Conclusion Note the following circumstances: Firstly, order of differential equations of attractors (boundary cycles of self-oscillating systems) is always first. That fact is not underlined in oscillations theory literature for some reason. Secondly, energetic invariants of kind 1fI; = 0 (7), (10), and (23) have been introduced to synthesize new types of nonlinear oscillation generators. Such invariants present natural properties of these generators. So on base of energetic invariants, using ADAR [1-3], new types of nonlinear oscillation generators are synthesized. These types include well-known generator equations as special cases.

References [1] Kolesnikov A.A. Synergetics control theory. Moscow: Energoatomizdat, 1994. [2] Kolesnikov AA Synergetics and Control Theory Problems. Moscow: PHYSMATHLIT,2004. [3] Kolesnikov A Synergetics Methods of Complex Systems Control: System Synthesis Theory. Moscow: URSS, 2006.

128

Nonlinear System's Synthesis - The Central Problem of Modern Science and Technology: Synergetics Conception. Part II: Strategies of Synergetics Control Anatoly A. Kolesnikov Technological Institute of Southern Federal University Synergetics and Control Processes Department 44, Nekrasovky str., Taganrog, 347928, Russia (e-mail: [email protected]) Abstract: Our environment, such as natural, social, economics and engineering ones are the world of complex supersystems of various natures. These systems are a collection of various subsystems providing defined functions and interconnected by processes of forced dynamics interaction and exchange of power, matter and information. These supersystems are nonlinear, multidimensional and multi linked. And in these systems are complex transients and have place of critical and chaotic modes. Problems of system synthesis, i.e. finding of common objective laws of control processes in such a dynamics system are much actual, complicated and, in many respects, practically inaccessible for present control theory. In the report we consider fundamental basis of nonlinear theory of system's synthesis based on synergetics approach in modern control theory as well as its application [1 ,2]. The report consists of three parts: Part I General statements; Part II Strategies of synergetics control ; Part III Synergetics synthesis of nonlenear systems with state observers. Keywords: synergetics, system's synthesis, invariants, nonlinear systems, regulator'S design, chaotic disturbances

Directed selforganization implies control strategy that forms and keeps dynamic invariants either belonging to the system or external ones. Depending on the control aim, the invariants can be constant or changing causing stabilization or a new dynamic state respectively. Talking Biology, in the first case the system's invariants realize stabilizing selection, and in the second case - dynamic selection. In other words, purposeful forming of dynamic invariants allows to perform purposive selforganization in the system. In order to apply the ideas of Synergetics in control theory, it is necessary to keep the conceptual correspondence to the main qualities of selforganization: nonlinearity - open systems - coherence. For control tasks the most important quality is that the system must be open.

Strategies of Synergetics Control

129

In the initial statement the control system is described by the differential equations of the object

x{t)=F{x,u,q,M) (1) It includes state coordinates x{t) and some external forces consisting of sought controls u{t), setting actions q{t) and possibly the disturbing actions M{t). In order to move from the system "object - external forces" to forming the self-organization equations we must exclude these forces in an appropriate way. To do that we should extend the initial equations of the system "object - external forces" in such a way that excluded forces would become internal for the system. So for the new extended system its equations can become the se/forganization equations. I.e. as a result of this extension we can move to organization of the system to its selforganization. Such an extension takes place in the existing formulation of the control system synthesis problems, i.e. finding the control laws as a function of state coordinates of the extended system. These laws are the equations of the regulator and they should ensure the desired dynamic qualities of the closedloop system "object - control law (regulator)". Then we can apply relations characterizing the self-organization processes to the system ("object - regulator") according to the qualities mentioned above. So in order to apply the synergetic approach based on cooperative processes of selforganization to the control problems. It is necessary to move from initial control task including the object's equations and external forces (as control, setting and disturbing actions) to extended task statement, where the mentioned forces become internal interactions of the closed-loop system. To do this we should present the external setting q{t) and disturbing M{t) actions as solutions of some additional differential equations describing the informational model. Doing this we perform their "submerging" into the general structure of the extended system. Then the control problem should be formed as a task of search for interaction laws between the components of the extended system ensuring appearing of selforganization processes. Specifically, this problem comes down to synthesis of appropriate closed-loop control laws U(XI' ... ' x n ' wI' ... ' w,ll) as a function of state coordinates of the

extended system. Here

WI' ... '

W,ll - coordinates of the corresponding infor-

mational models of the setting and disturbing actions written as additional differential equations. Then giving energy or matter to such system we can create an unbalanced situation necessary for emerging of directed self-organization processes. The mentioned extension of the initial system and forming the selforganization equations allows to set a connection between the ideas of Synergetics and the problem of synthesis of nonlinear control systems basing on invariant relations. This means that synergetic control theory is the theory of synthesis of closed-loop control systems based on forming the cohered cooperative processes in the systems of various nature.

130

A. A. Kolesnikov

According to the mentioned qualities of self-organization and to the state flow compression-decompression principle of the phase flow [1,2] the basic statements of the synergetic approach to the synthesis of nonlinear dynamic systems are: • firstly, forming the extended system of differential equations reflecting the processes of achieving the set values, suppressing the disturbances, optimization, coordinate observing etc. • secondly, synthesis of such "external" controls that ensure the reduction of the extra degrees of freedom of the extended system with respect to the final manifold where the motion of the representing point is described by the equations of the system's "internal" dynamics. • thirdly, synthesis of "internal" controls by means of forming the links between the "internal" coordinates of the system. These links ensure the reaching of the control aim. The stated basic principles of the synergetic approach lead to the it's stages shown below. First we write the initial differential equations of the object, e.g. in the following form: Xk (t) = Ik (XI , ... ,xn) + M k (t);k = 1,2, ... ,m -1; m:::; n, xk+1 (t) = 1k+1 (XI , ... ,xn ) + uk+1 + M k+1 (t); (2)

where: xI"",x n -

object's state coordinates,

uk+i"",u n -

controls,

M k (t), ... ,M n (t) - disturbances,

k = 1,2, ... ,m-l, m:::; n .. Then we add /1 equations connected to the problem of prediction and suppression of disturbances in the system (2). Wj (t) = g;C WI , .•. , W J1 ,XI , ... ,xn ), j = 1, ... ,/1 When we build these equations, we face two independent and important tasks. First, the task of describing the real disturbances M k (t), ... ,M n (t) as specific solutions of some differential equations. Second, the task of forming the links between the equations of the initial object (l) and the equations of the disturbances. After the selection of the links equations we get the extended system of differential equations Wj (t) = g i (WI , ... , WJ1' XI"'" Xn); Xi (t) = Ii (XI"'" Xn)

+ Wi;

Xi+1(t) = li+1 (XI"'" Xn)

+ Wi+1 + Ui+l ;

xn(t)=ln(xI"",Xn)+W n +u n '

where j

= 1, ... ,/1; i = /1 + I, ... ,m -1.

(3)

Strategies of Synergetics Control

Equations (3) allow to set the synthesis task of control laws

131

ui+l,,,,,u n

allowing to suppress the disturbances M k (t), ... ,M n (t) and ensuring the set dynamic qualities of the closed-loop system. It is necessary to synthesize such a control vector u(u l , .•. ,u m ) that would ensure motion of the representing point (RP) of the extended object (3) from and arbitrary initial state (in some allowed area), first, to some manifolds 'l'S(Xl"",Xn,Wl" " 'W,u) ==0 and then to the set state, e.g. origin point of the extended state space. On the trajectories of the closed-loop system motion minimum of some optimizing functional can be reached or some prime performance criteria can be satisfied. Asymptotic stability in some area or in the large must be ensured. In the report we mention that RP motion of the synthesized system must satisfy the following system of functional equations: Tslfrs (t) + CPs ('I's) == 0, s == 1,2, ... ,m. (4) Functions CPs('I's) in (4) are selected in such a way that: asymptotic stability in the large with respect to 'l's == 0 and 'l's == B is ensured in the system (4); the desired performance criteria of RP's movement to the attracting manifolds 'l's (Xl , ... ,Xn , WI , ... , W,u) == 0, s == 1,2, ... ,m where 'l's - some aggregated variables. There are no special limitations put on the selection of functions (4). It is important to underline that macro-variables 'l's reflect synergetic (cooperative, coherent) qualities of the synthesized multi-level systems. This means that it would be rather reasonable to use the known laws of the natural systems, which manifest the qualities of coherence and cooperative activities. We will be using synergetic ideology in order to solve the stated task. This method means that the RP of the system gets to the intersection of manifolds '1'1 == O,· ··,'I'm == 0 as a result of action performed by "external" controls u i+ l "",u n .

Moving along the intersection is described by the following equa-

tions of "internal" dynamics WjIl'U) ==

g j(W11I"'''' w,ull" vi+I"'"

VII' XIII"'''' Xm-III');

(5)

Xill'(t) == fi(X11I"'''' xm-lll" Vi+l , ... , VII)'

Vi+l , .•. , vn - "internal" controls, j == l, ... ,,ll, i == ,ll + l, ... ,m -1 . Considering the decomposed system (5) having the order of n +,ll - m , we synthesize "internal" controls Vi+l , .•. , VII ensuring the dynamic

where

qualities of RP's motion along the intersection of manifolds

'1'1==

0, .. .,'1'm== 0 .

Synthesis of controls vi+l , •.. , VII is a separate task of controlling the subobject (5). Consecutive-series totality of invariant manifolds is used for this purpose.

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A. A. Kolesnikov

According to the principle of control preservation [2,4], the internal controls vk have a constant dimension dim v k = m that coincides with the dimension of the external controls. Internal controls influence the subobject (5) decomposing it to the object of dimension n - 2m with its own controls. After that the process of consecutive decomposition continues until the RP of the object reaches the preset final manifold. As a result of the described procedure, we find the internal controls connected recursively. Knowing the controls Vi+l , ... , vn ' we can introduce the desired macrovariables that can have, for example, the following linear form: If/s = Y.d(X i+1 - VI) + ... + Ysm(x n - vn ), s = l, ... ,m (6) Basing on the functional equations of form (4) and on the desired macrovariabIes If/s (6) and with respect to the equations of the extended system (6) we find the external controls using the ADAR method Dl u i+l =-fi+l(xl,···,xn )-W;+I--;

D

(7)

where

YII

YI2

Ylm

D= Y21

Yn

Y2m

YmI

Ym2

Ymm

<1>1 <1>2

YI2

Ylm

Y22

Y2m

m Y m2

Y mm

Dl =

:;to,

(8)

:;to,

(9)

for s = 0,

Dll =

for s

:;t

Yll

Y12

YI.m-l

Y21

Y22

Y2,m-I

<1>1 <1>2

Yml

Ym2

Ym,m-l

<1>",

:;to

(10)

0, where

s = Yd VI (t) + Ys2 V2 (t) + ... + YsnVn (t) - ;

qJs(lf/s)

(1)

s

The relations (7)-(9) allow to find the vector control laws (7), which move the RP to the vicinity of the manifold's intersection If/l = O, ... ,lf/m = 0 . Motion along it is determined by the equations of internal dynamics (5). The control laws (7) together with link equations form the dynamic equations of an aggregated regulator that ensures the selective in variance of the closed

Strategies of Synergetics Control

133

loop system (3)-(7) to the disturbances M k (t), ... , M n (t). It also provides asymptotic motion stability and the desired qualities of transients. From the point of view of the problem of synthesis of control laws, the differences of the developed new approach are as follows. First, shift of the main attention to the behavior of the system on the attractors. This allows to decompose the system and therefore to simplify it. It lets us focus our attention on the stable asymptotic motion modes. Second difference is in cascade synthesis of parallel-consecutive group of internal controls, i.e. dynamically tied links of the synthesized system. When we use a synergetic approach, there is an internal process of self-control in the synthesized system, when the a cascade sequence of internal controls compressing the volume of the phase flow is formed. This is performed starting from the external maximal possible area and going through the enclosed one into the other internal areas until the RP gets to the desired state of the system. The new synergetic approach allowed to perform a certain breakthrough in the area of synthesis of multiply connected systems of continuous, discontinuous, discrete-time, selective-invariant, multicriterial, terminal and adaptive control for nonlinear dynamic objects of various nature [1-4]. This approach found a specific application for solution of complex problems of control of nonlinear technical objects (flying apparatus, turbogenerators, robots, electric drives, technological aggregates etc.) and in the control tasks of ecology, biotechnology etc.

References [1] Ko1esnikov AA Synergetics control theory. Moscow: Energoatomizdat, 1994. [2] A.A. Kolesnikov, et a!. Modern applied control theory. Under ediotion of A.A Kolesnikov. Moscow-Taganrog: Integracia-TSURE pub!, 2000. [3] Kolesnikov AA, Balalaev N.V. Synthesis of Nonlinear Systems with State Observers, II New Concepts of the General Control Theory: Collection of scientific worksl Under edition of AA Krasovsky. Taganrog: TSURE, pp. 101-115 . [4] Kolesnikov AA. Sequential optimization of nonlinear aggregated control systems . Moscow: Energoatomizdat, 1987.

134

Synergetic Approach to Traditional Control Laws Multi-Machine Power System Modification Anatoly A. Kolesnikov l and Andrew A. Kuzmenko Technological Institute of Southen Federal University Synergetics and Control Processes Department 44, Nekrasovky str., Taganrog, 347928, Russia I (e-mail: [email protected]) Abstract: Traditional algorithms for control power system that were proposed more than half a century ago are applied in our days. We propose principally new synergetics laws for frequency and power for power station units and unit groups. This approach requires development of technique for application. Moreover, directed implementation of synergetics control laws require global rebuilding of existing patterns of power units control. The simplest way of synergetics algorithms implementation is by using hierarchical principal of control system building. So we rate synergetics control laws as dynamical desired values for ordinary algorithms or as correcting signals [1]. Keywords: power system, synergetics control, regulator, modificated control law.

1. Object model Exploring multi-machine system is consisted of turbogenerator's group powering busses of infinity power (constant voltage) through power transmission line. As initial nonlinear model of this power system that consists of one turbogenerator connected with big power busses we explore following system of differential equations [2]:

j(t}=s .. S (t) = bl.(~r· -

E2y .. sin (a. ql

EJt} = b" (- E

q,

i i '

I

I

I

/I

II

}- E

I

0J - a)+ M.(t}l.,f, Ij

I

+ b3, • s, sin(~ - aij )+U1, + N, (t)},

Pr, (t) = b q,{t} = b

6, (-

r, (q,) - bs,s, + h,);

h,(t} = b

7i ( -

h, +UJ;

4, (-

.EqJ.y/). sin(o -

ql

PTi + q, . C, );

(1)

z, (t) = 17, (U ~o, - U~,}, w,(t} = ~s" here i = 1,2 is number of power unit; j = 1,2, j *- i; 0, is angle of rotation for rotor of i-th synchronous generator (SO) relatively synchronous axis of rotation; Si = (wo - W i )/ Wo is slip; Wi is SO's frequency of rotation; Wo is synchronous frequency of rotation; Pr , is mechanical power of i-th turbine; E q , is SO's synchronous EMF; C,

= const

is steam pressure before turbine;

Synergetic Approach

135

q; is actuating valve solenoid shift controlling power medial (steam) entry into turbine; r, (q;) is function accounting limitation of solenoid shift; h; is signal from turbine speed secondary controller; VI; is voltage applied to SO excitation wiring; V 2 ; is control action to turbine speed secondary controller; b];, b2 ;, b3; ; b4 ;, bs;, b6 ; , b7 ; are constant factors; Yu = y p' au = a p are SO mutual conductance and its additional angles; w; is estimation of unmeasured disturbances M; (t) = M 0;; z; is estimation of unmeasured disturbances N; (t) = No; . In model (1) two last equation are models of external piecewise constant disturbances acting SO excitation subsystem N ;(t) and power system from load's side. According to terms of Synergetics control theory the model (1) is design model (system's extended model). Conventional algorithms of control for groups of power units are built by decentralized approach: control loops for excitation and turbine control are independent and define by following: - excitation controller for stabilizing of input voltage for i-th SO [3]: V APE; = V /0; + kou;llV c; + k lU;

dllV c; dllOJ; -----:it + kofillf»; + klfi ----;;;(2)

= V /,,;

dllV c;

ds;

+ kou; llV c; + k lU; -----:it + kOfiOJOs; + kl fi OJO dt'

- PI-control for i-th turbine rotation frequency stabilization: fin; =

=

where

ll~

= OJo -

~

= OJos;

fllOJ; dt + k llPr; k,,;OJos; + ~: f dt + k,,;llP

k,,;llOJ; + ;,;

p;

S;

(3)

TP

is deviation of frequency, llV r;

=V

GO; -

V G;

-

is

deviation of SO's input voltage, Mr; = PT~ - PI; is deviation of turbine power.

2. Modification of control laws The common problem of plant (I) control is: We need to build control laws v];, V 2 ; providing system (1) asymptotic stability and generating engineering attractors into system (1) phase space. These attractors reflect engineering tasks that appointed to multi-machine power system and are presented by following correlations: 1) stabilizing of SO output voltage (4) U GO; - U G; = 0, where V GO; is desired SO voltage; 2) stabilizing of turbogenerator ~ = OJo rotation frequency. For model (l) this follows

136

A. A. Kolesnikov and A. A. Kuzmenko Si

(5)

=0.

In details the procedure of synergetics synthesis for power system turbo generators control laws is presented in [1]. Let us introduce modified law of SO excitation control as: (6) Vii = U APBi + U '

Sl'l1i

here U APBi is defined by (2), and U,yni is synergetics additive component providing implementation of engineering attractor (4): F,s i + /32lr(U~Oi U"ni=E-b's.sin(O-a)ql 12 31

Where: F; F, U Gi

I

-U~,)+;

F2 b2i

I

Iffli

(7)

I,

= 2Eqi V'Y12 Xdi (sin (<5, - a ,2 ) - YiiX di sin(oi - a '2 + a,,}),

= -2AiE'/i -

2Bi (Oi)' Iffli

J

= U~Oi - U~i + /32iZi '

= E'~iAi + 2Bi (<5, )E"i + Di

, Ai

= 1- 2Yii Xdi cos(aJ+ (YiiXdJ '

Bi (<5,) = U,Y,2Xd,(COS(<5, - a,J- YiXdi cos(<5, - a '2 + aJ), Di = (U,Y,2 XdJ . Turbine control modified law is control law (3) in which we used following dynamics equation as power preset value PT~:

PT~ = E:iY ii sin(aii )+ EqiEqj Y,2 sin(<5, - OJ - a ,2 )-

*

-W(~-IJ-S.(_1 , ';i bli T2i 'bliT"

+&J,

(8)

b li

here j = 1,2, j i . Computer simulation results for closed-loop system (1) at i = 1,2 with modified control algorithms (2), (6), and (3), (8) (each turbo generator is under action of external piece-constant perturbation, see fig. 1) are shown at of regulators are: U GOI = 1,05; U G02 = 1,1; fig. 2-4. Parameters

= 50;klUi = 7; kof' = 30; kif' = 8; k ni = 20; ,; = /3, = 1; 1J = /32 = 1 ; ~i = 1; T" = 5 . kOUi

= 2,5;

T,'i

= 1;

k pi

U roi

= 0;

,.61 61(1)'

I 1.6- -

I -1- -

1.6,--- -

-,- -

1.4

:

-1

2==1=::=J

::f . I

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1 - - - - - - - ;- - - - - -: - - !

1.21

I

Pt1(t)

I

I

Pt2(t)

0_8~---------'-------

o

Fig. 1. External perturbation

50

100

I

150

200

250

300

350

400

Fig. 2. Transients of mechanical power PTi (t) and angles <5, (t)

Synergetic Approach

:0::1:: _~~' ~2(~; 001 r- -

- I- -

0.005 -

-

_I _

0 005--

-

I - I -"

0,01 "

-

_I _

:

-

137

I

-

_

-

-

1-

-

-

-

_

-0.015 -

300

250

Fig. 3. Transients of slip s; (t)

Fig. 4. Transients of output voltages U G; (t)

0,03

-

1

0,02 l-

52(t)

I

, - - - - , - - - --------.--

_: _ _ _

~

__ __ :

-------,

__

j

-O, 04~ -~~-

Fig. 5. Transients of slips s; (t)

Fig 6. Transients of output voltages U r; (t)

According simulation results we can conclude that modified conventional control algorithms for power unit group provide rotation frequency stabilizing (s; = 0, see fig 3) as well as stabilizing of each SG output voltage (see fig. 4). We must underline that static error relating voltage is eliminated, and we provide suppression of piece-constant perturbations M; (t) = M 0;' N ;(t) = No; , acting to power system. For comparison we provide on fig. 5 and 6 closed-loop system behavior (1) with conventional control laws for exiting SG U ,; = U APB; (2) and turbine U 2; = fipr; (3), under action of the same external piece-constant perturbation as shown on fig. 1. It seems that in case of conventional laws the output voltage is stabilized with static error (see fig. 5), and transients has oscillation manner with long duration.

Conclusion Using obtained controls we built principally new classes of automatic controller providing asymptotical stability in a whole for closed-loop system of "turbogenerator's group and controllers", its robustness to parameters and invariance to external disturbances. Modified synergetics controllers are

138

A. A. Kolesnikov and A. A. Kuzmenko

significantly exceed existing typical controllers, based on ideology of linear control theory, by its dynamical properties, because of linear control theory does not reflect real physical processes in turbogenerators. Implementing of modified synergetics controllers for turbo generators control will provide principal improvement of power systems static and dynamic properties in emergency and extreme modes of operation.

References [1] Kolesnikov A.A., Kuzmenko A.A., at all. Synergetics Methods of Complex Systems Control: Power Systems. Moscow: URSS, 2006. [2] Koalov V.N., Shashihin V.Y. Synthesis of coordinating robust control of interconnected syncronous generators II "Electrichestvo" magazine, 2000, #9. -Po 2026. [3] Michnevitch G.V. Synthesis of control system structure for synchronous machines exitation. Moscow: Nauka, 1964.

139

Dynamic Stability Loss of Closed Circled Cylindrical Shells Estimation Using Wavelets V. A. Krysko l , J. Awrejcewicz 2 , M. Zhigalov l , V. Soldatov l , E. S. Kuznetsova l , and S. Mitskevich l

I

Saratov State Technical University, Department of Mathematics and Modeling, 410054, Polytechnical 77, Saratov, Russia (e-mail: [email protected])

2

Technical University of Lodz, Department of Automatics and Biomechanics, 1/15 Stefanowski St., 90-924 Lodz, Poland (e-mail: [email protected])

Abstract: In this work dynamic stability loss of closed cylindrical shells is studied. The hybrid type PDEs governing cylindrical shells dynamics and regarding deflection (Airy's) function and stresses are first derived. Then they are reduced to ODEs and algebraic equations (A E) applying the Bubnov-Galerkin high order approximations method. Dynamic stability loss of the cylindrical shells subject to sign-changeable loading using wavelets is investigated. In addition, the convergence of the Bubnov-Galerkin method and results validation are addressed. Keywords: chaos, closed cylindrical shells, wavelets, nonlinear dynamics, dynamic stability loss.

1. Introduction Closed cylindrical shells are members of various constructions and machines including airplanes, ships, civil engineering constructions, measurement devices, etc. For many years thousands of researchers attacked a key problem regarding safe and durable properties of thin- and thick-walled constructions addressing and formulating the so called stability loss criteria (see [1]-[4]). Note that even a particular case of this topic, i.e. analysis of free and forced nonlinear vibrations of circled cylindrical shells attracted and still attracts many engineering oriented researchers (see for instance [5-10] and the cited references therein). In this work, among others, we propose a novel stability loss criterion with a help of the wavelet-based analysis.

2. Problem statement We study the closed circled cylindrical shell of finite length with constant stiffness and density subject to action of non-uniform sign-changeable load and being embedded in a temperature field (Fig. 1). The following coordinates are introduced: axis y refers to circled co-ordinate, whereas axis z

140

V. A. Krysko et al.

overlaps a normal one of the shell's middle surface. Therefore, our shell is defined as Q = {x, y,z I (x, y) E [O;L] x [0;2;r],-h:S; z:S; h}.

Fig. 1. Computational scheme Its governing dynamic non-dimensional equations have the following form [9,10]: I

-------:2c-

12(1- V

4 [-2 A a w+2 --4

ax

)

4 a w

-2--2

ax ay

2

4 w] + A -a- 4

ay

-

L( w, F)

(1)

2 2 aF aw aw 2 -k, - 2 - - - 2 - - E - + kyq(x, y,t) = 0', . ax at at .

{ A

~

a4F -4-

ax

a 4F

+ 2 -2- -2 + A

ax ay

2

a 4F -4-

ay

1

+ -L(w, w) + ky

2

i W} = o. ax

--2

Equations (I) have been transformed to the non-dimensional form using the following relations:

-

3-

w=2hw F=EO(2h) F, t=

_ 2h ky = ky - 2 ' q R

_ Eo(2h) =q L2 R2

RL c;:-t, A=LIR; x=Lx, y=Ry; 2hVgEo

4

. Here Land R

=R

are shell length and y

radius, respectively; t - time, E - damping coefficient of a shell surrounding medium, F - stress (Airy's) function, w - shell's deflection, h - shell's thickness, v - Poisson's coefficient, g - gravity acceleration, Eo - shell material Young modulus, ky - shell's curvature regarding axis y and finally

Dynamic Stability Loss of Closed Circled Cylindrical Shells Estimation

iwiF iwiF

L(w F)= - - + ,

dX

2

dy

2

dy

2

-

dX

2

141

iw iF

-2 - - -- '

dxdy dxdy

'

iw 2] -( [iwiw J ax ay axay

L(w, w) = 2 ~2- -~ 2

are widely known nonlinear operators (note that bars over non-dimensional quantities are omitted). Let us focus on investigation of one of the boundary conditions, i.e. hinged shell's support with flexible ribs on the shell's edges: w

= M x = N x = c y = 0 for

x

= 0', 1, y = o·" 27C

(2)

and with the following initial conditions:

w(x, y) It=O= 0,

dWI t=O = o. a;

(3)

The boundary value problem formulated so far is solved via the Bubnov-Galerkin method with high order approximations. In this work the being sought functions wand F are approximated by analytical expressions having a finite number of parameters arbitrarily taken through the following functions depending on time and spatial co-ordinates: N, N2

F=

L LB .. (t)If/ " (x , y) ,

;=0 j=O

IJ

(4)

IJ

where rjJij (x, y) and If/ij (x, y) are certain functions of x, y. After the Bubnov-Galerkin method application to system (1), a system of ODEs is obtained regarding functions ~/t) and Bij (t) , and its matrix form is: (5)

where D,

G

= Ilcijklll,

= IIDUjrSkl l1 ,D 2

=

s = IISijrskl ll,

IID2ijrski li ,P =

II~jkl l

- squared matrices of dimensions

142

V. A. Krysko et al.

dimensions 2· N] . N2 xl. Second equation of (5) is solved regarding matrix B using an inversed matrix method on each time step:

B

= [-P-I D2A - P- 1C 2]A.

(6)

Multiplying first equation of (5) by G -1 and denoting A = R, one achieves the Cauchy's problem for the following first order ODEs:

{~ =

-eR + G-1D1AB - G-]SA + q(;)G-1Q

(7)

A=R

These transformations are allowed since matrices G-] and p-l exist assuming linear independence of the co-ordinate functions. We add to equations (7) boundary and initial conditions and the yielded Cauchy problem is solved using the fourth order Runge-Kutta algorithm (computational step is chosen via the Runge principle). In order to satisfy boundary conditions (2), functions 9ij' lfIij appeared in (4) are sought in the form of a product of two functions, where each of them depends only on one argument: NI N2

W

=I I

i=l j=O

A.(t)sin(i.7rx)cos(jy), lJ

(8) NI N2

F

=I I

i=l j=O

B .. (t)sin(i.7rx) cos(jy) lJ

3. Numerical results We study our shells' vibrations character using the following fixed parameters: k =112.5 for A = 2, c =9, and subject to action of transversal )'

load q(t) = qo sin(OJ/) distributed on the whole shell's surface (0 OJP

~ x ~

1,

= 2.3), and we take numbers of functions N] = 1, N 2 = 9 (note that

increase of those numbers do not increase the results accuracy).

Dynamic Stability Loss of Closed Circled Cylindrical Shells Estimation

143

Qualitative analysis is carried out using the well-known characteristics: w(0.5; 0; t) , wavelets in 2D and 3D, phase portraits, Poincare maps, FFT (Fast Fourier Transform), Lyapunov exponents and the autocorrelation functions. In particular, we focus on dynamic stability loss estimation of our investigated shell. Further, only the following characteristics will be reported: time histories, phase portraits, frequency power spectra, wavelet-spectra, and shell buckling forms for o ~ x ~ 1; 0 ~ y ~ 21r and for x = 0.5; 0 ~ y ~ 21r . the computations In order to construct the dependence w max (q), 0 have been carried out for qo

E

[0; 0.4975] . In addition, for each element qo

a maximum shell deflection in point (0,5; 0) has been estimated. Results are reported in Fig. 2, where one may observe the so-called shell dynamic stability loss (small change of qo is associated with a rapid shell deflection increase). Inflexion point of the curve given in Fig. 2 allows to define the critical value q~r. We study the vibration character in the shell pre-critical state (qo

~

0.225 , see Figures 3-8). Analysis of shell vibrations (for instance,

in point A) allows to conclude that they are harmonic.

A: qo = 0.1 - pre-critical

12

shell state; B: qo = 0.3225 - after first shell buckling; C: qo = 0.3975 - after second shell buckling.

Fig. 2. Relation w max (q) 0

'''a

.n, 0.17

w(ll&5

0.16

('-155

0. 15

.,~

100

105

,

110

liS

120

Fig. 3. w(0.5;0;t)

Goo..

,(10

.:10

I. . . ..,

XIII

Z»

Fig. 4. 2D Morlet wavelet

Fig. 5. 3D Morlet wavelet

144

V. A. Krysko et al.

o Fig. 6. Phase portrait w(O.5 ;O;t) ,

Fig. 7. Shell deflection form

Fig. 8. Shell transversal cross-section I-I

w(O.5;O;t)

Observe that when our dynamical system passed through the critical point q~r (we take qo

= 0.3225 ), chaotic dynamics appears (see Figures 9-14).

:.~ cw , Fig. 9. Time history w(0.5 ; 0;t)

qo

Fig. 10. 2D Morlet wavelet

Fig. 11. 3D Morlet wavelet

= 0.3225

o Fig. 12. Phase portrait w(0 .5; 0; t) ,

Fig. 13. Shell deflection form

Fig. 14. Shell transversal cross-section I-I

w(0.5 ; O; t)

In point C associated with second shell buckling (qo

= 0.3975 ) the

wavelet-spectrum allows to detect an interesting qualitative change of the shell dynamics in a very short time interval (t ~ 200 ).

Dynamic Stability Loss 0/ Closed Circled Cylindrical Shells Estimation

Figure 15. Time history w(0.5;0;t)

qo

145

Figure 16. 2D Morlet wavelet

Figure 17. 3D Morlet wavelet

Figure 19. Shell defl ection form

Figure 20. Shell transversal cross-section I-I

= 0.3975

~. :~'JI

•. 'r

='l

~

r

, , " "

Figure 18. Phase portrait w(0.5; 0; t), W(O.5; 0; t)

o

4. Conclusions Owing to our computational technique used to study non-linear vibrations of closed circled cylindrical shells subject to action of transversal signchangeable load the following conclusions are carried out: (i) our investigated shell exhibits dynamics stability loss during a transition from regular to chaotic vibrations; (ii) we have detected the second type shell buckling associated with chaotic intermittency behavior.

References 1.

2.

3. 4.

A. C. Shian, T. T. Soong and D. S. Roth. Dynamic buckling of conical shells with Imperfection. AlAA Journal, 12(6):24-30,1974. B. Budiansky and D. S. Roth. Axisymmetric Dynamic Buckling of Clamped Shallow Spherical Shells. NASA TN D-151O, 597-606, 1962. B. Ya. Kantor. Nonlinear Problems of Theory of Non-Homogenous Shallow Shells. Naukowa Dumka, Kiev, 1971, in Russian. V. A. Krysko and A. V. Krysko. Problems of bifurcations and stiff stability loss in nonlinear theory of plates. Mechanics of Plates and

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V. A. Krysko et al.

Shells in XXI century. The Saratov Technical University Press, 5067, 1999, in Russian. 5. H. N. Chu. Influence of large amplitude on flexural vibrations of thin cylindrical shell. 1. Aerospace Sci. , 28(8):602-609, 1961. 6. J. L. Novinski. Nonlinear transverse vibrations of orthotropic cylindrical shells. A1AA J., 1(3):617-620, 1963. 7. D. A. Evensen. Some observations on the nonlinear vibrations of thin cylindrical shells. A1AA J, . 1(12):2857- 2858, 1963. 8. M. Amabili, F. Pellicano and M.P. Paidoussis. Non-linear vibration of simply supported circular cylindrical shells coupled to quiescent fluid. Journal of Fluids and Structures, 12:883-918, 1998. 9. V. A. Krysko, J. Awrejcewicz, E. S. Kusnetsova and A. V. Krysko. Chaotic vibrations of closed cylindrical shells in a temperature field . International Journal of Bifurcation and Chaos, 18(5):1515-1529, 2008. 10. V. A. Krysko, J. Awrejcewicz, E. S. Kusnetsova and A. V. Krysko Chaotic vibrations of closed cylindrical shells in a temperature field. Shock and Vibration, 15:335-343, 2008 .

147

The Application of Multivariate Analysis Tools for Non-Invasive Performance Analysis of Atmospheric Pressure Plasma V. J. Law!, J. Tynan 2, G. Byrne 2, D. P. Dowling2, and S. Daniels! ! Dublin City University, National Center of Plasma Science and Technology Collins Avenue, Glasnevin, Dublin 9, Dublin, Ireland 2 School of Electrical, Electronic and Mechanical Engineering, UCD Belfield, Dublin 4, Ireland (e-mail: [email protected]) Abstract: This paper describes the development and use of real-time non-invasive Multivariate Analysis tools for the performance monitoring of atmospheric pressure plasma. The MVA tools (acoustic spectrogram analysis, principal component analysis (PCA) and non-parametric cluster analysis (NPCA) are embedded within a LabVIEW software program. The software program is used to probe a parallel-plate atmospheric pressure process system. It is found that the acoustic frequency spectrum distribution provides a signature of the plasma mode of operation. The signatures are modeled as overtones of the fundamental drive frequency and combination signals (intermodulation distortion). Within these spectrums syncopated patterns are observed. The acoustic signatures are correlated with changing electrical parameters. Using appropriate scaling factors, PCA of the current and voltage waveform are used to generate data set clusters that are deterministic of the acoustic signals. Non-parametric cluster analysis is used to identify and classify the modes. Keywords: multivariate analysis, frequency analysis, principal component analysis, non-parametric cluster analysis, atmospheric pressure plasma, acoustic emission and electrical waveform identification.

I. Introduction The monitoring and control of plasma processes is essential during manufacture to ensure uniform and predictable treatments. Many tools are available for monitoring vacuum-based plasma systems; however, the use of atmospheric pressure plasma systems is increasing due to their ability to process large areas in a reel-to-reel manner [1, 2], or, where mobile hand-held plasma devices treat small area surfaces [3]. With this increase in atmospheric pressure plasma processing a new set of metrology tools must be developed to analysis the plasma processing conditions and enable real-time decision on input parameters to be made such that factors like coating thickness, morphology and chemistry can be adequately controlled. The plasma equipment used in these series of experiments was the SE-J 100 atmospheric pressure plasma system developed by Dow Corning. The SEll 00 is a stand-alone processing tool, which is capable of plasma treating and coating flexible polymer webs. It consists of two chambers, each measuring 320mm x 320 mm and associated web handling equipment through which the polymer web is passed. Using the frequency matching power supply, powers of up to 2000 W, or 1000 W per chamber, is applied.

148

V. 1. Law et al. He & 0 2

1'a"

"0

~ Qi

"0

c

:::>

eOJ

-

PET web -+

rollers

-

PET web -+

Figure 1: parallel-plate APP system.

This paper describes how freely available plasma generated acoustic emission (microphone position at the output of the chamber) and non-sinusoidal arbitrary current and voltage waveforms are captured and processed for real-time noninvasive performance analysis. The data is collected using a multivariate analysis tool kit, (developed in LabVIEW software) where the most critical parameters of the process are extracted and decisions on input parameters are made based on this data [4]. The process is monitored during plasma treatment of materials without affecting the actual operating conditions. 2. Multivariate analysis software The software is placed within a laptop computer. For ease of use, reduced equipment setup cost, and portability, the data collection probes (fast digitizers and microphones) are connected to the computer through USB ports and a inline microphone sockets. The MV A tools comprise a short-term Fourier transformation [5] to produce acoustic spectrogram waterfall, phase space analysis and principal component analysis (PCA) of electrical waveform data [4], and non-parametric cluster analysis packarge built from NI Vision Builder software [6] . The basic methodology of the Short-Term Fourier Transform (STFT) is to extract N samples of the acoustic signal by windowing and compute the DFT (Discrete Fourier Transform) of the signal without a significant spectral leakage. In this case a Gaussian window function is used because we are detecting a repetitive signal, rather than speech patterns. The window length is 4096. This program cuts the recorded sound into frames and rebuilds them into a block of n frames. Principal component analysis is performed on the current and voltage waveforms and dispalyed in the from of a Loading plot. The X-axis is the voltage and the Y-axis the current. The scores are construct from the rms values

Multivariate Analysis Tools for Non-lnvasive Performance Analysis

149

of current and voltage wavefoms plus the addition of the frequency in unit of kHz. This process is represented in equations 1 and 2. The high dimensional space of the PCA Loading plot is used to track the plasma performance. Current wavefonn

-+

Inns -+ Inns+ f == score

(1)

Voltage waveform -+ Vrms -+ V rms+ f == score

(2)

To perform NPCA, the Loading plot image is exported into NI Vision builder. This software contains a suite of hieratical particle measurement tools and is set to look for bright objects within a local threshold. A background correction method within the 'detect object step' is used. The local threshold algorithm works by calculating the intensity (I) of the pixel under consideration (I = 1) with its neighbouring pixels (In == 1 or 0, where 0 has no data point). For a data point that is sized as a single pixel this would mean that 8 neighbouring pixels would be examined. If the 8 neighbouring pixel have an I == 0, then the pixel under consideration will be classiJied as a cluster. The algorithm allows the local threshold window to be varied in both height and width in steps of one pixel. This process is schematically illustrated in figure 2. ,...--,---._-.--....--. Pixel (particle) under consideration

,~+--j}

1--+-__,

\.'-_ _

~

Local threshold w; odow

_____.1

y-

Pixel frame Figure 2: Schematic of window used in the local threshold algorithm.

3. SE-llOO Results Acoustic: In this section acoustic and PCA representations of the current waveforms are presented and the interpretation discussed. The data was taken from a helium plasma driven at -25 kHz and applied power of 500W. Visually, t11is plasma is characterised by a uniform although spatially non-homogeneous plasma, with some visible features called column micro discharge (CMD) present [4] . Figure 3 shows two acoustic spectrums for these plasma modes.

150

V. 1. Law et al.

Figure 3: plasma acoustic spectrum 500 W He plasma. The top spectrum con·esponds to tbe bottom band cluster in figure 4. The bottom acoustic spectrum coaesponds to tbe top cluster in figure 4.

A tone analysis of the acoustic spectrums reveals a number featmes. Firstly the fundamental drive frequencies are slightly different at (fa = 24.5 and 25 kHz, respectively) and hence their second overtones 49 and 50 kHz. Secondly the combinations (or, intennodulation distortion) is present but is most marked in figure 3b as compared to figure 3a. This should be is excepted as we are measuring sound energy, rather than electromagnet frequencies. Mathematically this can be expressed as fo ± fl-3 to generate a comb of sums and differences combination tones around fo and its second overtone. Table 1 and 2 reveal the result of this analysis for figure 3a and 3b. In Table 1, the tabulated results indicate a partial intermodulation with the difference being present for f~ and the sums present of the second overtone. Table 1: Intermodulation distortion analysis for figure 3a.

Figure 3a 24.5 kHz ±fl 24.5 kHz±h 49.5 kHz ±fl 49.5 kHz ±f2

fl orh or.f3 -3 kHz -5kHz -3 kHz -5 kHz

-f. -21.5 kHz -18.5 kHz weak weak

+fx -27.5 kHz No -53.5 kHz -54.5 kHz

Table 2: lntermodulation distortion analysis for figure 3b.

Figure 3b 25 kHz ±fl 25 kHz±h 25 kHz ±f3 50 kHz ±fl SOkHz±h 50 kHz ±.f3

.il orh orf3

-j~

+.f~

-1 kHz -4kHz -9.5 kHz -1 kHz -4kHz -9.S kHz

-24 kHz -21 kHz -15.5 kHz -49 kHz -46 kHz -40.5 kHz

No -29 kHz -34.5 kHz No -50 kHz -59.S kHz

Multivariate Analysis Tools for Non-Invasive Performance Analysis

151

Table 2 shows that the strong intermodulation distortion relationship. Indeed the distortion can be found for nearly all combinations up to fo ±il-3 Furthermore the displayed spectral densities for these frequencies are evident with respect to figure 3a. The measured sum and difference need to have a physical origin, generally this is thought to be a non-linear mechanism, and in this case the reactance of the plasma is the likely cause. PCA results: To find the deterministic link between the acoustic data and the plasma, PCA of the plasma generative current and voltage waveforms has been performed. Figure 4 shows the exploration of this link for the first minute of the 500 W helium plasma generation. The process starts at the bottom left and progresses to the top right with data points added at are rate of 1 point per 0.5 seconds. The red-line highlights the trajectory between the clusters. Here it is seen that two clusters are formed, the clusters elongate in the direction of the arrow and process time. It is observed that the acoustic emission spectral density profile is synchronised with the individual clusters. The bottom cluster corresponds to the acoustic emission in figure 3a and the upper cluster is associated with figure 3b. Under these conditions the two clusters are selfevident. However when the process is run at high powers the cluster number becomes complex making the classification and interpreted uncertain. To check that the clusters do not arise from the measurement software or from the multiplicity of chambers two tests have be performed. The software test used a dual function generator that produced similar input waveforms: the outcome being a signal cluster with a random spread of data point. The multi-chamber effect was eliminated as the cluster effect was found when only one chamber was used. With this knowledge a series of peA plots as a function of applied power steps (400, 700, 1000, 1100, 1300 and 1500 W) were made, the outcome of this experiment is show in figure 5. The data reveals the cluster number increases with through the 500 to 1300 W power ranges. Above this power the clusters begin to merge and visual inspection becomes difficult. At these high powers it known that the plasma is operating in the spatially homogeneous pesudoglow discharge mode. For further cluster analysis non-parametric cluster analysis was performed.

152

V J. La,\, d al.

Figure 4: peA loading plot of helium plasma driven at 500 W as a function of time.

iB 0

..

~7,5

los..

•

.:

•

'I,;···.

"'s ..

' .... 8 ... '

II

Voltage score range

Figure 5: Cluster formation for the helium plasma as a function of 6 power steps.

NPCA results: To match the captured duster data NPCA on the data sets have been performed. With the data images exported into Vision builder software, the local adaptive threshold window XY kernel was iterated so that 2 clusters at low power where registered and increase in number in the mid power range before falling to one or two clusters at high power. The best fit was found when the kemels equaJ; X = 17, and Y:= 17 pixels. The cluster fit is shown in figure 6. Here it is seen that the algorithm does reproduce to a high degree the visual perspective of the data. One or two outlying data points not registered can be attributed to transients that do not belong to anyone of the man Clusters.

.

Multivariate Analysis Tools for Non-Invasive Performance Analysis

.

~

,

153

iii

, ill

. .. - .....

400W (2)

700 W(2)

+

1000 W (3)

II

.

.,

I I

300 W (2)

.1..

.

.

,

1100W (4)

A

i

I

1500W (1)

.'

Figure 6: Cluster calculation of Helium plasma at: 400, 700, 1000, 1100, 1300 and 1500 W.

4. Conclusion This abstract has described the application of data-driven multivariate analysis (real-time frequency spectral analysis, PCA and NPCA) to atmospheric pressure plasma. Three mathematical tools were applied to the acoustic emission discharge and generative analogue current and voltage waveforms. The result is a linear transformation of the input data into high dimensional space that is visible to the operator as a function of process time. Using this format a relationship between the acoustic emission and electrical data is found . In the lower power mode (uniform although spatially non-homogeneous plasma) two alternating clusters with a time periods of -10 seconds are found . When scaling and the sampling rate are fixed, NPCA can be performed. The NPCA was performed using Euclidian distance, rather than angle measurement [7]. The reason for this approach is to enable the local threshold algorithm within the National Instruments LabVIEW Vision builder software to be employed with ease. A local threshold algorithm was utilised. It was found that the best fit to the applied power to cluster has a symmetrical kernel of X = 17: Y =17. The outcome of instrumentation tests and chamber geometry studies point to a plasma influence in the formation of the observed cluster partitioning. With regard to the input-output data relationship in real-time, the spatial orientation of the displayed data points will allow a surrogate model of the underlying physical process to be generated; this aspect of the work is under investigation.

154

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Acknowledgements This work is partly supported by Enterprise Ireland grant CDT17/IT/304 and the Science foundation Ireland under grant 08/SRC11141. References [1] B. Twomey, M. Rahman, O. Byrne, A. Hynes, L.-A. O'Hare, L. O ' Neill and D. Dowling. Effect of plasma exposure on the chemistry and morphology of aerosol-assisted, plasma-deposited coatings. Plasma Processes and Polymers. 5, 737-744, 2008. [2] Evaluation of real-time non-invasive performance analysis tools for the monitoring of pilot plant atmospheric pressure plasma system. 1. Tynan, V.I. Law, B. Twomey, A. M. Hynes, S. Daniels O. Byrne, D. P Dowling. Meas, Sci, Technol20, 115703,2009. [3] V.I . Law, V. Milosavljevic, N. O'Connor, 1. F. Lalor, and S. Daniels Hand-held flyback driven coaxial dielectric barrier discharge: development and characterization. Rev, Sci, Inst. 79(9) 094707-1-10, 2008. [4] V.I. Law, N. O'Connor, B. Twomey, D. P. Dowling and S. Daniels. Visualization of Atmospheric pressure plasma electrical parameters. Topics of Chaotic Systems: Selected Papers of Chaos 2008 International Conference. World Scientific Publishing 2009. [5] H. Wu, M . Siegel and P. Khosla. Vehicle sound signature recognition by frequency vector principal component analysis. IEEE Trans, Inst, Meas. 48(5) 1005-1009. 1999. [6] Information on National LabVIEW and Vision builder AI3 .5 can be found at httpll:www.ni.com [7] N. F. Thornhill, H. Melb0 and 1. Wilko Multidimensional visualization and clustering of historical process data. Ind . Eng. Chern. Res. 45, 5971-5985, 2006.

155

Chaos Communication: An Overview of Exact, Optimum and Approximate Results Using Statistical Theory Anthony 1. Lawrance Department of Statistics University of Warwick Coventry CV 4 7 AL, UK (e-mail: [email protected]) Abstract: This paper overviews exact, optimum and approximate decoding and performance results for antipodal chaos shift-keying (CSK) in which a bit is transmitted by modulating a chaotic segment and decoded by use of the corresponding unmodulated reference segment. Both single~ and multiple-user versions with both known- and transmitted-reference segments are considered, the so-called coherent and non-coherent cases. There are three main themes to the paper (i) the use of statistical likelihood theory for deriving optimum or improved decoders, (ii) the availability of mathematically exact theory for BER performance of decoders, (iii) qualitative statistical insights provided by simple Gaussian approximations to BER. Keywords: communication systems, chaos shift keying, correlation decoding, likelihood-based decoding, exact calculation of bit error rate, Gaussian approximations, statistical theory.

1. Introduction Chaos-based communication involves using segments of chaotic waves to carry messages rather than the traditional sinusoidal waves. There are perceived advantages of chaos-based communication in terms of spread spectrum, security and robustness. Research activity in the area can be traced back to before 2000, but the set of papers in a special issue of the IEEE proceedings, Hasler, Mazzini et aI., 2002[3], was definitive at the time. Since then there has been much activity in the journals, a pair of linked monographs, Lau and Tse, 2003[9], Tam, Lau et aI., 2007[16], and three edited collections of chaos applications in telecommunications, Kennedy, Rovatti et aI., 2000[5], Stavroulakis, 2006[14] and Larson, Liu et aI., 2006[8]. The aim here is to give an overview of exact, optimum and approximate results from the point of view of statistical theory. There is much engineering leverage to be obtained from this perspective. Earlier results were transfered from conventional communication systems or derived with simple Gaussian approximations which did not fully incorporate the new joint chaotic-statistical interplay and sometimes tended to under-represent or over-complicate theoretical matters. A particular feature of chaos communications systems is the interplay of dynamical and statistical behaviour, and more particularly, the interplay of the

156

A. 1. Lawrance

x2 =(x2i l

Channel

x, =(x"l

Figure I. Block diagram of multi-user chaos shift-keying communication system.

statistical aspects of chaotic spreading and channel noise. The exact results give useful insights as well as opening computational paths, while the reworked and additional Gaussian approximations, although inaccurate, direct attention to key quantities determining performance. Modelling in the paper is restricted to employing discrete rather than continuous time, and thus is discrete time baseband. The models used are mathematical extractions of real systems, without electronic intricacies, but never-the-Iess allow theoretical investigation of performance and design issues.

2. Chaos Shift Keying Communication Chaos shift-keying (CSK) is perhaps the most developed form of chaosbased communication, probably inspired by earlier spread spectrum systems, and can be traced back to Dedieu, Kennedy et aI., 1993[2] and Kocarev and Parlitz, 1995[6]. An engineering perspective on the area is given in the monographs by Lau and Tse, 2003[9] and Tam, Lau et aI., 2007[16]. To appreciate the structure of a CSK communication system, the block diagram in Figure I illustrates for L-users. The binary messages hi = ±l,l = 1, 2, .. ,L are to be simultaneously transmitted in a memoryless channel by the L users to L receivers. The focus is on a single active user, in the presence of interference by other-users. To transmit a binary message, user I has a chaotic generating map rO defined over (-c, c) , such as logistic or Bernoulli-shift, which provides a spreading segment, given by the row vector xJ = (xlI ,···, XIN ) ,where X l t --

r ( X lt _ 1 )

--

) r (I - I ) (XII'

r (I) (.) -- r { r ( t-I) (.) },

t -1 - " 2 ... , N .

(1)

The binary bit hi is multiplied into each value of the spreading segment and is then transmitted as the message segment, hi XI j ' j = 1,2, ... , N . All the L bit-modulated spreading segments are transmitted additively in the same time slot and arrive at all receivers but with additive individual

Exact, Optimum and Approximate Results Using Statistical Theory

channel noise, with variance

Cli ,

a;.

157

j = 1,2, ... , N for the til user; this is white Gaussian noise

In the simplest model, described as coherent, additionally

the individual spreading segment of the til user is available without noise at the [," receiver, either by chaotic synchronization or by identical chaotic generation, and called the reference segment. This segment is compared with message segment in order to decode bl , or in statistical language, to estimate bl . In the non-coherent or differential case, the spreading segments are separately and additionally transmitted and are received with channel noise or synchronization error. The main motivation of all the subsequent modeling is calculation of the bit error rate. For the [," and active-user transmitting a single bit, the message segment containing the bit information arriving at the [," receiver is L

'ii

= blxli

+

I

bkxki

+ c li ' j = 1,2, ... , N

(2)

b'I=1

for I = 1,2, ... , L. The summation term represents interference by the message bits of other users. The decoding of the bit bl in the received [," message segment is done in conjunction with knowledge of the t" spreading reference segment, but without knowledge of the other spreading segments at the receiver. The message bit bl is treated as a statistical parameter and the optimum decoder is the likelihood-based estimate of bl ; on the other hand, the bits and spreading segments of the other users are not of interest to the t" user and are treated as unknown random variables. A statistically-based view of bit error rate is that it is the sum of the conditional probabilities of decoding a +1 as -1 and vice versa weighted by the ± 1 transmission proportions. For most systems and decoders, by symmetry the conditional probabilities are equal, and so no weighting is necessary.

3. Single-User CSK: Likelihood Decoding

Developing the single-user case L = I is a useful way to unfold the primary approach without the complication of interference by other users; the suffix I = 1 is henceforth omitted in this case. The received message segment is, by taking L = I in (2), (3)

With the Gaussian white noise channel, the likelihood of the transmitted bit b and the unknown noise variance is obviously

a;

1 N } l(b,a;lr)=(aE5) N exp { -~I(ri-bxy .

(4)

2a, ]=1 Maximum likelihood estimation implies choosing the value of b which has the greater likelihood, and the appropriate likelihood ratio simplifies as

158

A. 1. Lawrance N

LX/ =C j

(5)

rx '

j~ 1

Hence b , the maximum likelihood estimate of b, is given by ,

,

0 => b = + 1, C rx < 0 => b = -I . (6) This is known as a correlation decoder. From a statistical point of view, it is proportional to the standard estimate of the regression coefficient b in the line through the origin (3); the twist is that the regression coefficient can only take one of the two values ±1. Thus, the form (6), positive if b is +1 , negative if r; is -1 , is intuitively sensible; it does not depend on the type of spreading, chaotic or not, but is conditional on spreading. C rx

~

4. Single-User CSK: Exact and Approximate Bit Error Rate This is particularly a topic where careful statistical argument yields simple insightful and computational results which were not otherwise available, first derived in the author's paper Lawrance and Ohama, 2003[11]. The emphasis is on an exact result for the bit error rate. Unlike the approximate results quoted in several key early papers, such as Kolumban, Kennedy et aI., 2002[7], Sushchik, Tsirnring et aI., 2000[15], which strangely did not involve the type of spreading, the exact result involves all aspects of the system, in particular, the bit energy sum-of-squares. Interestingly, its lower bound for extensive spreading yields the corresponding BPSK result. By involving the spreading, the exact result gives a pathway to design of spreading for maximum performance, Yao, 2004[18] and Papamarkou Chaos, 2009[1]. Following earlier remarks in Section 2, the overall bit error rate can be taken as the conditional probability BER = P ( r; = -11 b = 1)

(7)

and so by (6) - (7), BER = p(Crx <

0Ib = 1) = p(~ Ex + L..J ~ ~ j~1

)

J

X2 J

<

0)

(8)

j~1

which involves both the statistical noise and the chaotic spreading as random variables. In the exact approach, this probability is evaluated exactly, while in the simple Gaussian approximation approach (SGA), the random variables in the probability are first approximated with a Gaussian distribution. The exact approach begins by conditioning on the spreading values and using the independent Gaussian distributions of the E j and the Gaussian distribution of their linear combination in (8); then continues by averaging over the chaotic spreading segments. The exact BER result becomes

Exact, Optimum and Approximate Results Using Statistical Theory

159

(9) where E, is the expectation over the chaotic spreading segments, marginally with means zero and variances O"~, (-) is the cumulative distribution function of a standardized Gaussian distribution, SNR is the signal-to-noise ratio defined as N a~ and the second part of the square root is the standardized bit energy. In order to calculate BER the expectation in (9) has to be evaluated numerically. In the case of chaotic spreading this requires a one-dimensional integral; with chaotic map 1'0 according to (I) and its invariant distribution cpO , the result (9) becomes

/a: '

L{,+

SNR

~

," "

(x)' /

NO";

J}9'(X)dx

(10)

This can be exactly numerically evaluated in cases where the map has a explicit and computable iterated form. Illustrations of BER as a function of SNR are given in Figure 2 for logistic, Bernoulli-shift and Gaussian spreading. Logistic is better than Bernoulli-shift and Gaussian is by far the worst. For large N, the standardized bit energy term in (9) converges to I, leaving the result for extensive spreading as lim N~oo

BER = <1>(-.) SNR) .

(11)

This result is in fact the lower bound of (9) for all N, the proof being from applying Jensen's expectation inequality to (9), see Figure 2. The exact result can be used until the result (11) is acceptably accurate. Note that the spreading limit (11) and lower bound do not depend on the type of spreading, chaotic or not. In early work, (11) was used as the exact BER result. It is exact for binary phase shift-keying (BPSK) in which spreading takes one of two oppositely signed values with fixed probability and thus bit energy is constant. This has led to the view that CSK cannot improve on BPSK, but it is very different, e.g., with the advantages of a spread spectrum and security. When the chaotic spreading map does not have an explicit iterated form, exact numerical computations from (9) are intractable. The approach suggested in Lawrance and Ohama, 2005[12] was to approximate the bit energy sum of squares by a chi-squared distribution with matching mean and variance. In current work, Kaddoum, Charge et a!., 2007[4], Chaos, 2009[1], have developed an approach in which the bit energy distribution is either simulated or matched by a Rician distribution, leading to analytical results. The SGA approach when applied to (8) yields (12)

160

A. 1. Lawrance

as first derived in Lawrance and Balakrishna, 2001 [10]. While this result is seriously inaccurate, partly shown by its incorrect limiting form as SNR --7 00 , the roles of kurtosis and quadratic autocorrelation of the spreading are revealed. They are illustrated by Figure 2 in that logistic spreading with its zero quadratic autocorrelations is better than Bernoulli-shift spreading with its geometric one-quarter quadratic autocorrelations. The corresponding SGA for the non-optimum correlation decoding in non-coherent CSK is

{

2 N BER=-¢ - [ - + - 2 SNR SNR

+N-I~{1+2L. (I __N ri"x, (j)}] I) .(13) (72,

N-I

(7,

;=1

k

-2

0.1

0.01

~

O.cXJl

-5

5 SNl(1II)

10

15

Figure 2. BER plotted against SNR in dB for coherent CSK and Jensen lower bound. Comparison of logistic (long dashes), Bernoulli-shift (short dashes), and Gaussian (dot-dashes) spreading, spreading factor N = 5.

Both (12) and (13) give statistical structure and simplicity to the computational results in Lau and Tse, 2003[9]; the additional terms in (13) indicate the loss in performance due the transmission of reference segments in place of their exact replication at the receiver. An exact BER result for correlation decoding in single-user non-coherent CSK is given in Lawrance and Ohama, 2003[11] in terms of non-central F-distributions, and optimal likelihood-based decoding is has been developed by T. Papamarkou, Chaos, 2009[1]. An alternative direction for the non-coherent case avoids the transmission of a reference segment by use of a particular chaotic attractor which itself carries the necessary demodulation information; this approach is currently being developed, Xu, Charge et al., 2008[17], Chaos, 2009[1].

5. Multi-User CSK: Coherent and Non-Coherent In any multi-user communication system the new feature relative to the corresponding single-user system is the interference of other users when decoding a bit, and the effect this has on the BER. As far as an individual user is concerned, the other-users constitute additional non-Gaussian noise. A

Exact, Optimum and Approximate Results Using Statistical Theory

161

correlation decoder could still be used but would not be as good in BER terms as a likelihood-based decoder, as will be seen. The likelihood decoder of the multi-user CSK system requires a careful analysis which significantly departs from that of the single-user case, Lawrance and Yao, 2008[13], and leads to the generalized correlation decoder

, {+l

b = I

x;I.~lr>O

-1

(14)

otherwise

where Lx is the circulant autocorrelation matrix of the spreading segment. There is a pleasing statistical simplicity to this result from its obvious generalization of the correlation decoder; it comes from a Gaussian approximation to the likelihood. A key new point is that the decoder now makes use of any autocorrelation in the spreading. In the interests of brevity, theoretical developments leading to multi-user BER results are omitted; they involve conditional Gaussian approximation (CGA) and give near exact results. Accurate Gaussian approximations are made during calculation of bit error probability which are conditional on spreading. Concise expressions can be obtained for both correlation decoders and generalized correlation decoders, and have been given in Lawrance and Yao, 2008[13] for coherent CSK. For the generalized correlation decoder, its bit error has been derived as BER ged

1[

= Ex

-

..

[() xTI.~lx

( X/I.-Ix x / NC7x) 2

xTI.-II.- IX "

I SNR

1 SIR

--+-

]_1])

(15)

where SIR is a new quantity, the 'spreading to other user interference ratio' (SIR), defined as N/ (L - 1) ; this will be important in all multi-user results. For the ordinary correlation decoder there is correspondingly

BER",

~ H+[- C~:~~~ ;~J(x:~:.) S~R +

S;R

r,]}

(16)

The advantage of the generalized correlation decoder is clear from Figure 3. When spreading is uncorrelated, as it is for logistic spreading, both (15) and (16) become much simpler in terms of standardized bit energy as BERUllcor=E . {[_ (XTX/NC7 2 ,d

,

x

)(_1 SNR

+_1

SIR

)-1]1

(17)

and consequently have the lower bound uIlcor ;::::= BERsprdg


(18)

The result may be a lower bound over correlated spreading as well , as suggested in Figure 3, but this has not been proved. Note also that the BER

162

A. J. Lawrance

0.126 0.100 0.079

10 SN«d8)

·5

15

20

25

Figure 3. Three-user (L = 3) coherent eSK with spreading factor N = 4 and SJR=2. BER comparison of SGA approximation for correlation decoding with Bernoulli-shift spreading (long-short dashes), with near exact eGA results for Bernoulli-shift spreading with correlation decoding (long-dashes), Bernoulli-shift spreading with optimum decoding (short-dashes), Jensen lower bound for uncorrelated spreading (solid curve) and universal lower bound ( horizontal line at 0.(7865 ).

curves in Figure 3 level out rather than continue to decrease; this effect is evident from (15) - (18) by setting SNR to infinity, effectively saying there is only interference noise. There are also SGA results for multi-user CSK with correlation decoding which are informative as to the key statistical quantities involved; under less general assumptions and presented more computationally, they have been used in the approach adopted by Lau and Tse, 2003[9] and Tam, Lau et at., 2007[16]. In the coherent case, (]"2, { N-I ( k} } I BER::::.cf> { - [ --+N - I~ 1+2L 1-- ,, (k) +

SNR

(]",

N

k =1

_1 {I +2I(I-~){Px (k)}2}]-~} SIR

(19)

N

k=1

and in the non-coherent case

N

k} }+

2 (]" 2, { N-I( BER::::.cf> { - [ -+--+N-I~ 1+2L 1--

SNR

SNR 2 N

SNR.SIR

(]":

+ {1+2~(11 SIR

N

k =1

N- I

, (k)

x-

k

N }Px(k)} 2}]

I}

-2

. (20)

The aspects which are not involved in the corresponding single-user results (12) and (13) are SIR and linear autocorrelations, and both are seen to increase BER, an increase being intuitively inevitable because of

Exact, Optimum and Approximate Results Using Statistical Theory

163

interference. The results (19) and (20) may insightfully be compared, with the extra terms in (20) suggesting increased BER, again intuitively inevitable. Exact results for correlation decoding are much more complicated than SGA results, and involve the distribution function of a doubly non-central-F random variable; some computational progress has been made in Yao and Lawrance, 2006[ 19]. SGA results for generalized correlation decoding have not yet been developed.

6. Conclusions This paper has been a selective and compacted overview of performance modelling in chaos shift-keying communication. The emphasis has been to present mathematical staistically-based exact and approximate structural results which are useful for computation and insightful for qualitative understanding. The paper could not cover the whole field, even for chaos shift-keying, and hence topics such as optimality in choice of chaotic map, interference jamming, optimality in spreading factor for non-coherent systems, and laser-based chaotic systems, were omitted. Some of these and others too may be covered elsewhere in Chaos, 2009[1] .

References [I] [2]

[3]

[4]

[5] [6]

[7]

Chaos, Presented at The 2nd Chaotic Modelling and Simulation International Conference, Chania, Crete, Greece, 2009. H. Dedieu, M.P. Kennedy and M. Hasler, "Chaos shift keying: modulation and demodulation of a chaotic carrier using selfsynchronizing Chua's circuits," IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, vol. 40, pp. 634-642, 1993. M. Hasler, G. Mazzini, M. Ogorzalek, M. Rovatti and G. Setti, "Scanning the special issue: Applications of nonlinear dynamics and information engineering," Proceedings of the IEEE, vol. 90, pp. 10, 2002. G. Kaddoum, P. Charge, D. Roviras and D. Fournier-Prunaret, "Comparison of chaotic sequences ina chaos based DS-CDMA system," in International Symposium on Nonlinear Theory and its Applications (NOLTA). Vancouver, Canada, 2007. M.P. Kennedy, R. Rovatti and G. Setti, Chaotic Electronics in Telecommunications. London: CRC Press, 2000. I. Kocarev and U. Parlitz, "General approach for chaotic synchronization with application to communication," Physical Review Letter, vol. 74, pp. 5028-5031,1995. G. Kolumban, P. Kennedy, Z. lako and G. Kis, "Chaotic communications with correlator receivers: theory and performance limits," Proceedings of the IEEE, vol. 50, pp. 711-732, 2002.

164 [S]

[9] [10]

[11]

[12]

[13]

[14] [15]

[16] [17]

[IS]

[19]

A.l. Lawrance

L.E. Larson, J.-M. Liu and L.S. Tsirnring, "Digital Communications Using Chaos and Nonlinear Dynamics." New York: Springer, 2006. F.C.M. Lau and c.K. Tse, Chaos-based Digital Communications Systems. Heidelberg: Springer Verlag, 2003. AJ. Lawrance and N. Balakrishna, "Statistical aspects of chaotic maps with negative dependence in a communications setting," Journal Royal Statistical Society, Series B, vol. 63, pp. 843-853, 200l. A.J. Lawrance and G. Ohama, "Exact calculation of bit error rates in communication systems with chaotic modulation," IEEE Trans. Circuits Systems I: Fund. Theory Appl., vol. 50, pp. 1391-1400,2003. A.J. Lawrance and G. Ohama, "Bit error probability and bit outage rate in chaos communication," Circuits, Systems, and Signal Processing, vol. 24, pp. 519-534,2005. A.J. Lawrance and J. Yao, "Likelihood-based demodulation in multiuser chaos shift keying communication," Circuits, Systems, and Signal Processing, vol. 27, pp. 847-864, 2008. P. Stavroulakis, Chaos Applications in Telecommunications. New York: CRC Press, 2006. M. Sushchik, L.S. Tsirnring and A.R. Volkovskii, "Performance analysis of correlation-based communication schemes utilizing chaos," IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, vol. 47, pp. 1684-1691,2000. W.M. Tam, F.C.M. Lau and c.K. Tse, Digital Communications with Chaos. Oxford: Elsevier, 2007. Y. Xu, P. Charge and D. Fournier-Prunaret, "Chaotic cyclic attractors shift keying," in International Conference Neural Networks and Signal Processing. Zhenjiang, China, 2008, pp. 62-66. J. Yao, "Optimal chaos shift keying communications with correlationa decoding," in IEEE International Symposium on CIrcuits and Systems (ISCAS2004), vol. IV. Vancouver, Canada, 2004, pp. 593-596. J. Yao and AJ. Lawrance, "Performance analysis and optimization of multi-user differential chaos shift-keying communication systems," IEEE Transactions on Circuits and Systems-I: Regular Papers, vol. 53, pp.2075-2091,2006.

165

Dynamics of a Bouncing Ball Shiuan-Ni Liang and Boon Leong Lan School of Engineering, Monash University 46150 Bandar Sunway, Selangor, Malaysia Abstract: The dynamics of a bouncing ball undergoing repeated inelastic impacts with a table oscillating vertically in a sinusoidal fashion is studied using Newtonian mechanics and general relativistic mechanics. An exact mapping describes the bouncing ball dynamics in each theory. We show, contrary to expectation, that the trajectories predicted by Newtonian mechanics and general relativistic mechanics from the same parameters and initial conditions for the ball bouncing at low speed in a weak gravitational field can rapidly disagree completely. The bouncing ball system could be realized experimentally to test which of the two different predicted trajectories is correct.

1. Introduction For dynamical systems where gravity does not playa dynamical role, it is conventionally believed [1-4] that if the speed of the system is low (i.e., much less than the speed of light c), the trajectory predicted by special relativistic mechanics is well approximated by the trajectory predicted by Newtonian mechanics from the same parameters and initial conditions. This conventional belief was however recently [5-7] shown to be false. In particular, it was shown that the two predicted trajectories could rapidly diverge and bear no resemblance to each other. For dynamical systems where gravity does play a dynamical role but only weakly (i.e., gravitational potential « c2 [8]), it is believed that the trajectory predicted by general relativistic mechanics for a slowmoving dynamical system is well approximated by the trajectory predicted by Newtonian mechanics from the same parameters and initial conditions. For instance, according to Einstein [1]: If we confine the application of the theory [general relativity] to the case where the gravitational fields can be regarded as being weak, and in which all masses move with respect to the co-ordinate system with velocities which are small compared with the velocity of light, we then obtain as a first approximation the Newtonian theory. Ladipus [9] also wrote: For weak fields and low velocities the Newtonian limit is obtained. We show in this paper with a counterexample dynamical system that this conventional belief is also false.

166

S.-N. Liang and B. L. Lan

2. Bouncing ball Our counterexample dynamical system is a bouncing ball [10,11] undergoing repeated impacts with a table oscillating vertically in a sinusoidal fashion with amplitude A and angular frequency OJ . The impact between the ball and the table is instantaneous and inelastic, where the coefficient of restitution a (0::; a < 1 ) is a measure of the energy lost of the ball at each impact. The table is not affected by the impact because the table's mass is much larger than the ball's mass. In between impacts, the ball moves in a uniform gravitational field since the vertical distance travelled is much less than the Earth's radius. Following [10,11], we will use the ball's velocity v and the table's phase just after each impact to describe the motion of the bouncing

e

ball. The table's phase

e is given by (OJ[ + eo)

modulus 2n. We will

refer to the table's phase just after each impact as the impact phase. In the Newtonian framework, the dynamics of the bouncing ball is [10,11] exactly described by the impact phase map

A[sin(Bk )+l]+Vk[~(Bk+I-Bk)]-lg[~(Bk+I-Bk)r -A[sin(Bk+I)+l]=O (1)

and the velocity map V k"

= (1 + a)m cOS(ek+ l ) -

a{ vk - g[ ~ (e,+, - ek )]}

(2)

where g is the acceleration due to gravity. In the general relativistic framework, the dynamics of the bouncing ball is (our derivation will be given elsewhere) exactly described by the impact phase map

~{~{I+sin[2Bkg (Bk+l-Bk)+E>k]}-I}-A[sin(Bk+l)+I]=O 2g 2Bk cOJ (3)

where

Dynamics of a Bouncing Ball

167

fJk =2 ' C

and the velocity map

- c2

a[ C

V k+l

J+

V'k+l-Uk+l

2,

U k+l

- V k+l U k + 1

= -----[~-------'------,--

1- au

V'k+l-Uk+l 2 , C - V k+1 U k +1

k+l

J

(4) where Uk+l

= AU) cos (Bk + 1 )

is the table's velocity just after the (k+l)th impact, and ,

V k+l

C [2B g ( Ok+l = --cos - -k -

2Bk

-

Ok

) + 0] k

CO)

is the ball ' s velocity just before the (k+ 1)th impact. Our derivation of the general relativistic map [Eqs. (3) and (4)] for the bouncing ball, which is similar to the derivation [10] of the Newtonian map [Eqs. (1) and (2)], utilized Ladipus' result [9] for the general relativistic motion of a body in a uniform gravitational field. The impact phase maps, Eq. (1) and Eq. (3), which are implicit algebraic equations for

0k+l'

must be solved numerically by finding the

zero of the function on the left side of the equation given

Ok

and v k

.

We use the izera function in MATLAB for this purpose. Numerical accuracy of the solutions was carefully checked by varying the tolerances.

168

S.-N. Liang and B. L. Lan

3.

Results

In the example given here, the parameters of the bouncing ball system are: g = 981 cmls2, C = 2.998x103 cm/s (we have to use an artificially smaller c value for accurate numerical calculation of the

= 60 Hz, table's coefficient of restitution a = 0.5 . The

general relativistic map), table's frequency (OJ/2;r)

amplitude A = 0.012 cm, and initial conditions are 8.17001 cmls for the ball's velocity and 0.12001 for the normalized impact phase (i.e., impact phase divided by 2n.

The Newtonian and general relativistic trajectories are plotted in Figure 1. The figure shows that the two trajectories are close to each other for a while but they are completely different after 21 impacts although the ball's speed and table's speed remained low (about 1O-3c) and the gravitational field is weak (gravitational potential gy is about 1O-6c2). We have carefully checked that the breakdown of agreement between the two trajectories is not due to numerical errors.

4.

Conclusion

For a slow-moving dynamical system where gravity is weak, we have shown, contrary to expectation (see, for example, [1,9]), that the trajectories predicted by Newtonian mechanics and general relativistic mechanics from the same parameters and initial conditions could rapidly disagree completely. The bouncing ball system could be realized experimentally [10-12] to test which of the two completely different trajectory predictions is physically correct.

References 1.

2. 3. 4.

5. 6.

7.

A. Einstein, Relativity: the special and the general theory, Random House, New York, 1961. J. Ford and G. Mantica, Am. 1. Phys. 60, 1086-1098 (1992). W. D. McComb, Dynamics and relativity, Oxford University Press, Oxford, 1999. J. B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Addison-Wesley, San Francisco, 2003. B. L. Lan, Chaos 16,033107 (2006). B. L. Lan, in Topics on chaotic systems: selected papers from Chaos2008 international conference, eds. C. H. Skiadas, I. Dimotikalis, and C. Skiadas, World Scientific, 2009, pp. 199-203. B. L. Lan, Chaos, Solitons and Fractals 42, 534-537 (2009).

Dynamics of a Bouncing Ball

169

8.

P. Davies, in The new physics, ed. P. Davies, Cambridge University Press, 1992, pp. 1-6. 9. I. R. Ladipus, Am. 1. Phys. 40, 1509-15lO (1972). lO. N. B. Tufillaro, T. Abbott, J. Reilly, An experimental approach to nonlinear dynamics and chaos, Addison-Wesley, California, 1992. 11. N. B. Tufillaro, T. M. Mello, Y. M. Choi and A. M. Albano, 1. Physique 47,1477-1482 (1986). 12. N. B. Tufillaro and A. M. Albano, Am. 1. Phys. 54,939-944, 1986.

I

.

'.

.'

0.8

'

II.)

'" 0.6 ~

...

Po

<J

~ 0.4

.§

impact

.~

~~

..' .

10

A '.

'.

'.

'

'

8

'.

<J

o

Q) >

7

", I

I' I I

.' I I

III

,

11

Il

t:.

.' t

I I

I

11

4 L-______________- 4_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

o

10

20

30

40

~

50

impact Figure 1. Comparison of the Newtonian and general relativistic trajectories: normalized impact phases (top) and velocities (bottom). Newtonian (general relativistic) values are plotted with triangles (diamonds).

170

Symmetry-Break in a Minimal Lorenz-Like System Valerio Lucarini I and Klaus Fraedrich2 1

2

Department of Mathematics, University of Reading Whiteknights, PO Box 220, Reading, RG6 6AX, UK Department of Meteorology, University of Reading Earley Gate, PO Box 243, Reading, RG6 6BB, UK Department of Physics, University of Bologna Viale Berti-Pichat 6/2 40127 Bologna, Italy (e-mail: [email protected]) Meteorologisches Institut, KlimaCampus, University of Hamburg Grindelberg 5,20144 Hamburg, Germany

Abstract: Starting from the classical Saltzman 2D convection equations, we derive via spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled, generalized Lorenz system. The consideration of this process breaks an important symmetry, couples the dynamics of fast and slow variables, and modifies the structural properties of the attractor. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of 0(Ec·· 1) is introduced in the system. Moreover, the system is ergodic and hyperbolic, the slow variables feature long term memory with 11/ 312 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics.

1. Introduction The Lorenz system [1] has a central role in modern science as it has provided the first example of low-dimensional chaos [2], and has literally paved the way for new scientific paradigms. The Lorenz system can be derived with a minimal truncation of the Fourier-modes projection of the 2D Boussinesq convection equations introduced by Saltzman [3], where a specific selection of the spatial symmetry of the fields is considered. Extensions of the Lorenz system taking into account higher-order spectral truncations in the 2D case have been presented, see e.g. [4,5,6], whereas in [7] the standard procedure has been extended to the 3D case. The mathematical properties of the Lorenz system have been the subject of an intense analysis; For several classic results, see [8]. Recently, at more theoretical level, the investigation of Lorenz-like systems has stimulated the introduction of the family of singular hyperbolic systems as extension of the family of hyperbolic systems [9]. Moreover, moving from the theory developed for non-equilibrium systems [10-12], in a recent paper a careful verification on the Lorenz system of the Kramers-Kronig dispersion relations and sum rules has been performed [13]. In spite of its immense value, the Lorenz system does not provide an efficient representation of several crucial phenomena typically associated to

Symmetry-Break in a Minimal Lorenz-Like System

171

complex, chaotic systems. When looking at finite time predictability properties, the Lorenz system features an unrealistic return-of-skill in the forecast, i.e. there are regions in the attractor within which all infinitesimal uncertainties decrease with time [14]. Additionally, the Lorenz system does not feature an interplay between fast and slow variables, so that it cannot mimic the coupling between systems with different internal time scales. Palmer [15] introduced artificially ad hoc components to the Lorenz system in order to derive a toy-model able to generate low-frequency variability. Moreover, as discussed in [16], the Lorenz system does not allow for a closed-form computation of the entropy production. In this work, we wish to propose a minimal dynamical system, possibly with some paradigmatic value, able to overcome some of the limitation of the Lorenz system, and endowed with a much richer dynamics. Starting from the 2D convection equations, we derive with a truncation a la Lorenz a minimal 10 d.oJ. ODEs system which includes the description of the thermal effect of viscous dissipation. We discuss how, while neglecting this process lead us to a dynamical system which includes a decoupled (generalized) Lorenz system, its consideration breaks an important symmetry of the systems, couples the dynamics of fast and slow variable, with the ensuing modifications to the structural properties of the attractor and in the spectral features of the system.

2. Rayleigh-Benard convective system We consider a thermodynamic fluid having kinematic viscosity V, thermal conductivity k, thermal capacity at constant volume Cl" and linearized equation

P = Po (1- cff). The 2D (x, z) Rayleigh-Benard top-heavy convective system confined within z E [0, H] can be completely described, when adopting

of state

the Boussinesq approximation, by the following system of PDEs:

d,V2V+J(V,V2V)= gad/J + vV 4V d/J+J(V,e)= I1T dxV+kV2e+~diiVdijV H

where

g

is

(la-lb)

C, . .

the

gravity

acceleration,

T = To - zl1T/ H + e , with H uniform depth of the fluid and I1T imposed temperature difference. The suitable boundary conditions in e(x, z = 0) = e(x, z = H) = 0 and the case of free-slip system are

V(x, z = 0) = V(x, z = H) = V 2V(x, Z = 0) = V 2V(x, Z = H) = 0 for all values of x. These PDEs can be non-dimensionalized by applying the linear transformations

x=Hx, z=HZ, t=H 2 /kt, v=k¥/,and e=kv/(gaH 3 )B:

V. Lucarini and K. Fraedrich

172

ai V2v/ + '(v/, V2v/)= ciJ/)+ cfi4v/

(2a-2b)

ai)+ '(Iii, V2B+ aEca,v/a,v/' t r B) = Ra.v/+ x IJ

I)

where fJ=v/k is the Prandl number, R=gaH 3f:,.T/(kv) is the Rayleigh number,

and

Ec =

e /(C/y..TH

2)

is

the

Eckert

number,

and

the

B(x, z= 0) = B(x, z= 1) = 0 and v/(x, z = 0) = v/(x, z= 1) = V2v/(x, z= 0) = V2v/(x, z= 1) = O. The Eckert number

suitable

boundary

conditions

are

in Eq. 2b quantifies the impact of viscous dissipation on the thermal balance of the system. As this number is usually rather small in actual fluids, the corresponding term in the 2previous set of PDEs is typically discarded. We perform a truncated Fourier expansion of

x

z

v/

and

B,

assuming that they are

periodic along the and with periodicity of 2/ a and 2, respectively. When boundary conditions are considered, we then derive a set of ODEs describing the temporal evolution of the corresponding (complex valued) modes 'Pm . and 1I

Om which are associated to the wave vector R = (J/Zln,mn). See [3] for a detailed derivation in the case Ec = 0, considering that the sign of the term proportional to R is wrong in both Eqs. 34 and 35. In the case Ec = 0, the seminal Lorenz system can be derived by severely truncating the system, considering the evolution equation for the real part of 'Pl.l and for the imaginary II '

parts of

°

0 ,2

°

1,1

and

°

0 ,2'

and performing suitable rescaling (see below). While

has no real part because of the boundary conditions [3], neglecting the

imaginary (real) part of 'Pl.l (° 1,1) amounts to an arbitrary selection of the phase of the waves in the system. An entire hierarchy of generalized Lorenz models, all obeying to this constraint, can be derived with lengthy but straightforward calculations. See, e.g., [4-6] for a detailed discussion.

3. Symmetries of the extended Lorenz system In this work we include the modes 'PI,I' 'P2,2' 01.1' 02,2 in our truncation, and retain both the real and the imaginary parts, whereas the considered horizontally symmetric modes 00,2 and 00.4 are, as mentioned above, imaginary. Finally, we assume, in general, a non vanishing value for Ec. If we define 'PI,I = a(X I+iXJ, O1,1 = f3(~ +iyJ, 'P2,2 = a(AI +iAJ, 0 2.2 = f3(B I+iBJ,

0 0,2=iJi.lI'

0 0 ,4=iJi.l2'

and

f3 = JZ'3 (a 2 + 1) j {2.J2a 2 ),

r=

JZ'3

following system of real ODEs:

with

a={a 2 +1)/{2.J2a),

(a 2 + 1JI{2a 2 )' set a = I/.J2

[1], we derive the

Symmetry-Break in a Minimal Lorenz-Like System

173

XI = O"Y2-(JX I

X2 =

~

-(J~

-(JX 2

= X 2 Z 1 -rX 2 -~ +s/(.J2;r)(JEc(AI X I +~X2)

1'2 = - XIZI + rX I - Y2 + s/(.J2;r )oEc(A2X I - AI X 2)

AI = (J/2 B2 -

4(JAI

(3a-3j)

,12 = -(J/2BI -4(JA2

BI = 4A2Z 2 - 2rA2 -4BI -1/(2.J2;r )oEC(XI2 - xi) B2 = -4AIZ 2 + 2rAI - 4B2 -1/(.J2;r )(JEcX I X 2

X2~

ZI = X I Y2 -

-bZ I

Z2 = 4AIB2 -4A2BI -4bZ2

where the dot indicates the derivative with respect to r=;r2(a 2 +1}=3/2;r2 t

r = R/ Rc

= R/(27;r4 /4)

,

is the relative Rayleigh number, and b is a geometric

factor. Note that if we exclude the faster varying modes

tp2,2

and O 2,2 in our

truncation (as in the Lorenz case) the Eckert number is immaterial in the equations of motion, the reason being that viscous dissipation acts on small spatial scales. Therefore, the ODEs (3a-31) provide the minimal system which includes the feedback due to the thermal effect of viscosity. If we retain all the variables in the system (3a-3j) and Ec is set to 0, the following symmetry is obeyed. Let

Set, So) = (X (t), y(t), A(t), B(t}, z(t)Y - with - (t) = (-I (t}'-2 (t)) - be a

solution of the system with initial conditions So = (X(O),Y(O),A(O),B(O),Z(O)Y. We then have that: y-I (m,¢ )S(t, T(m, ¢ )So) =

for

all

(m,¢)E 9\2,

(y-I (m, ¢) =

TT (m,¢))

(4)

T(m,¢)

where

is

a

real,

linear,

orthogonal

transformation:

0

0

0

0

X

0

0

y

0

RaJ 0

0

0

0

A

0

0

R¢ 0

0

B

0

0

0

RI/J 0

I

Z

RaJ 0

T(m,¢)S =

S(t, So)

(S)

174

V. Lucarini and K. Fraedrich

with I being the 2X2 identity matrix and R;; being the 2X2 rotation matrix of angle ~. The T(OJ,¢) -matrix shifts the phases of both the \f'1,1 and 0 1,1 waves by of the same angle OJ and, independently, the phases of both the \f'2,2 and 02.2

waves by the same angle ¢. The symmetry (4-5) gives a solid framework

to and generalizes the results given in [17].

4. Statistical properties of the system - symmetric case In the Ec = 0 case, we expect the presence of degeneracies in the dynamics, leading to non-ergodicity of the system. Since, from (4), we have that S(t,T(OJ,¢)SeJ=T(OJ,¢)S(t,So), we readily obtain that the statistical properties of the flow depend on the initial conditions, and that by suitably choosing the matrix T(OJ, ¢) , we can, e.g., exchange the statistical properties of XI and r; with those of X 2 and Y2 by setting OJ = ;r/2 (and similarly for the A and B variables by setting ¢ = ;r/2). Whereas, in actual integration, numerical noise coupled with sensitive dependence on initial conditions (see below) breaks the instantaneous identity S(t,T(OJ, ¢ )So) = T(OJ,¢ )S(t, So) after sufficiently long time, the statistical properties of the observables transform according to the symmetry defined in Eq. (5). The dynamics of the variables (Xi' X 2' r;, Y2 , ZJ defines an extended Lorenz system [17]. The classical three-component system results from a specific selection of the phase of the waves of the system, which is obtained by setting vanishing initial conditions for X 2 and r;. With such as choice, we have that X 2 =

r; = 0

at all times, whereas the x and y variables of the Lorenz system

are Xl and Y2 , respectively. As a consequence of the symmetry given in Eqs. (4-5), the statistical properties of ZI do not depend on the initial conditions, and agree with those of the z variable of the classical Lorenz system. Moreover, the statistical properties of the quadratic quantities X2 == X I2 + xi, y2 == r;2 + y22, and XY == X I Y2 - X 2r; do not depend on the initial conditions (whereas those of each term in the previous sums do!), and agree (for all values of r, a, b !) with those of x 2 and l, and xy of the classical Lorenz system, respectively. Analogously, since the symmetry defined in Eq. (5) is obeyed, the statistical properties of AI' A2, BI , B2 depend on the initial conditions, whereas the statistical properties of the quadratic quantities A 2 == AI2 + Ai

B2 == Bl2 + Bi '

AB == AIB2 - A2BI ' and of Z2 are well defined. The symmetry (4-5) implies that the attractor of the system (3a-3j) can be expressed as a Cartesian product of a 2torus times a "fundamental" attractor. The initial conditions will define where on the torus the system is located.

Symmetry-Break in a Minimal Lorenz-Like System

175

From a physical point of view, the symmetry (4-5) is related to the fact that the system does not preferentially select streamfunction and temperature waves of a specific phase, and, does not mix phases. Therefore, in a statistical sense, the relative strength of waves with the same periodicity but different phase can be arbitrarily chosen by suitably selecting the initial conditions. Instead, the statistical properties of the space-averaged convective heat transport, which is determined by a linear combination of Z, and Z2' as well as those of the total kinetic energy (determined by a linear combination of X 2 and A 2) and available potential energy (determined by a linear combination of y2 and B2) of the fluid [3] must not depend on the initial conditions - but only on the system's parameters - as they are robust properties of the flow. In general, since the dynamics of the variables (X"X2'~'Y2'Z,) is entirely decoupled from that of the variables {A" A2 , B" B 2 , ZJ, the attractor of the system (3a-31) will be strange at least for the same values of the parameters r, a, b providing the classical Lorenz system with a chaotic dynamics. Additionally, strange attractors could result from choices of the values of r, a, b determining a chaotic dynamics for the (A" A2, B" B 2, Z2) variables and a trivial or periodic behavior for the

(X" X 2' y, ' Y2 ,Z,) variables.

With the "classical" Lorenz parameter values r

= 28,

a

= lO,

b

= 8/3 ,

the variables (X"X2'~'Y2'Z,) obviously have an erratic behavior, whereas the variables (Ai' A2, B" B2, Z2)' which describe the faster spatially varying wave components, do not feature any time variability when asymptotic dynamics is considered, as they converge to fixed values above, the some

r

(A,=, A;, B,=,B;)

manipulations

on

(A,=, A; ,B,=, B;, Z; ) . As discussed

values depend on the initial conditions, while, after Eqs

(3a-3j),

we

obtain

(A2

r

=

b/8{r/2-8) ,

(B2 = 8b{r/2-8) ,

Z; = r/2-8, and (ABt = b{r/2-8). In general, since the fundamental properties of the system do not depend on where in the 2-torus the system converges, despite the fact that ergodicity is not obeyed, it makes sense to compute the Lyapunov exponents [2,18] of the system (3a-3j). In fact, if we compute the Lyapunov exponents (,11' ... '..110 ) using the algorithm by Benettin et al. [19] with the classical Lorenz parameter values, we obtain the same results independently of the initial conditions. The sum of the Lyapunov exponents is given by the trace of the jacobian J of the system (3a-3j) and yelds Tr(J) = LAj =-5{2+2a+b)"" -123.333. In particular, we obtain one positive exponent (AI "" 0.905 ), which gives the metric entropy h of the system [2] and matches exactly the same value as the positive Lyapunov exponent in the classical Lorenz system, and three vanishing exponents Az = ~ = ,14 = o. The non-hyperbolicity of the system, as described by the

176

V. Lucarini and K. Fraedrich

presence of two additional vanishing exponents, is related to the existence of two neutral directions due to the symmetry property (4-5). Moreover, we have 6 negative Lyapunov exponents, one of which agrees with the negative exponent of the Lorenz system (As "" -14.572). Given the very structure (and derivation) of the model discussed above, the fact that we recover the classical results of the 3-component Lorenz system is quite reinsuring. The Kaplan-Yorke dimension [20] of the system is: (6)

where k is such that the sum of the first k (4, in our case) Lyapunov exponents is positive and the sum of the first k+ 1 Lyapunov exponents is negative.

5. Symmetry-break and phase mixing Considering a non-vanishing value for Ec amounts to including an additional coupling between the temperature and the streamfunction waves. Such a coupling breaks the symmetry (4-5), so that a dramatic impact on the attractor properties is expected. Using standard multi scale procedure, we can emphasize the emergence of the additional time scale Ec- 1 • We introduce a slow time variable r = £t with c = Ec . Then we take the expansion Vi(t)=Voi(t,r)+c~j(t,r)+ .. +cl7v,/(t,r)+ ... where

Vi,j=l, ... ,lO is a generic

variable of the ODEs system considered, we redefine the time derivative d/ dt as d/ dt == d/dt + liJ/dr , we plug these new definitions in Eqs. (3a-3j), and group together terms with the same power of £. At zero order - the r variable is frozen - we obtain that the variables V0' obey the Eqs. (3a-3j) with Ec = 0 . Therefore, for time scales shorter than 0(c- 1 ), the symmetries (4-5) described in the previous section are approximately obeyed, which confirms that our multiscale expansion is well-defined. We hereby consider the specific case where we select the classical values for the other 3 parameters of the system, so that r = 28, (j = 10, b = 8/3 . When large values of Ec are considered (Ec > 0.045), the system loses its chaotic nature as no positive Lyapunov exponents are detected (not shown), whereas periodic motion is realized. The analysis of this transition and of this regime is beyond the scopes of this paper, as we confine ourselves to studying the properties of the system when chaotic motion is realized, thus focusing on the Ec ~ 0 limit. We consider Ec E [0,0.02]. In physical terms, the coupling allows for a mixing of the phases of the thermal and streamfunction waves: after a 0(c- 1 ) time the system "realizes" that the symmetry (4-5) is broken, so that the previously described degeneracies

Symmetry-Break in a Minimal Lorenz-Like System

177

are destroyed and ergodicity is established in the system. Note that, since also in the case of Ec > 0 the convection does not preferentially act on waves of a specific phase, we have that, pairwise, the statistical properties of XI and X 2 ' and of ~ and -Y2 are identical and do not depend on the initial conditions. The same applies for the A and B pairs of variables, respectively. If, in particular, we consider the long term averages of the observables

(A2), (B2),

and

(AB),

same contribution, e.g.

(X 2), (y2), (Xy) ,

we obtain that in all cases the two addends give the

(Xn = (xi) = 1/2(X

2 ).

In this case, long is considered

with respect to [,-I = Ec- I : these results can be obtained by first and second order £ -terms in the multi scale expansion envisioned above. As an example, the

(Xn = (Xi) = 1/2 ( X2)

identity is obtained by taking the long term average of

the first order £ -term of Eq. (3g). Whereas having Ec > 0 is crucial for mixing the phases of waves, the impact of the symmetry break on the long term averages of the physically sensitive observables 0 = X2 , y2, A2, B 2, ZI' and

Z2 is relatively weak in the considered Ec -values range.

6. Symmetry-break and hyperbolicity We hereby want to take a different point of view on the process of symmetry break of the dynamical system and on its sensitivity with respect to Ec, by analyzing the Ec -dependence of the spectrum of the Lyapunov exponents. Again, the classical values for r, (J, b are considered. The first, crucial result system is that the system is hyperbolic for any finite positive value of Ec: two of the vanishing Lyapunov exponents branch off from 0 with a distinct linear dependence on Ec, so that ~ '" 5.IEc and ,14 '" -6.0Ec for Ec < 0.008, whereas

~

=0

corresponds to the direction of the flow. The largest Lyapunov

exponent also decreases linearly as ~ '" 0.905-9.0Ec. Time scale separations emerges again, as the two globally unstable directions define two characteristic times II ~ and 1/ ~ , which are 0(1) and O(£c- I ), respectively. All the other Lyapunov exponents feature a negligible dependence on Ec, except As '" -14.572 + 1O.0Ec, which ensures that the sum of Lyapunov exponents is independent of Ec. Therefore, we derive that Ec < 0.008 the metric entropy decreases with Ec as h(Ec) = ~ + ~ '" 0.905 - 4.0Ec , whereas the Kaplan-Yorke dimension can be approximately written as dKy{Ec) = 4+(~ +~ +~ +A,4)J1AsI '" 4.l83-2.01Ec. For larger values of Ec, linearity is not obeyed (except for ~ and ,14)' whereas a faster, monotonic decrease of both the metric entropy and the Kaplan-Yorke dimensions are found.

178

V. Lucarini and K. Fraedrich

See Figs. la)-lc) for results for values of Ec up to 0.02. We then conclude that, in spite of introducing a second unstable direction, which is responsible for mixing the phases of the waves, the inclusion of the impact of the viscous dissipation on the thermal energy balance acts with continuity on the dynamical indicators, by reducing the overall instability, increasing the predictability, and by confining the asymptotic dynamics to a more limited (in terms of dimensionality) set. "'-~--~--~--~------~--~--~--~--~

0.9

1 - -_____

0,8 0,7 0,6 0 ,5

0,4 0,3 0,2

O,'~===========~

-O.l~ ~ -0_20

0.002

0.004

0.006

0.008 '

0.01

0.01 2 - 0.014

0.016

0.018

0.02

Be

(a) 0.95 0.9 0.85 0.8

------------------------------------------

~075 r

07r 0.65 0.6,

0.551

°

(b)

0.002

0.004

0.006

0.008

~12

0.014

0.016

0.018

0.02

Eo

4.2 1-

~ --~--,

-~--~

4.19

41B f 4. 17

"

"t::

4.15 416 4.14

1

413 4.1 2

1

4-11 0-

(c)

0.002

0.004

0,006

0.008

O~----O_014

0.016

0.0 18

0.02

Ec

Figure I: Four largest Lyapunov exponents (a), metric entropy h (b) and Kaplan-Yorke d KY dimension (cl as a function of Ec. Note the continuity for Ec = 0 of all parameters and the distinct linear behavior - see specifically the dashed lines in (b) and (cl - for Ec < 0.008. The linear behavior of the second and fourth Lyapunov exponents branching off zero in (a) extends throughout Ec '" 0.018. Details in the text.

Symmetry-Break in a Minimal Lorenz-Like System

179

7. Symmetry-break and low-frequency variability We now focus on spectral properties of the system. We first observe that the variables AI' A2 , B I , B 2 , Z2 feature mostly ultra-low frequency variability. In order to provide some qualitative highlights on these variables, in Fig. 2 we show, from top to bottom, the projection on Al of three typical trajectories for Ec = 10-3 , Ec = 10-4 , and Ec = 10-5 , respectively. Going from the top via the middle to the bottom panel, the time scale increases by a factor of 10 and 100, respectively. The striking geometric similarity confirms that the time scales of the dominating variability for the slow variables can be estimated as O(Ec- l ) As we take the Ec ~ 0 limit, such a time scale goes to infinity, which agrees with the fact that for Ec = 0 these variables converge asymptotically to fixed values. Moreover, as (A 2 )

= (A12 + An ~ b/8{r/2 - 8) = 2

when &.-1 -long time

averages are considered, Fig. I suggests that the system switches (on short time scales) back and forth between Al - and A2 -dominated dynamics. Same applies for the BI and B2 pair of variables. On the other hand, the dynamics of XI' X 2 ,

~, Y2'

ZI is basically

controlled by the 0(1) time scale 1/ ~ «1/ ~ , so that a clear-cut separation between fast and slow variables can be justified. In Fig. 3 we depict, from the top to bottom panel, some trajectories of the fast variable XI for Ec = 10- 3 , Ec = 10-4 , and Ec = 10-5 , respectively. Similarly to Fig. 2, the time scale of the x-axis is scaled according to Ec- I • Since XI is a fast variable, the dominating high-frequency variability component is only barely affected by changing Ec, because Al has a weak dependence on Ec, as discussed in the previous section (see also Fig. la). Nevertheless, in Fig. 3 we observe an amplitude modulation occurring on a much slower 0(&·-1) time scale of the order of

1/ ~ . Note that,

as in Fig. 2, such a slow dynamics of the "coarse grained" XI variable for various values of Ec is statistically similar when the time is scaled according to Ec- I • As in the previous case, the slow amplitude modulation determines the changeovers between extended periods of XI - and X 2 -dominated dynamics, thus ensuring the phase mixing and ergodicity of the system. In the Ec ~ 0 limit, as discussed above, the relative strength of the two phase components of the (l, I) waves is determined by the initial conditions, as mixing requires ~ Ec- I time units. The same considerations apply for the ~ and Y2 pair of variables. Further insight can be obtained by looking at the spectral properties of the fast and slow variables; some relevant examples of power spectra for Ec = 10-3 ,

V. Lucarini and K. Fraedrich

180

Ec = 10-4 , and Ec = 10-5 are depicted in Fig. 4, where we have considered the low frequency range by concentrating on time scales larger than 10/ A, . The

white noise nature of the fast variable X 1 is apparent and so is the overall lack of sensitivity of its spectral properties with respect to Ec. Note that any signature of the amplitude modulation, which is responsible for phase mixing observed in Fig. 3, is virtually absent. This shows how crucial physical processes are masked, due to their weakness, when using a specific metric, like that provided by the power spectrum. Instead, the slow variable Al has a distinct red power spectrum, which features a dominating

1-3/ 2

scaling in the

low frequency range (black line) between the time scales ~ 0.5 Ec- I and ~ 10 Ec- I • Such a scaling regime dominates the ultra-low frequency variability described in Fig. 2 and is responsible for the long-term memory of the signal. For higher frequencies, the spectral density p(p) decreases - see the sharp

I"" O.SEc

corner in the for DC

1-

2

- as 1-4 , and then a classical red noise spectrum

is realized for very low values of the spectral density. Note also that, in

agreement with our visual perception of Fig. 2, the spectral density

p(p) scales

according to Ec - I in a large range of time scales. 7

2,---,----,----~--_,--_,----,_--_,----,_--,_--_,

~:~

"'"

0

1

2

3

4

5

6

7

8

9

10

X 10'

1

2 , - - -, _- - - ,- -- - , -- -, -- -- ,- - - -, -- -- ,- -- , - -- -, - - -_,

~:~

"'"

0

1

2

3

4

5

6

7

8

9

10

X 10 5

~:~ "'" -20

1

2

3

4

5

t

6

7

8

9

10

X 10 6

Figure 2: Impact of the viscous-thermal feedback on the time scales of the system. From top to bottom: typical evolution of the variable A I for different values of the Eckert number (Ec = 10-3 • Ec = 10-4 , Ec = 10-5 , respectively). Note that the time scale is magnified by a factor of I, 10 and 100 from top to bottom. Details in the text.

Symmetrv-Break in l.l Minimal Loreflz-Liki' S:vstem

181

"

S 'i

~ ~-2m---'---·c_----~-

----:~-----~----~---.-.-~---~~-----~-----.-. . ~

Figure 3: Impact of the viscous-thennal feedback on the time scales of the system. From top to bottom: typical evolution of the variable Xl for different values of the Eckert number (Ee '" 1O-J, Be '" 10-.1, Ee = 10-', respectively). Note that the ljme scale is magnified by a factor of 1, 10 and 100 from top to bottom. Details in the text.

Figure 4: Power spectrum of 1'1 1 and Xl (multiplied times 10-8) for Ee = 10', Be = 10-", and Ee ",10-5 in units of power per unit frequency. The black straight lines corre,pond to 312 scalings. Details in th(, texL

r

8. Summary and Conciusions III this work we have derived and thoroughly analyzed a IO-variable system of Lorenz-like ODEs obtained by a severe spectral truncation of the 2D x-z convective equations for the streamfunction ljf and temperature {} in Boussinesq approximation and subsequent nondimensionaIization. In the tmncation we retain the terms corresponding to the real and imaginary part of the modes 'P"w (streamfunction) and em.n (temperature) associated to the x-z

182

V. Lucarini and K. Fraedrich

wave vector

K = (nan,nm)

with

(m,n)= (1,1)

imaginary part of the x-symmetric modes

and

(m,n) = (2,2)'

gO,2

and

gO,4'

and the

As modes

characterized by a faster varying spatial structure and different parity are added to the Lorenz spectral truncation, which describes the dynamics of the real part of P 1,] and of the imaginary part of g1,1 and gO,2 only, the system we consider in this work represents the thermal impact of viscous dissipation, controlled by the Eckert number Ec. The presence of a forth parameter - together with the usual Lorenz parameters r (relative Rayleigh number), (j (Prandl number), and b (geometric factor) - marks a crucial difference between this ODEs system and other extensions of the Lorenz system proposed in the literature - see, e.g., [46]. Moreover, as can be deduced following the argumentations of Nicolis [16], this system specifically allows for the closed-form computation of the entropy production, as the Eckert number enters into its evaluation. Up to our knowledge, this is the first ODE system obtained as a truncation of the convective equations presented in the scientific literature featuring all these additional properties. The following results are worth mentioning: 1. When the Eckert number is set to 0, as common in most applications, the dynamical system is invariant with respect to the action of a specific symmetry group, which basically shifts the phases of the streamfunction and thermal waves with m > 0 . At a physical level this implies that the initial conditions set the relative strength of waves with the same spatial structure but n12 -shifted phases. In particular, it is found that the Lorenz system is included in the ODEs system analyzed here when specific initial conditions, which set the parity of the slower spatially varying modes, are selected. The absence of phase-mixing implies a lack of ergodicity, and the degeneracy of the dynamics is reflected also in the non-hyperbolicity of the system. When the classical parameters' values r = 28, (j = 10, b = 8/3 are selected, three of the Lyapunov exponents vanish, one corresponding to the direction of the flow, the other two accounting for the toroidal symmetry. The value of the only positive exponent coincides with that of the Lorenz system, and the value of one of the six negative exponents agrees with that of the negative Lyapunov exponent of the Lorenz system. Therefore, the Lorenz system contains already all the interesting unstable dynamics described by this extended ODEs system, and features exactly the same value for the metric entropy. Correspondingly, while the five variables (fast) describing the modes Pl,l , gl,l , and

2.

gO,2

have an erratic behavior, the other five variables (slow)

converge to fixed values. When Ec 0, the symmetry of the system is broken, and coupling occurs

'*

between the fast and slow variable over a time scale O(Ec -1). This is clarified by adopting multi scale formalism. If we select, as in the original Lorenz system, r = 28, (j = 10, b = 8/3, the system is chaotic for

Symmetry-Break in a Minimal Lorenz-Like System

183

o< Ec ~ 0.045 , whereas for higher values of Ec a quasi-periodic regime is realized. In the chaotic regime, the symmetry-break is accompanied by the establishment of a hyperbolic dynamics: two Lyapunov exponents branch off from zero (one positive, one negative) linearly with Ec. Therefore, a second unstable direction with a second

3.

o (Ec-

I)

time scale

11 ~ » 11 AI is established. The impact of the thermal-viscous feedback is stabilizing, as indicated by the metric entropy and the Kaplan-Yorke attractor dimension monotonically decreasing with Ec, with a marked linear behavior for Ec ~ 0.008. The coupling establishes dynamics on time scales of the order of Ec- I responsible for the changeovers between extended periods of dominance of waves of specific phase, both for the slow and for the fast variables. Such processes, which result from small terms in the evolution equations, correspond to the phase mixing of the waves and ensures the ergodicity of the system. In particular, the slow variables have a non-trivial time-evolution and are characterized by a dominating

1-312

scaling in the

low frequency range for time scales between 0.5 Ec- I and 10 Ec- I . Instead, in the case of fast variables, the phase mixing appears as a slow amplitude modulation occurring on time scales of Ec - I which superimposes on the fast dynamics controlled by the 0(1) time scale 11 AI . The system introduced in this paper features very rich dynamics and, therefore, may have prototypical value for phenomena generic to complex systems, such as the interaction between slow and fast variables and the presence of long term memory. Moreover, analysis shows how, neglecting the coupling of slow and fast variables only on the basis of scale analysis - as usually done when discarding the Eckert number - can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing its ability to describe intrinsically multiscale processes. We have shown that a standard multiscale approach allows for understanding the role of the small parameter controlling the coupling. This may suggest that a careful re-examination of the scaling procedures commonly adopted for defining simplified models, especially in the climate science community, may be fruitful in the development of more efficient modeling strategies. Note that in a recent study [21] we have tested that refeeding the kinetic energy lost to dissipation as positive thermal forcing to the fluid (which corresponds to considering Ec > 0) brings the long-term global energy budget of the system closer to zero by an order of magnitude. While the numerical values presented in this paper refer to a specific choice for the parameters r, (J, b, analogous properties - like the symmetry break and the ensuing emergence of ergodicity and multiscale processes, the linearity of the Lyapunov exponents with respect to Ec- I , the low frequency variability are expected for all values of r, (J , b leading to a chaotic dynamics.

184

V. Lucarini and K. Fraedrich

Bibliography [I] E.N. Lorenz, J. Atmos. Sci. 20,130 (1963) [2] D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, Cambridge, 1989 [3] B. Saltzman, J. Atmos. Sci. 19, 329 (1962) [4] J.H. Curry, Commun. Math. Phys. 60, 193 (1978) [5] Z. Wenyi and Y. Peicai, Adv. Atmos. Sci. 3, 289 (1984) [6] D. Roy and Z.E. Musielak, Chaos, Solitons and Fractals 31 (2007) 747 [7] P. Reiterer, C. Lainscsek, F. Schtirrer, C. Letellier, and J. Maquet, J. Phys. A 31,7121 (1998) [8] C. Sparrow, The Lorenz Equations. B!furcations, Chaos. and Strange Attractors, Springer, New York, 1984 [9] c. Bonatti, L.J. Diaz, and M. Viana, Dynamics Beyond Un!form Hyperbolicity: A Global Geometric and Probabilistic Perspective, Springer, New York, 2005 [10] V. Lucarini, l. Stat. Phys. 131,543 (2008) [II] D. Ruelle, Phys. Lett. A 245, 220 (1998) [12] D. Ruelle, Nonlinearity 11, 5 (1998) [13] V. Lucarini, J. Stat. Phys. 134,381 (2009) [14] L.A. Smith, C. Ziehmann, and K. Fraedrich, Q. J. R. Metero!. Soc. 125,2855 (1999) [15J T.N. Palmer, Bul!. Am. Met. Soc. 74, 49 (1993) [16] C. Nicolis, Quart. J. Roy. Meteor. Soc. 125, 1859 (1999) [17] Z.-M. Chen and W.G. Price, Chaos, Solitons and Fractals 28 571 (2006) [18] V. Oseledec, Moscow Math. Soc.19, 197 (1968) [19] G. Benettin, L. Galgani, A. Giorgilli and J .M. Stre1cyn, Meccanica 15, 9 (1980) [20] J.L. Kaplan and l.A. Yorke, Chaotic Behavior of Multidimensional Difference Equations, in Functional Differential Equations and Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, (Springer, Berlin, 1979) [21] V. Lucarini, K. Fraedrich, F. Lunkeit, Thermodynamic Analysis of Snowball Earth Hysteresis Experiment: Efficiency, Entropy Production, and Irreversibility, submitted to Q. l. Roy. Met. Soc. (2009); also at arXiv:0905.3669vl [physics.ao-ph]

185

Sub-Critical 'Transitions in Coupled Map Lattices P. Manneville Hydrodynamics Laboratory Ecole Poly technique 91128 Palaiseau, France (e-mail: [email protected]) Abstract: Sub-critical transitions between two collective regimes with different periods have been observed in a 4-dimensional hypercubic lattice of coupled logistic maps. Statistical findings can be interpreted in terms of macroscopic quantities (measuring global properties) escaping from potential wells in the presence of noise induced by chaos in the microscopic dynamics (local variables). The relevance of these results to a "thermodynamic" understanding of the dynamics of distributed complex systems is briefly discussed. Keywords: coupled map lattices, sub-critical transitions, escape from attractors.

1

Introduction

Large assemblies of interacting nonlinear units give examples of complex systems in which collective behaviour emerges, characterised by the ordered evolution of averaged quantities in spite of strong chaos persisting at the scale of the individual units. The acronym NTCB, meaning non-trivial collective behaviour, has been coined to label this dynamics, emphasising the fact that conventional thermodynamic arguments seem to rule them out. In the present study, we consider coupled map systems [1] in the form: XHI -

.i

-

_1_ " 2d + 1

~ j'EVj

f r (xt) J'

,

where t denotes (discrete) time and Vj the neighbourhoods of sites j at nodes of ad-dimensional hypercubic simple lattice. Here, Vj is limited to nearest neighbours and the local evolution is governed by the logistic map fr(X) = rX(l-X). Such models have already been studied in various space dimensions [2]. They are known to display NTCB for r between the accumulation point of the period-doubling cascade of the isolated map, roo ~ 3.57, and its crisis point rmax = 4. Thc nontrivial features arise from the fact that the mechanisms at the origin of the different macroscopic states and the transitions from one to another are not known. In particular, the mean-field approximation obtained by replacing variables by mean values and neglecting correlations is unable to give hints on the emergence of a collective behaviour apparently unrelated to the properties of the isolated map.

186

P. Manneville

Bifurcations between different NTCB has been observed upon varying r. Supercritical period-doubling cascades for d = 2 and 3 and the Hopf bifurcation toward quasi-periodic behaviour for d = 5 are well documented [3,2]. The size of the system as measured by the number of sites N = n d , where n is number of sites along a space direction, is an important parameter and, remarkably, NTCB is better and better defined as N increases (the so-called thermodynamic limit). Accordingly, it is believed that the asymptotic evolution in the infinite-time limit is accounted for by a genuine attractor in the infinite-size limit. Whereas size effects on continuous transitions, especially period-doubling bifurcations, can be understood within appropriately adapted standard approaches [4], the role of the noise that they introduce in discontinuous transitions needs further study. We focus here on the subcritical bifurcation between two NTBC periodic regimes T2 and T4, with respective periods 2 and 4, coexisting in the range 3.88 < r ::; 4 [2]. Only preliminary results are presented; a more complete analysis is in progress.

2

Results

NTCB is identified from the evolution of the mean value of the Xj over the lattice at time t, X t = N- 1 ~j Xj. Size effects alluded to above are easily observable at r = 4 as seen in Fig. 1 which displays time series of X t for systems with n = 16, 19, 23, 27, and helical boundary conditions,! the numbers of sites increasing by factors of 2 from 6.510 4 to 5.310 5 . The T4 regime corresponds to 4 bands and the T2 regime to 2 bands only with the 't = 4k + 3' and' 4( k + 1)' series partly masking the' 4k + l' and' 4k + 3' series. For n = 16, switchings between T4 and T2 are frequent and it may happen that a phase change occurs upon return to the T4 regime, e.g. after the T2 episodes at t :::::; 1.3105 , :::::; 1.42105 and:::::; 1.82105 . For n = 19, switchings are less frequent, even less for n = 23, and for n = 27 regime T2 seems sustained (notice the change of time scales for n = 23, 27). This situation no longer changes as n is increased further. These observations helps us understanding that bifurcation diagrams strongly depend on the experimental protocol. In [2], regimes T2 and T4 were found to exist in the range r E [3.90,3.97] with n = 27 in sweepup/sweep-down adiabatic simulations with 2,000 iterations and detection of the state obtained after elimination of 1,500 iterations. Considering instead 10 4 iterations, after elimination of 8 10 3 transients, regime T2 is obtained in sweep-up simulations up to r :::::; 3.94 where a transition to T4 is observed; regime T4 is then kept up to rmax = 4. On the other hand, in sweep-down 1

The lattice is arranged as a one-dimensional array of size N = n 4 with periodic boundary conditions. The coupling then reads ~ (Yj + Yj+l + Yj -1 + Yj+n + Yj-n + YJ+n2 +YJ-n2 +YJ+n3 +YJ- n3) for j = 1, ... ,n4, where Yj = Jr(Xj). Other kinds of boundary conditions, e.g. periodic, give the same results provided that the system is large enough.

Sub-Critical Transitions in Coupled Map Lattices 0.9

187

~----~--~-------,

o'r:"-<;;-';-";' '1 0.8

0.7

n=27

: ~ lli"_~~ ••

2

6

10

Fig. 1. Time series of X t at r = 4 for systems of increasing sizes. Color indicates time modulo 4; blue: 4k + 1, red: 4k + 2, green: 4k + 3, magenta: 4(k + 1).

simulations T2 is obtained from T = rmax down to r ~ 3.98, next T4 in the range [3.90,3.98]' and T 2 again for T < 3.90. Turning to n = 54 (presumably closer to the thermodynamic lilnit) using the same protocol, the sweep-up T2---T4 bifurcation is not observed and the T2 regime obtained for r = 3.88 is kept up to r' = 4. In sweep-down experiments starting frorH an initial condition belonging to regime T2, no spontaneous T2---T4 transition is observed so that it is kept all the way down. If the initial condition belongs to regime T4, it persists down to r ~ 3.90 where the T4---T2 transition takes place. These apparently ill-matched results can in fact be rationalised by taking into account the role of the microscopic noise induced by the local chaotic evolution within a few assumptions of thermodynamic flavour as analysed below.

3

Interpretation and discussion

The findings described above can be quantit.atively understood by taking a stochastic approach in terms of master equation [5] and assuming that, at the macroscopic level, we are dealing with a two-state system T2 and T4 characterised by two variables Pz and P4 = 1 - P2 which are the probabilities for system of being found in state T2 or T4, respectively, and two transition

188

rates

P Manneville P24

=

RT2->T4

and

P42

=

RT4->T2,

which immediately leads to

(1) We next assume that the two states represent local minima of some potential V, V2 = V(2) and V4 = V( 4), and that the transitions are noiseactivated processes, the states being separated by a barrier to which a potential Vb = V(B) is associated, Vb - V2 and Vb - V4 being the potential barriers to overcome when the system goes from state T2 to T4 and T4 to T2, with an Arrhenius law in the form

(2) where W is the noise amplitude. Parameters in (1) can be estimated consistently by (i) considering the steady state -!ft P2 == 0 (long after the transient have decayed):

-P2 = -P42 = -V4 P4

P24

exp(-(V2 - V4 )/W)

V2

and (ii) making the assumption that the noise affecting variable X follows from a law of large numbers, that is Wex 1/(n d )1 /2 = 1/n 2 , where n is the linear size of the network and d = 4 is the dimension of space. Setting just W = 1/n 2 fixes the potential scale. This assumption is in line with early observations [2] and can be checked by fitting log(P2 / P 4 ) against a law linear in n 2 , i.e. 10g(Pd P 4 ) = a(r)n2 + b(T) with a(r) = V4 - V2 and b(r) = log(v4 /v2). All these assumptions have been validated by measuring P2 and P4 as the fractions of time spent in states T2 and T4 during very long simulations (tens of millions of iterations) at given r (here r = rmax) for a series of values of n (here n = 13, 16, 19,21,23,25). Within that framework, it is not difficult to understand the differences between the bifurcation diagrams according to the corresponding protocols and values of n by varying r in order to find which state is most likely to be visited. Results for several values of n are displayed in Fig. 2, in which it is seen that the amplitude ratio gets larger as n increases. Extrapolating for n large, one obtains that P4 > P2 for for rinf '::::' 3.917 < r < rsup '::::' 3.997. When n is small, the amplitude of the noise is sufficient to trigger transitions between T2 and T4, but when n is large, depending on initial conditions, the system may remain stuck in one or the other state for arbitrarily long times since noise becomes too small. Regime T4 is strongly preferred for T ~ 3.95 and regime T2 for r < 3.90. For r ~ rinf or r ~ Tsup the two regimes are equally probable. They are both observed when n is small and which one is observed when n is large strongly depends on the experimental protocol, i.e. whether or not one has been able to construct initial conditions depending in one or the other basins of attraction. As an example, for n = 54 uniform distributions of initial states Xj are in the basin of attraction of regime T2

Sub-Critical Transitions in Coupled Map Lattices

189

-l 0

-2 -4 -6

-8

3.88

3.9

3.92

3.94

3.96

3.98

4

Fig. 2. Preferred states at given r as obtained from the ratio P2/ P4 . n = 13: blue; n = 14: green; n = 15: red ; n = 16, cyan.

for r < 3.9290 and of regime T4 heyond. on the other hand if the initial stat e is a 1'2 solution at T ~ 3.88 it renmins in the T2 attraction hasin for all values of l' > 3.88 whether r is increased adiabatically as in sweep-up simulations or abruptly (in fact it has vanishingly small probability to turn to T4) . The ca..'le n = 27 close to r = 4 can be discussed in t he same way, thus explaining the presence of the small interval of T2 regime, the width of \vhich depends OIl how slowly r is decreased in sweep-down simulations. The data can be exploited further to give a = V4 - V2 and b = log(v4/v2) as functions of r. In fact, the distributions of times spent in regimes T2 and T4 are available for a whole set of values of n.and r and not only the ratio PdP4. These waiting times turn out to be exponentially distributed (Fig. 3) and from the decay rates of the distributions, one could in principle determine the transition rates V2 and V4 and t he value of the potential barriers Vb - V2 and Vb - V4" but this post-treatment remains to be done. To conclude, we have shown that coupled logistic maps OIl a 4-dimensional hyper-cubic lattice display ordered macroscopic behaviour though no explanation from first principles is available yet . Local chaos implies t hat the standard bifurcation framework has to be made stochastic, and that the discontinuous transitions observed as the control parameter is varied are empirically well accounted for in terms of a master equation written for a two-level system and an accompanying activated transition process with noise intensity controlled by the size of the system. Better understanding of the global dynamics of such systems could also shed some light on the st atistics of transients observed in discontinuous turbulent-turbulent transitions for which small scale turbulence would play the role of local chaos and large scale flow patterns would correspond to the different regimes obtained [6].

190

P. Manneville

0.0 -0.5

-1.0 -1.5

-2.0

-2.5 -3.0

Fig. 3. Quench experiments from regime 1'4 at Ti = 3.98 down to various values of r'f were performed with n = 27. From 200 to 400 transients were examined, the length of a transient was determined from the sudden decrease of the quantity X 4k + 2 X 4 (k+l), finite in regime 1'4 and statistically zero in regime 1'2. As understood from the semilog plot of the fraction of systems still in regime 1'4 at time t as a function of t itself, the distribution of lifetimes displays an exponential tail in the form pet' > t) ex exp( -tiT), where 7 - 1 varies rapidly with r from a small value for rf = 3.905 (upper curve with black dots) down to 3.895. Somewhat above rr = 3.92 regime 1'4 is essentially sustained, which corresponds to 7 - 1 = o. The author acknowledges the participation of L.A. Zepeda Nunez to the early developments of this study within the framework of his second year study programme at Ecole Polytechnique.

References 1.K. Kaneko. The Theory and Application of Coupled Map Lattices. Wiley, New York, 1993. 2.H. Chate and P. Manneville. Collective behaviors in spatially extended systems with local interactions and synchronous updating. Prog. Them'. Phys. , 87:1- 60, 1992. 3.H. Chate, A. Lema'itre, Ph. Marcq, and P . l'vlanneville. Non-trivial collective behavior in extensively-chaotic dynamical systems: an update. Physica A, 224:447- 457, 1996. 4.A. LemaItre and H. Chate. Renormalisation group for strongly coupled maps. J. Stat. Phys., 96:915 -962, 1999. 5.N.G. Van Kampen. Stochastic Proce8ses in Physics and ChemistTY. NorthHolland , Amsterdam, 1983. 6.F. Ravelet , L. Marie, A. Chiffaudel, and F. Daviaud. Multistability and memory effects in a highly turbulent flow: experimental evidence for a global bifurcation. Phys. R ev. Lett. , 93:164501.1- 4, 2004.

191

Quantum Scattering and Transport in Classically Chaotic Cavities: An Overview of Old and New Results Pier A. MeHol, Victor A. Gopar2, and J. A. Mendez-Bermudez 3 1

2

3

Instituto de Ffsica Universidad Nacional Autonoma de Mexico 01000 Mexico Distrito Federal, Mexico (e-mail: [email protected]) Departamento de Ffsica Teorica and Instituto de Biocomputacion y Ffsica de Sistemas Complejos U niversidad de Zaragoza Pedro Cerbuna 12, 50009 Zaragoza, Spain (e-mail: [email protected] ) Instituto de Ffsica Universidad Autonoma de Puebla Apdo. Postal J-48, Puebla 72570, Mexico (e-mail: [email protected])

Abstract: We develop a statistical theory that describes quantum-mechanical scattering of a particle by a cavity when the geometry is such that the classical dynamics is chaotic. This picture is relevant to a variety of physical systems, ranging from atomic nuclei to mesoscopic systems and microwave cavities; the main application to be discussed in this contribution is the electronic transport through mesoscopic ballistic structures or quantum dots. The theory describes the regime in which there are two distinct time scales, associated with a prompt and an equilibrated response, and is cast in terms of the matrix of scattering amplitudes S. We construct the ensemble of S matrices using a maximum-entropy approach which incorporates the requirements of flux conservation, causality and ergodicity, and the system-specific average of S which quantifies the effect of prompt processes. The resulting ensemble, known as Poisson 's kernel, is meant to describe those situations in which any other information is irrelevant. The results of this formulation have been compared with the numerical solution of the Schri:idinger equation for cavities in which the assumptions of the theory hold. The model has a remarkable predictive power: it describes statistical properties of the quantum conductance of quantum dots, like its average, its fluctuations, and its full distribution in several cases. We also discuss situations that have been found recently, in which the notion of stationarity and ergodicity is not fulfilled, and yet Poisson's kernel gives a good description of the data. At the present moment we are unable to give an explanation of this fact. Keywords: quantum chaotic scattering, statistical S matrix, mesoscopic physics, information theory, maximum entropy.

192

1

P A. Mello, V A. COPGI; Gild 1. A. Mendez-Bemnidez

Introduction

The problem of coherent multiple scattering of waves has long been of great interest in physics and, in particular, in optics. There are a great many wave-scattering problems, appearing in various fields of physics, where the interference pattern due to multiple scattering is so complex , that a change in some external parameter changes it completely and, as a consequence, only a statistical treatment is feasible and, perhaps, meaningful [1,2]. In the problems to be discussed here, complexity in wave scattering derives from the chaotic nature of the underlying classical dynamics. Our discussion will find applications to problems like; i) electronic transport in microstructures called ballistic quantum dots, and ii) transport of electrom agnetic waves, or other classical waves (like elastic waves), through cavities with a chaotic classical dynamics. In particular, we shall study the statistical fluctuations of transmission and reflection of waves, which are of considerable interest in mesoscopic physics. The ideas involved in our discussion have a great generality. Let us recall that, historically, nuclear physics - a complicated many-body problemhas offered very good examples of complex quantum-mechanical scattering. Here , the typical dimensions are on the scale of the fm (1 fm = 1O-13 cm). The statistical theory of nuclear reactions has been of great interest for many years, in those cases where, due to the presence of many resonances, the cross section is so complicated as a function of energy that its detailed structure is of little interest and a statistical treatments is then called for . Most remarkably, one finds similar statistical properties in the quantum-mechanical scattering of "simple" one-particle systems -like a particle scattering from a cavity- whose classical dynamics is chaotic. The typical dimensions of the ballistic quantum dots that were mentioned above is 1 lun, while those of microwave cavities is of the order of 0.5 meters. Thus the size of these systems spans ;::::0 14 orders of magnitude! The purpose of this contribution is to review past and recent work in which various ideas that were originally developed in the framework of nuclear physics, like the nuclear optical model and the statistical theory of nuclear reactions [3], have been used to give a unified treatment of quantum-mechanical scattering in simple one-particle systems in which the corresponding classical dynamics is chaotic, and of microwave propagation through similar cavities (see Ref. [1] and references contained therein). The most remarkable feature that we shall encounter is the statistical regularity of the results, in the sense that they will be expressible in terms of a few relevant physical parameters, the remaining details being just "scaffoldings" . This feature will be captured within a framework that we shall call the "maximum-entropy approach" . In the next section we present some general physical ideas related to scattering of microwaves by cavities and of electrons by ballistic quantum dots. The formulation of the scattering problem is given in Sec. 3, where we

Quantum Scattering and Transport in Classically Chaotic Cavities

193

employ, to be specific, the language of electron scattering. The notion of doing statistics with the scattering matrix and the resulting maximum-entropy model are reviewed in Secs. 4 and 5, respectively. In Sec. 6 we compare the predictions of our model with a number of computer simulations. We also present a critical discussion of the results, based on some recent findings on the the validity conditions of our statistical model. The conclusions of this presentation are the subject of Sec. 7.

2

Microwave cavities and ballistic quantum dots

Consider the system shown schematically in Fig. 1. It consists of a cavity connected to the external world by two waveguides, ideally of infinite length.

_ _ bt,;) ... - - - -

a(2)

n

Fig. 1. The 2D cavity referred to in the text. The cavity is connected to the outside via two waveguides, each supporting N running modes. The arrows inside the waveguides indicate incoming or outgoing waves. In waveguide I = 1, 2 the wave amplitudes are indicated as a~), b~), respectively, where n = 1, ... , N.

We may think of performing a m'icmwave experiment with such a device [4]. A wave is sent in from one of the waveguides, the result being a reflected wave in the same waveguide and a transmitted wave in the other. One given point inside the system receives the contributions from many multiply scattered waves that have bounced from the inner surface of the cavity, giving rise to the complex interference pattern mentioned in the Introduction. An alternative interpretation of the same situation is that the outgoing wave has suffered the effect of many resonances that are present inside the cavity. The net result is an interference pattern which shows an appreciable sensitivity to changes in external parameters: for example, a small variation in the shape of the cavity may change the pattern drastically. In experiments on electmnic transport [5] performed with ballistic microstructures, or quantum dots, one finds, for sufficiently low temperatures and for spatial dimensions on the order of llLm or less, that the phase coherence length 11> exceeds the system dimensions; thus the phase of the singleelectron wave function - in an independent-electron approximation- remains

194

P. A. Melia, V. A. Gopar, and 1. A. Mendez-Bermudez

coherent across the system of interest. Under these conditions, these systems are called m esoscopic [6- 8]. The elastic mean free path lei also exceeds the system dimensions: impurity scattering can thus be neglected, so that only scattering from the boundaries of the system is important. In these systems the dot acts as a resonant cavity and the leads as electron waveguides. Experimentally, an electric current is established through the leads that connect the cavity to the outside, the potential difference is measured and the conductance G is then extracted. Assume that the microstructure is placed between two reservoirs (at different chemical potentials) shaped as expanding horns with negligible reflection back to the microstructure. In a picture that takes into account the electron-electron interaction in the random-phase approximation (RPA) , the two-terminal conductance at zero temperature understood as the ratio of the current to the potential difference between the two reservoirs- is then given by Landauer's formula [10,9 ,1] e2

G=2-T h '

(1)

where the total transmission T is given by

(2) t being the matrix of transmission amplitudes tab associated with the resulting self-consistent single-electron problem at the Fermi energy tF, with a and b denoting final and initial channels. The factor of 2 in Eq. (1) is due to the two-fold spin degeneracy in the absence of spin-orbit scattering that we shall assume here. In this "scattering approach" to electronic transport, pioneered by Landauer [10], the problem is thus reduced to understanding the quantum-mechanical single-electron scattering by the cavity under study. When the system is immersed in an external magnetic field B and the latter is varied , or the Fermi energy tF or the shape of the cavity are varied, the relative phase of the various partial waves changes and so do the interference pattern and the conductance. This sensitivity of G to small changes in parameters through quantum interference is called conductance fluctuations. The experiments on quantum dots given in Ref. [5] report cavities in the shape of a stadium, for which the single-electron classical dynamics would be chaotic, as well as experimental ensembles of shapes. The average of the conductance, its fluctuations and its full distribution were obtained over such ensembles. In what follows we shall phrase the problem in the language of electronic scattering; spin will be ignored, so that we shall deal with a scalar wave equation. However , our formalism is applicable to scattering of microwaves, or other classical waves, in situations where the scalar wave equation is a reasonable approximation.

Quantum Scattering and Transport in Classically Chaotic Cavities

3

195

The scattering problem

In our application to mesoscopic physics we consider a system of noninteracting "spinless" electrons and study the scattering of an electron at the Fermi energy EF by a 2D microstructure, connected to the outside by two leads of width W (see Fig. 1) . Very generally, a quantum-mechanical scattering problem is described by its scattering matrix, or S matrix, that we now define. The confinement of the electron in the transverse direction in the leads gives rise to the so called transverse modes, or channels. If the incident Fermi momentum kF satisfies N
(3)

there are N transmitting or running modes (open channels) in each of the two leads. The wavefunction in lead l (l = 1,2 for the left and right leads, respectively) is written as the N-dimensional vector

(4) the n- th component (n = 1, ... , N) being a linear combination of plane waves with constant flux (given by h), i.e.,

being the electron mass. In (5), k n is the "longitudinal" momentum in channel n, given by

Tn

(6) The 2N -dimensional S matrix relates the incoming to the outgoing amplitudes as (7) where a,ll), structure

a(2), b(1) , b(2 )

are N-dimensional vectors. The matrix S has the

t')

r S = ( t r'

,

(8)

where r, t are the N x N reflection and transmission matrices for particles from the left and r', t' for those from the right: this thus defines the transmission matrix t used earlier in Eq. (2). The requirement of flux conservation (Fe) implies unitarity of the S matrix [1], i.e., (9) sst = I. This is the only requirement in the absence of other symmetries. This is the so-called unitary case, also denoted by (3 = 2 in the literature on randommatrix theory. For a time-reversal-invariant (TRI) problem (as is the case

196

P. A. Melia, V. A. Gopar, and 1. A. Mendez-Bermudez

in the absence of a magnetic field) and no spin, the S matrix, besides being unitary, is symmetric [1]:

(10) This is the orthogonal case, also denoted by fJ = l. The symplectic case (fJ = 4) arises in the presence of half-integral spin and time-reversal invariance, and in the absence of other symmetries, but will not be considered here.

4

Doing statistics with the scattering matrix

In numerical simulations of quantum scattering by 2D cavities with a chaotic classical dynamics one finds that the S matrix and hence the various transmission and reflection coefficients fluctuate considerably as a function of the incident energy E (or incident momentum k), because of the resonances occurring inside the cavity [1]. These resonances are moderately overlapping for just one open channel and become more overlapping as more channels open up. An example is illustrated in Fig. 2, which shows the total transmission 2

1.5

Ll i

T 0.5

2.5

3

3.5

4

kWln Fig. 2. Total transmission coefficient of Eq. (2) as a function of the incident momentum, obtained by solving numerically the Schrodinger equation for the system indicated in the upper left corner (from Ref. [11]), whose classical dynamics has a chaotic nature.

coefficient T(E) of Eq. (2) [proportional to the conductance, by Landauer's formula of Eq. (1)], obtained by solving numerically the Schrodinger equation for a cavity whose underlying classical dynamics is chaotic. Since we are not interested in the detailed structure of T(E), it is clear that we are led to a statistical description of the problem, just as was explained in the Introduction.

Quantum Scattering and Transport in Classically Chaotic Cavities

197

As E changes, 5(E) wanders on the unitarity sphere, restricted by the condition of symmetry if TRI applies. The question we shall address is: what fraction of the "time" (energy) do we find 5 inside the "volume element" dp,(5)? [dp,(5) is the invariant measure for 5 matrices, to be defined below.] We call dP(5) this fraction and write it as dP(5)

= p(5)dp,(5),

(11)

where p(5) will be called the "probability density". Once we phrase the problem in this language, we realize that we are dealing with a mndom 5-matrix theory. In the next section we shall introduce a model for the probability density p(5). The concept of invariant measure dp,(5) for 5 matrices mentioned above is an important one and we now briefly describe it. By definition, the invariant measure for a given symmetry class does not change under an automorphism of that class of matrices onto itself [12,13]' i.e., (12) For unitary and symmetric 5 matrices we have

5'

=

uo5ul,

(13)

Uo being an arbitrary, but fixed, unitary matrix. Eq. (13) represents an automorphism of the set of unitary symmetric matrices onto itself and defines the invariant measure for the orthogonal ((3 = 1) symmetry class of 5 matrices. We recall that this class is relevant for systems when we ignore spin and in the presence of TRI. When TRI is broken, we deal with unitary, not necessarily symmetric, 5 matrices. The automorphism is then

5'

=

Uo5Va,

(14)

Ua and Va being now arbitrary fixed unitary matrices. For this unitary, or (3 = 2, symmetry class of 5 matrices the resulting measure is the well known Haar measure of the unitary group and its uniqueness is well known. Uniqueness for the (3 = 1 class (and (3 = 4, not discussed here) was shown in Ref. [12]. The invariant measure, Eq. (12), defines the so-called Circular (Orthogonal, Unitary) Ensemble (COE, CUE) of 5 matrices, for (3 = 1,2, respectively.

5

The maximum-entropy model

We shall assume that in the scattering process by our cavities two distinct t'/me scales occur [1,3,14 -17]: i) a prompt response arising from short paths; ii) a delayed response arising from very long paths. Historically, this assumption was first made in the nuclear theory of the compound nucleus, where the

198

P. A. Mello, V. A. Gopar, and 1. A. Mendez-Bernnidez

prompt response is associated with the so called direct processes, and the delayed one with an equilibrated response arising from the compound-nucleus formation. The prompt response is described mathematically in terms of 8, the average of the actual scattering matrix S(E) over an energy interval around a given energy E. Following the jargon of nuclear physics, we shall refer to 8 as the optical S matrix. The resulting averaged, or optical, amplitudes vary much more slowly with energy than the original ones. As explained below, the statistical distribution of the scattering matrix S(E) in our chaotic-scattering problem is constructed through a maximumentropy "ansatz" , assuming that it depends parametrically solely on the optical matrix 8, the rest of details being a mere "scaffolding". As in the field of statistical mechanics, it is convenient to think of an ensemble of macroscopically identical cavities, described by an ensemble oj S matrices (see a lso Refs. [18,19]). In statistical mechanics, time averages are very difficult to construct and hence are replaced by ensemble averages using the notion of ergodicity. In a similar vein, in the present context we idealize S(E), for all real E, as a stationary random-matrix Junction of E satisfying the condition of ergodicity [1 ,17]' so that we may study energy averages in terms of ensemble averages. For instance, the optical matrix 8 will be calculated as an ensemble average, i.e. , 8 = (S) , which will also be referred to as the optical S matrix. We shall assume E to be far from thresholds: locally, S(E) is then a meromorphic function which is analytic in the upper half of the complexenergy plane and has resonance poles in the lower half plane. From "analyticity-ergodicity" one can show the "reproducing " property: given an "analytic function" of S, i.e., J(S)

=

(15)

we must have: (1(S))

==

J

J(S) P(S)(S) dfL(S)

=

J((S)).

(16)

This is the mathematical expression of the physical notion of analyticityergodicity. Thus PeS) (S) must be a "reproducing kernel". Notice that only the optical matrix (S) enters the definition. The reproducing property (16) and reality of the distribution do not fix the ensemble uniquely. However, among the real reproducing ensembles, Poisson's kernel [13], i.e. , «(3)

PeS) (S) =

-1

V (3

[det(I - (S) (st) )](2N(3+2-(3 )/ 2 I det(I _ S(st)) 12N(3+2- (3 ,

(17)

Quantum Scattering and Transport in Classically Chaotic Cavities

199

(recall that 2N is the dimensionality of the 8 matrices, N being the number of open channels in each lead) is special, because its information entropy

[1,14] (18) is greater than or equal to that of any other probability density satisfying the reproducibility requirement for the same optical (8). We could describe this result qualitatively saying that, for Poisson's kernel, 8 is as random as possible, consistent with (8) and the reproducing property. As for its information-theoretic content, Poisson's kernel of Eq. (17) describes a system with: 1.- the general properties associated with i) unitarity of the 8 matrix (flux conservation), ii) analyticity of 8(E) implied by causality, iii) presence or absence of time-reversal invariance (and spin-rotation symmetry when spin is taken into account) which determines the universality class [orthogonal (f3 = 1), unitary (f3 = 2) or symplectic (f3 = 4)], and II.the system-specific properties -parametrized by the ensemble average (8)-, which describe the presence of short-time processes. System-specific details other than the optical (8) are assumed to be irrelevant.

6

Comparison of the information-theoretic model with numerical simulations

A number of computer simulations have been performed, in which the Schrodinger equation for 2D structures was solved numerically and the results were compared with our theoretical predictions. In Refs. [1,15,16,18]), statistical properties of the quantum conductance of cavities were studied, like its average, its fluctuations and its full distribution, both in the absence and in the presence of a prompt response. Here we concentrate on the conductance distribution, mainly in the presence of prompt processes. Fig. 3 shows the probability distribution of the conductance [which is proportional to the total transmission coefficient, according to Landauer's formula, Eq. (1) 1 for a quantum chaotic billiard with one open channel in each lead (N = 1), embedded in a magnetic field: the relevant universality class is thus the unitary one (f3 = 2) (reported from Refs. [1,15,16]). The numerical conductance distribution was obtained collecting statistics by: i) sampling in an energy window f1E larger than the energy correlation length, but smaller than the interval over which the prompt response changes, so that (8) is approximately constant across the window f1E. Typically, 200 energies were used inside a window for which kW/7r E [1.6,1.8] (where W is the width of the lead). That such energy scales can actually be found in the physical system under study is interpreted as giving evidence of two rather widely separated time scales in the problem. ii) sampling over 10 slightly different structures, obtained by changing the height or angle of the convex "stopper" , used mainly to increase the statistics.

200 P. A. Mello, V. A. Gopar, and 1. A. Mendez-Bermudez 3

(a)

low field no barrier

(b)

low field with barrier

(c)

high field no barrier

2

f='

!IJ

~

-->------------

o

low field

0 3

highfield

0~: .

.

en

en

(d)

low field no barrier

rnrnrn 0

!IJ en

en en

0.5 T

1 0

0.5 T

1 0

0.5 T

Fig. 3. The probability distribution of the total transmission coefficient T, or conductance, obtained by numerically integrating the Schri:idinger equation for the billiards shown on the left side of the figure for N = 1. The results are indicated with squares, that include the statistical error bars. The curves are obtained from Poisson's kernel, Eq. (17), with (8) extracted from the numerical data. The agreement is good. (From Ref. [15].)

Since we are for the most part averaging over energy, we rely on ergodicity to compare the numerical distributions with the ensemble averages of the theoretical maximum-entropy model. The optical S matrix was extracted directly from the numerical data and used as (S) in Eq. (17) for Poisson's kernel; in this sense the theoretical curves shown in the figure represent pammeter-free predictions. Several physical situations were considered in order to vary the amount of direct processes, quantified through (S). The upper panels in Fig. 3 correspond to structures with leads attached outside the cavity, whereas the lower panels show the conductance distribution when the attached leads are extended into the cavity; in the latter case, the presence of direct transmission is promoted. In some instances, a tunnel barrier with transmission coefficient 1/2 has been included in the leads (indicated with dashed lines in the sketches of the cavities in the figure), in order to cause direct reflection. The magnetic field was increased as much as to produce a cyclotron radius smaller than a typical size of the cavity, and about twice the width of the leads. The cyclotron orbits are drawn to scale on the left of Fig. 3 for low and high fields. In all cases, the agreement between the numerical solutions of the Schrodinger equation and our maximum-entropy model is, generally speaking, found to be good [15]. We should remark that in Figs. 3(e) and 3(f) four subintervals (treated independently) of 50 energies each had to be used, as each subinterval showed a slightly different optical S. This observation will be relevant to the discussion to be presented below on the validity and applicability of the information-theoretic model.

Quantum Scattering and Transport in Classically Chaotic Cavities

20 I

In order to elaborate further on the last remark, we investigated more closely a number of systems where the conditions for the approximate validity of stationarity and ergodicity are not fulfilled. Fig. 4(a) shows the statistical distribution of the conductance for a structure consisting of half a Bunimovich stadium conpled to a la.rger stadium. The system is time-

T

Fig. 4. (a) The conductance distribution obtained from the numerical integration of the Schr()dillger equation for the structure shown in the inset a.nd described in the text , compared with the prediction of Poisson's kernel, Eq. (17), with (8) extracted from. the lJumerica.l data. The system is time-reversal invariant, so that we are dealing with the universality class ;3 = 1. The agreement i.s reasonable. although one observes systematic deviations, as evidenced by the statistical error bars. (b) The square of the scattering wave function for a fixed energy inside the system considered in the analysis. \Ve interpret the concentration of the wave function along the wall of the small cavity as a "whispering gallery mode" , (From Ref. [20].)

reversal invariant, so that we a,re dealing with tbe universality class !3 = 1. The smaller stadium supports a "whispering gallery mode" . providing the short path in this problem· as illustrated in Fig. 4(b), which shows the square of the absolute value of the scattering wave function for 11 fi.'{ed mlergy. The larger stadium provides a "sea" of fine structure resonances and should be responsible for the longer t.ime seale in the problem. The two waveguides support N = 1 open channel each and the system is time-reversal invariant (8 =, 1) (Ref. [20]). Statistics were collected by: i) sampling in an energy window :J.E. It was found t.hat even for :J.E's contaiuing only ~ 20 points-for which the 8(E)'8 were approximately uneorrelated· the energy variation of (8(E)) inside 11E was not negligible. Ii) constructing an enseulble of 200 structures by moving the position of the circular obstacle (shown in Fig. 4) loca.ted inside the larger stadium. Again, the value of the optical 5' vms extracted from the dat.a and used as an input for Poisson's kernel, Eq. (17). \lVe observe that the agremnellt

202

P. A. Mello, V. A. Gopar, and J. A. Mendez-Bermudez

between theory and the computer simulation is reasonable. However, judging from the statistical error bars (and the results for other intervals as well), the discrepancies shown in the figure seem to be systematic. We interpret the situation described in i) above as giving evidence of two not very widely separated time scales in the problem. With the idea of decreasing the energy variation of (S(E)) inside the energy interval f1E used to construct the histogram, f1E was divided into two equal subintervals, giving the two conductance distributions shown in Figs. 5 (a) and (b). We notice the improvement in the agreement between

~~~

~(a) 200~

E=[22, 22.5]

I

T

T

BO{)r

(c)

~

~

"l,-

Lu

o{)

T

T

Fig. 5. The conductance distribution for the same structure as in Fig. 4, but using energy intervals twice as small for the construction of the histograms. Panels (a) and (b) show the same data of Fig. 4, but analyzed inside each of the two subintervals. Panels (c) and (d) show the data for two other similar subintervals. The agreement with theory has improved considerably. (From Ref. [20].)

theory and numerics. The results for two other similar subintervals are shown in Fig. 5(c) and (d), where a similar conclusion applies. We remark that, in contrast to the results shown in Fig. 3, now we are "mainly" averaging over the ensemble of obstacles described in ii) above. We wish to remark that, no matter how much f1E needs to be decreased in order to reduce the energy variation of (S(E)) across it, the improvement between theory and experiment would not be surprising if the number of resonances inside f1E were kept constant, so as to keep approximate stationarity. This could be achieved, for instance, by increasing the size of the big cavity. However, this is not what has been done: the size of the big cavity

Quantum Scattering and Transport in Classically Chaotic Cavities

203

has been kept fixed , so that decreasing iJ.E the number of resonances inside it decreases, and stationarity and ergodicity are ever less fulfilled . Yet , the agreement improves considerably: this we find surprising. The results of a recent calculation [21] are even more intriguing. The conductance distribution was obtained numerically for the same structure as in the previous two figures , but now reducing the energy window iJ.E literally to a point -so that the distribution actually represents statistics collected across the ensemble at a fixed energy- and the histogram was compared with the theoretical model. The results are shown in Fig. 6 for two different energies: they show a very good agreement with theory. 100 300

(a) E

= 22.5

200

~ 100

00

0.2

0.4

T

T

Fig. 6. The conductance distribution for the same structure as in Figs. 4 and 5, but using an energy window iJ.E = 0 for the construction of the numerical histogram, which then represents statistics collected across the ensemble of obstacles at a fixed energy. Panels (a) and (b) correspond to two different energies. The agreement with theory is, in each case, very good.

In Fig. 7 we present the conductance distribution for the same twostadium structure as in the previous figures, again for the universality class (3 = 1, but now at an energy such that the waveguides support N = 2 open channels. Just as in Fig. 6, the numerical analysis was done collecting data across the ensemble for a fixed energy. The agreement, although not perfect, is reasonable.

6.1

Discussion

The theoretical model that we have used to compare with computer simulations, i.e., Poisson's kernel of Eq. (17), is based on the properties of analyticity, stationarity and ergodicity, plus a maximum-entropy ansatz. The analytical properties of the 5 matrix are of general validity. On the other hand, ergodicity is defined for stationary random processes, and stationarity, in turn, is an extreme idealization that regards 5(E) as a stationary random (matrix) function of energy: stationarity implies, for instance, that the optical (5(E)) is constant with energy and the characteristic time associated

204

P A. Mello. V. A. Gopar, and J. A. Mendez-Bermudez

Fig. 7. The conductance distribution for the structure shown in the inset, at an energy such that N = 2 channels are open. The system is time-reversal invariant , so that we are dealing with the universality class {3 = 1. The energy interval iJ.E was literally reduced to a point (after Ref. [20]). The agreement between theory and the simulation is reasonable .

with direct processes is literally zero. Needless to say, in realistic dynamical problems like the ones considered in the present section, stationarity is only approximately fulfilled, so that one has to work with energy intervals ,dE across which the "local" optical (S(E)) is approximately constant, while, at the same time, such intervals should contain many fine-structure resonances. Above we saw examples in which this compromise is approximately fulfilled, and cases in which it is not. Yet, we have given evidence that reducing the size of ,dE, even at the expense of decreasing the number of resonances inside it, improves significantly the agreement between theory and numerical experiments. We went to the extreme of reducing ,dE literally to a point, which amounted to leaving the energy fixed and collecting data over the ensemble constructed by changing the position of the obstacle: in the example shown in Fig. 6 we found an excellent agreement of the resulting distribution with Poisson's kernel. An essential ingredient to construct Poisson's kernel of Eq. (17) is the reproducing property of Eq. (16). This is a property of the ensemble, defined for a fixed energy: at present we only know how to justify it through the properties of stationarity and ergodicity. However, the results we have shown (Fig. 5, which is "almost" an ensemble average, and Fig. 6 which is literally an ensemble average), seem to suggest that Poisson's kernel is valid beyond the situation where it was originally derived: i.e., it is as if the reproducing property of Eq. (16) were valid even in the absence of stationarity and ergodicity. We do not have an explanation of this fact at the present moment.

7

Conclusions

In this contribution we reviewed a model that was originally developed in the realm of nuclear physics, and applied it to the description of statistical

Quantum Scattering and Transport in Classically Chaotic Cavities

205

Quantum Mechanical scattering of one-particle systems inside cavities whose underlying classical dynamics is chaotic. The same model is also applicable to the study of scattering of classical waves by cavities of a similar shape. In the scheme that we have described, the 5(E) matrix is idealized as a stationary random (matrix) function of E, satisfying the condition of ergodicity. It is assumed that we have two distinct time scales in the problem, arising from a prompt and an equilibrated component: the former one is associated with direct processes, like short paths, and is described through the optical matrix (5;.

The statistical distribution of S is proposed as one of maximum entropy, once the reproducing property described in the text is fulfilled and the systemdependent optical (5; is specified. In other words, system-specific details other than the optical matrix (5; are assumed to be irrelevant. The predictions of the model for the conductance distribution in the presence of direct processes should be experimentally observable in microstructures where phase breaking is small enough, the sampling being performed by varying the energy or shape of the structure with an external gate voltage. They have also been observed in microwave scattering experiments. Comparison of the theoretical predictions with computer simulations indicate that the present formulation has a remarkable predictive power: indeed, it describes statistical properties of the quantum conductance of cavities, like its average, its fluctuations, and its full distribution in a variety of cases, both in the absence and in the presence of a prompt response. Here we have concentrated on the conductance distribution, mainly in the presence of prompt processes. In Ref. [15] we found that, with a few exceptions, it was possible to find energy intervals L1E containing many fine-structure resonances in which approximate stationarity could be defined; the agreement was generally found to be good. In contrast, in Ref. [20] we encountered cases in which this was not possible and we undertook a closer analysis of the problem. We found that reducing L1E , even at the expense of decreasing the number of fine-structure resonances inside it, improved the agreement considerably. In a number of cases, L1E was reduced literally to a point and the data were collected over an ensemble constructed by changing the position of an obstacle inside the cavity: in other words, Poisson's kernel was found to give a good description of the statistics of the data taken across the ensemble for a fixed energy. As a result, Poisson's kernel seems to be valid beyond the situation where it was originally derived, where the properties of stationarity and ergodicity played an important role. It is as though the reproducing property of Eq. (16), which is a property of the ensemble defined for a fixed energy, were valid even in the absence of stationarity and ergodicity, which were originally used to derive it. At the present moment we are unable to give an explanation of this fact.

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P. A. Mello, V. A. Gopar, and I. A. Mendez-Bermudez

References l.P. A. Mello and N. Kumar, Quantum Transport in Mesoscopic Systems, Complexity and Statistical Fluctuations (Oxford University Press, New York, 2004). 2.P. Sheng, Scattering and Localization of Classical Waves in Random Media, (World Scientific, Singapore, 1990). 3.H. Feshbach, C. E. Porter, and V. F. Weisskopf, Phys. Rev. 96, 448 (1954); H. Feshbach, Topics in the Theory of Nuclear Reactions, in Reaction Dynamics (Gordon and Breach, New York, 1973). 4.J. Stein, H.-J. Stockmann, and U. Stoffregen, Phys. Rev. Lett. 75, 53 (1995) and references therein; A. Kudrolli, V. Kidambi, and S. Sridhar, Phys. Rev. Lett. 75, 822 (1995) and references therein; H. Alt, A. Backer, C. Dembowski, H.-D. Graf, R. Hofferbert, H. Rehfeld, and A. Richter, Phys. Rev. E 58, 1737 (1998) and references therein; U. Kuhl, M. Martinez-Mares, R. A. Mendez-Sanchez, and H. J. Stockmann, Phys. Rev. Lett. 94, 144101 (2005); S. M. Anlage, Phys. Rev. E 71, 056215 (2005). 5.C. M. Marcus et al., Phys. Rev. Lett. 69, 506 (1992); Phys. Rev. B 48, 2460 (1993); A. M. Chang et al., Phys. Rev. Lett. 73, 2111 (1994); M. J. Berry et al., Phys. Rev. B 50, 17721 (1994); 1. H. Chan et al., Phys. Rev. Lett. 74, 3876 (1995); M. W. Keller et al., Phys. Rev. B 53, R1693 (1996); Huibers et al., Phys. Rev. Lett. 81, 1917 (1998). 6.B. L. Altshuler, P. A. Lee, and R. A. Webb, eds., Mesoscopic Phenomena in Solids (North-Holland, Amsterdam, 1991). 7.C. W. J. Beenakker and H. van Houten, in Solid State Physics, edited by H. Eherenreich and D. Turnbull (Academic, New York, 1991), Vol. 44, pp 1-228. 8.C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). 9.M. Biittiker, IBM Jour. Res. Develop. 32, 317 (1988). 1O.R. Landauer, Phil. Mag. 21, 863 (1970). 11.P. A. Mello and H. U. Baranger, Physica A 220, 15 (1995). 12.F. J. Dyson, Jour. Math. Phys. 3, 140 (1962). 13.L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domain, (Amer. Math. Soc., Providence RI, 1963). 14.P. A. Mello, P. Pereyra, and T. H. Seligman, Ann. Phys. (N.Y.) 161,254 (1985); W. A. Friedman and P. A. Mello, Ann. Phys. (N.Y.) 161, 276 (1985). 15.H. U. Baranger and P. A. Mello, Europhys. Lett. 33, 465 (1996) 16.P. A. Mello and H. U. Baranger, Waves Random Media 9, 105 (1999). 17.P. A. Mello, Theory of Random Matrices: Spectral Statistics and Scattering Problems, in Mesoscopic Quantum Physics, E. Akkermans, G. Montambaux, and J.-L. Pichard, eds. (Les Houches Summer School, Session LXI). See also references contained therein. 18.H. U. Baranger and P. A. Mello, Phys. Rev. Lett. 73, 142 (1994). 19.R. A. Jalabert, J.-L. Pichard and C. W. J. Beenaker, Europhys. Lett. 27, 255 (1994); R. A. Jalabert, and J.-L. Pichard, Journal de Physique 15, 287 (1995); P. Brouwer and C. W. J. Beenaker, Phys. Rev. B 50, 11263 (1994); P. Brouwer, Phys. Rev. B 51, 16878 (1995). 20.E. N. Bulgakov, V. A. Gopar, P. A. Mello, and 1. Rotter, Phys. Rev. B 73, 155302 (2006). 21.V. A. Gopar and J. A. Mendez, unpublished.

207

Parametric Excitation of a Longwave Marangoni Convection Alexander B. Mikishev 1 , Alexander A. Nepomnyashchyl and Boris 1. Smorodin 2 1

2

Department of Mathematics Technion - Israel Institute of Technology 3200 Haifa, Israel (e-mail: [email protected]) Department of Phase Transitions Physics Perm State University 614990 Perm, Russia

Abstract: The development of long-wave Marangoni instability under the action of a heat flux modulated in time is studied. The critical Marangoni number for the deformational instability is obtained as a function of frequency. Keywords: instabilities, Marangoni convection, thin films.

1

Introduction

A dependence of the surface tension on the temperature can lead to the development of a thermo capillary (Marangoni) instability in a layer with a free surface (Davis (1987) [1]). The Marangoni instabilities can be of two types. The first one is a short-wave instability which does not need surface deformation. The second one is based on surface deformation. This latter kind of instability dominates for very thin layers, and it is characterized by large values of the wavelength. The impact of external constraints changes the onset of the Marangoni instability. Among these constraints can be gravity, temperature modulation, laser irradiation, gravity jitter etc. This work is devoted to the investigation of the Marangoni convection in a long-wavelength under influence of heat flux modulation applied to the bottom of the layer.

2

Formulation of the problem

Consider a two-dimensional layer of an incompressible liquid of horizontally infinite extent and of the mean thickness H lying in gravity field on a rigid plane and exposed to the ambient gas phase at its deformable free surface. The layer is subjected to a uniform heat flux harmonically varying with time

(1)

208

A. B. Mikishev, A. A. Nepomnyashchy, and B. L. Smorodill

Here A is the heat conductivity of the liquid; Qa and [2 are the amplitude and the frequency of the heat flux, respectively. A set of two-dimensional governing equations in the plane (x, z) is given by

V·v = 0, Vt + (v· V)v = _p-1Vp + vV 2 v - ge z , Tt

+ V· VT =

XV 2 T.

(2) (3) (4)

Here v = (u, w), T and p are fields of the fluid velocity, temperature and pressure, respectively, v, p, X are, respectively, kinematic viscosity, density and thermal diffusivity, e z is the unit vector in the z-direction, V = (ox, oz). In addition to the harmonically changing heat flux on the layer bottom, (1), we assume the no-slip, no-penetration condition for the velocity

z

=

0:

v

=

(5)

O.

At the free surface z = h(x, t), the boundary conditions are, respectively, the kinematic boundary condition, heat insulation condition and the balance of normal and tangential interfacial stresses given as

h t + uhx = w, VT· n = 0, -p

(6) (7)

2/l- [ 2 + N2 ux(h x -

1) - hx(uz + wx)]

=

~[2(wz - ux)hx + (u z + wx )(I- h;)]

CJ

=

hxx N3 '

(8)

osCJ.

(9)

Here n is the unit vector normal to the interface and directed outward, N = (1 + h~)1/2, /l- = vp is a fluid viscosity and s is arc length along the interface. To account the Marangoni effect assume, that the surface tension varies linearly with the temperature CJ = CJa -

where

CJa

CJTT,

(10)

is he reference value of surface tension. Let us choose scales

for coordinates, time, velocity, temperature and pressure, respectively. Then the governing equations in the nondimensional form are

V·v

=

p-l [Vt

Tt

0,

+ (v . V)v]

(11 ) = -

+ v . VT = V 2 T,

Vp + V 2 v - Ge z ,

(12) (13)

Parametric Excitation of a Longwave Marangoni Convection

209

where P = v Ix is the Prandtl number and G = gH3 I (vX) is the Galileo number. The boundary conditions are rewritten as

z=0: z

=

1:

v

= 0,

OtT = cos(2wt),

ht

+ Uh'E = w,

(14) (15) (16)

VT· n = 0,

2 [ 2 -P + N2 u 3 ;(hx - 1) - hx(uz + Wx)] 1

N [2(w z - ux)hx + (u z =

+ w x )(1 -

=

hxx E N3 '

(17)

2

h x )] =

M(Tx + hxTz).

(18)

Here the following dimensionless parameters appear: E = (T HI (fiX) - the inverse capillary number and M = (TTQH 2 I (A/-lX) - the Marangoni number, W = f!H 2 /chi is the non-dimensional frequency of heat flux modulation. For the steady equilibrium state the upper surface is planar and the liquid is motionless, i.e. v = O. The pressure and temperature fields are:

Po

= -G(z -

r,-,

_

10-

-

(19)

1),

cosh[ex(1 - z)] 2iwt . 2exsm . l() e +c.c., 1 ex

where 'c.c.' denotes the complex conjugate. Here ex = use notations (To)z = ozTo and (To)zz = ozzTo.

3

(20)

J2iW.

Later on, we

Long-wavelength linear stability analysis

Introducing perturbations of the dependent variables of the problem, like v = vo + v(x, z, t), T = To + B(x, z, t), P = Po + p(x, z, t), we obtain the set of the equations and boundary conditions in terms of the perturbation functions (tildes will be omitted). Eliminating the horizontal component of velocity, pressure disturbances and omitting nonlinear terms in the system we formulate the linearized problem around the steady state in terms of normal velocity component, w, temperature distubances, B, and deviation, (, from the unperturbated flat surface z = 1:

+ WX.Tt) = Wzzzz + 2w xxzz + Wxxxx , Bt + w(To)z = Bxx + Bzz , P- 1 (W zz t

(21 )

(22)

The boundary conditions become

=0: z = 1: z

= W z = 0, c't = W,

W

Bz = 0,

(23) (24)

210

A. B. Mikishev, A. A. Nepomnyashchy, and B. L. Smorodin

8z

+ ((To) zz = 0,

(25)

Wxx - Wzz = M[8 xx + (xx (To)z] , -p-1Wzt + Wzzz + 3w zxx + G(xx - IXxxxx = O.

(26)

(27)

For linear analysis of the system we introduce p erturbations

8 ( W)

=

(W(Z' t)) iJ(z,t)

(

(28)

exp(ikx+,t) ,

((t)

where k and, are, respectively, the dimensionless wave number and growth rate of the disturbances, functions W, iJ and ( are periodic in time with the period 7r /w For the amplitudes we obtain the following system (hats are omitted):

p-l[V/'

+ ,w" -

e+,8 + w(To)z

k 2 1V -,k 2 w] = = wIV _ 2k 2w"

+ k 4w,

(29)

8" - k 2 8,

(30)

w

= w' = 8' = 0,

(31)

(

+, ( =

(32)

=

and boundary conditions z

= 0:

z

=

1:

w,

8' + ((To)zz = 0, w" + k 2w = Mk2[8 + ((To)z], p-l(W' + , w') - w"' + 3k 2w' + (G

(33) (34)

+ Ek 2 )k 2 (

=

o.

(35)

Here the prime denotes the coordinate derivative and the dot denotes the derivative with respect to time. To analyze the system (29)-(35)in the longwavelength limit we assume that all dimensionless parameters, namely the Prandtl, Marangoni, Galileo and capillary numbers, are of order unity. For asymptotic analysis we employ the standard scaling for the long-scale instability: k = EK , where E is a small scaling parameter serving as a measure of smallness of the disturbance wave number. The growth rate of the disturbances is expanded as , =

E2 (

,0+ E ,2 + E ,4+ ... 2

4

)

and appears as a function of the physical and modulation parameters. The amplitudes of the perturbation functions are expanded in the form

W = E2 ( Wo + E2 w2 + E4 W4 + . .. ) , 8 = 80 + E 2 82 + E 4 84 + ... , ( =

(0

+ E2(2 + E4(4 + .. .

Parametric Excitation of a Longwave Marangoni Convection

211

and substituted into (29)-(35). At zero order the solution is given by

= Co, = 8 0 + 8t(z)e 2wit + 8o(z)e - 2wit, Wo = wot(z) + wt(z)e 2wit + wo(z)e - 2wit

(36)

(0 80

(37)

(38)

Here we denote by " +" thc pulsating part of the variable , in this case altering with 2w frequency and by " - " - its complex conjugated, latter in the same manner we denote parts of (To)z and (To) zz. The term with" st" corresponds to the part independent on timc. Co and 8 0 are constants yet to be determined. At this order of approximation the coefficients are

=

8+ ( ) o z

(0 cosh( az) 2 2 sinh (a)

'

st() K 2(M80 - G(0)Z2 Wo Z = 2 w+(z) o

GK2(0

+ --6- z

MPK 2(ocoth(a)sinh 2( =

3

,

7P)

2

a 2 sinh(a) cosh( Jp)

We can find Co and 8 0 from solvability conditions. For this purpose we need the equations at the second order

. + w 2IV - p-1·" W2 = - P - 1K2 Wo +(P- 1,O + 2K2)WO", 82 " - 8.2 = (,0 + K2)eO + wo(To)z

(39) (40)

with boundary conditions

z

= 0

Z

=

1

W2

=

w/

(41)

= 82 ' = 0

+ ,0(0 = wo, 82' + (2(To)zz = 0, W2" + K2wo = MK2[8 2 + (2 (To)z], p - l[W·2' + 10WO']- W2 + 3K2wo' +GK 2(2 + EK4(0 = o. (2

(42)

(43) (44)

111

(45)

We integrate over the time the condition (42). Taking into account the time periodicity of functions , we find the first solvability condition which can be written as

(46) Obtain the second solvability condition we can if we integrate (40) over the interval 0 ::::: z ::::: 1 and resulting integral for the stationary part of 82 is

(8~t") = (,0

+ K2)80 + (wt(To-)z + wo(To+)z)),

(47)

212

A. B. Mikishev, A. A. Nepomnyashchy, and B. L. Smorodin

where (1) = relation

Jo

1

Jdz. Using the boundary conditions (41) and (43) and the

(48) we obtain the second solvability equation in the form (49) where S(w, P) is a function depending only on the frequency and the Prandtl number Now using these two solvability conditions, we find that /0 : /o± = -

(G +63 )K 2 ±

~2 )(G _ 3)2 + 18M2S(w, P).

(50)

The lines of the constant value of the growth/decay rate /0 on the plane (M.K o ) are determined by the relation

M(/o,K) = ±

2(/0

+ K2)(3/0 + GK2) 3K4S(W, P)

(51)

To find the neutral curve of the long wavelength convection we take /0 = 0 in (51), then we recover the same expression as obtained in the paper of Or and Kelly (2002) [2]. It is interesting to compare this result with the similar one which has been obtained in the case when the heat flux modulation is applied onto the upper free layer, see Smorodin et al. (2009)[3]. In latter case, the minimum of neutral curve lies lower ( 14.01 at w = 2.65), than in the case which is considered here, where the minimum equals to 105.26 at w = 11.29, see Fig. 1. The instability domains are situated above the neutral curves. The crucial point is whether the critical Marangoni number M(O, 0) corresponds to a minimum or a maximum of the neutral curve. To answer that question, - we need to consider the next order in the expansion with respect to the parameter E. For the deviation of the interface from the flat surface, in the second order, we obtain: (52) The temperature and velocity disturbances in this order are sought as

+ e~t(z) + et,2(z)e 2wit + et,4(z)e 4wit + c.c. w~t(z) + wt(z)e 2wit + C.C

e2 = fh

(53)

W2

(54)

=

Parametric Excitation of a Longwave Marangoni Convection

213

M{2G13)-1/2 104.--rr---r--,r-~r---r---r---r---r---r---,,-,

10

5

15

20

25

ro Fig. 1. The first long-wavelength regions of Marangoni instability. 1 - heating from the surface, 2 - heating from the bottom.

and the explicit expressions for these functions are very cumbersome. In these expressions indices like {2,2} indicate that this terms correspond to the perturbation harmonic with frequency 2w of the temperature or velocity disturbance of the second order. Here C 2 and 8 2 are constants yet to be determined. To obtain how /2 depends on Marangoni number we need to apply the solvability criteria to the fourth order system, which is

wiv - P- 1W4"

=

+p-l/2WO/l 8 4/1

-

84 = ho +

+ (p-l/O + 2K2)W2/' K2(K2 + p-l/O)WO,

_p-1K2W2

K2)8 2 + /280 + w2(To)z

(55) (56)

with boundary conditions

= 0 W4 = W4' = 84 ' = 0 z = 1 (4 +/0(2 +/2(0 = W2,

(57) (58) (59) (60)

z

84 ' + (4(To)zz = 0, W4/1 + K 2w2 = MK2[8 4 + (4 (To)z], p-l [W4' + /OW2' + /2WO'] - W4/1' +3K2W2' + GK2(4 + EK4(2

= o.

(61)

The solvability conditions here are constructed in the same way as in the previous order they are based on (58) and (56) and include functions obtained for the second order. Substituting the expressions for the second order functions into the solvability conditions we obtain a linear system for C 2 and 8 2 .

214

A. B. Mikishev, A. A. Nepomnyashchy, and B. L. Smorodin

Its solution gives us a cumbersome expression for "/2. The value "/2 in the point M = M o, corresponding to "/0 = 0, is of the major importance. This expression depends on G, P, E and w. We calculate this expression numerically for different ranges of parameters. Figure 2 shows the changing of "/2 depending on the frequency w for the parameter values P = 7, G = 1400 and E = 1580 .

y2 LO

5

Y2 50 -5

-10

100

W

150

20

250

n

Fig. 2. T he behavior of 12 for different w at P = 7, G = 1400, E = 1580.

The long-wave nonlinear equation that governs the spatiotemporal dynamics of the thin layer is derived using the results of the linear analysis. Acknowle dments A.M. and A.N. acknowledge partial support from the Israel Ministry of Science through Grant No. 3-5799. B.S. acknowledges a partial support by the Russian Basic Research Foundation (proj ects Nos. 06-01-72031 and 06-08-00289). The research was partially supported by the European Union via the FP7 Marie Curie scheme [PITN-GA-2008-214919 (MULTIFLOW)] .

References 1.8. H. Davis. Thermocapillary instabilities. Annu. Rev. Fluid Mechanics , 19:403, 1987. 2.A.C. Or and R.E . Kelly. The effects of thermal modulation upon the onset of marangoni-benard convection. Journal Fluid Mechanics, 456:161 , 2002. 3.B.L. 8morodin , A .B. Mikishev , A .A. Nepomnyashchy, and B.I. Myznikova. Thermocapillary instability of a liquid layer under heat flux modulation. submitted to Physics of Fluids, 2009.

215

Symmetry Broken in Low Dimensional N-Body Chaos David C. Nil and Chou Hsin Chin 2 1

2

Direxion Technology 9F, 177-1 Ho-Ping East Road Section 1 Taipei, Taiwan, R.O.C. (e-mail: [email protected]) Department of Electro-Physics National Chiao Tung University Hsin Chu, Taiwan, R.O.C. (e-mail: [email protected])

Abstract: In this paper, we explore domains in conjunction with the defined parametric space of function f = h(z) Il{exp(gi(z)) [(ai - z)/(l - aiz)]}, which represents a N-body system in complex form. Interested observations on the computed results reveal limited levels of Herman ring fractals, symmetry broken, and transition to chaos based on phase parameter exp(gi (z)) and ai. Keywords: N-body chaos, Herman ring, fractals, meromophic, symmetry broken.

1

Introduction

Herman Rings represent a class of fractals with hierarchical structure in meromorphic dynamical systems. Due to the richness of mathematical elaboration and computing sophistication, we have observed limited efforts on simple rational functions [1-4]. In the previous efforts, we explored complex function: f = h(z) I14{exp(gi(Z)) [(ai - z)/(l - (liZ)]), and revealed some interested characteristics of this function [5]. We extended our observations to the applications in the antenna areas [6,7] and elaborated the mathematical foundations [8]. We have then classified the function f = h(z) I1i{ exp(gi (z)) [( ai - z) / (1 (liZ)]), and revealed the characteristics of this function [9]: 1. There are at most five levels of partial and localized Herman Ring domains at different complex number z scales. 2. For the orders, which are equal and higher than fourth orders, there exist five and only five level domains. 3. The normalized phase and energy level parameters can induce symmetry broken. In this paper, we further examine in details about the symmetry broken and transition from stable fractal domains to chaotic domains based on two

216

D. C. Ni and C. H. Chin

parameters: exp(gi (z)) and ai in the paramtric space as listed above. These two parameters define thestable and chaotic domains of initial condition of functions.

2 2.1

FUnction and Parametric Space Function

The function set is re-write as the following third-order form:

f = zq IIi Ci

with h(z) = zq, and may have

(1) Where z is a complex variable, q is an integer and Ci has following form:

Ci

=

exp(gi(z))[(ai - z)/(l - (liZ)]

(2)

Here (li is the complex conjugate of complex number ai. We propose this form based on the following form known in the theory of special relativity by A. Einstein:

(3) The zq term represents the interaction of Ci . Here, we set q = -1 representing potential energy of interaction is propotional to the reverse of distance. The term exp(gi(z)) represents the phase, where gi(Z) is a complex function. A given domain can be a domain of complex numbers, z = x + yi , with (x 2 + y2)1/2 ::; R. Here, R is a real number. We also use a simpler domain, (x 2 + y2)1/2 = R, to observe directly a mapping or a transformation from original domain to the set of x + yi after function iteration. There are two sets of domains in conjunction with the function: the converged set remained from original domains after several orders of function iteration and the mapped domains of original domains after several orders of functional iterations. The function f will go through iteration as:

r(z)

=

f

0

r-

1

(4)

Here, n is a positive integer indicating the order of functional iterations. 2.2

Parametric Space

In order to obtain the remained domains and mapped domains, we have to determine the criteria of convergence, divergence, and oscillation of functional values after several iterations. We adopted function values of several consecutive iterations for this purpose. We define the parametric space or normalized space including the following parameters:

Symmetry Broken in Low Dimensional N -Body Chaos

217

1. z : original or mapped domains 2. a : normalized energy level of interaction

3. exp(gi(z)) : normalized phase factor 4. q : normalized distance factor of interaction 5. Iteration These parameters can transform stable domains into chaotic domains.

3

Original and Mapped Domains

Figure 1 (a) through (e) show the Z-lCIC2C3C4 original domains. Figure l(a), (b), (c), (d) and (e) show a original domain z = x + yi , with (x 2 + y2)1/2 :s; R I . forming a set of 5-level of co-center partial Herman-Ring fractal domains at different scales from a set of parameters in parametric space. Figure l(a) shows a partial Herman-Ring domain at scale of R rv 10- 6 .. Figure 1 (b) shows domain a partial Herman-Ring domain at scale of R rv 10- 2 . Figure l(c) shows a partial Herman-Ring domain at scale of R = 1, the unit circle. The domain in Figure 1 (b) can be seen locating around z = 0 point at Figure l(c). Figure l(d) shows a partial Herman-Ring domain at scale of R rv 10 2 . Figure 1 (c) can be seen locating around z = 0 point at Figure 1 (d). Figure 1 (e) shows a partial Herman-Ring domain at scale of R rv 106 . Figure 1 (d) is locating around z = 0 point at Figure 1 (e). The partial Herman-Ring domain at scale of R rv 10- 6 is topologically similar to that at R rv 10 2 and the domain at scale of R rv 10- 2 is topologically similar to that at R rv 10 6 . We call this as skip-level symmetry. For higher functional order than the fourth order (i.e., f = Z-lCIC2C3C4), the number of levels keeps the same as f = Z-ICI C 2 C 3 C4 , namely, only 5 levels are observed at 11th order or 20th order as examples. The skip-level symmetry is also observed in the higher-ordered domains. Figure 2 (a) and (b) show mapped domains offunction f = Z-ICI C 2 C 3 C4 . Figure 2 (a) shows the quasi-perodic orbit becoming chaotic orbit as shown in Figure 2 (b) when the parameters in parametric space change.

4

Symmetry Broken and Chaos

The parameters in the parametric space correlate to one another when an original domain becoming a chaotic domain. We define the transition from a stable to a chaotic domain based on the change of domain boundries by increasing the order of iteration. For example, when the iteration order changes from 10 to 1000 iterations and the low-iterated fractal bondaries of the original domain remain the same, we then claim that this is a stable domain. On the opposite, when the fractal boundaries change with iteration order, then original domain becoming a chaotic domain with the given set of parameters. In the following, we will examine the impact of normalized energy level and

218

D. C. Ni and C. H. Chin

Figure 1(a) R -10-6

Figure l(b) R -10-2

Figure l(c) R =1

Figure 1(d) R -102

Figure lee) R-10 6

Symmetly Broken in Low Dimensional N -Body Chaos

Figure 2 (a)

219

Figure 2 (b)

normailzed phase factor on the symmetry broken of original domains and transition from stable doma.ins to chaotic doma.ins. The compound influence of these two parameters on the transitions is also illustra.ted.

4.1

Normalized Energy Level ( parameter: a )

Figure 3 (a) through (d) show the original domains of nnit circle (i.e., R = 1) with normalizpd energy level a = 0.01 in Figure 3 (a), a = 0.1 in Figure 3 (b), a = 0.2 in Figure 3 (c) and a = 1 in Figure :3 (d) of the function f = Z- l C\C2 C;j. \Ve observe that the original domain showing symmetrical characteristics when the value of normalized energy level is small ( a = 0.01 ) as shown in Figure :3 (a). As the value of the normalized energy level increases from 0.1 to 1; the origina.l domain becomes gradually to a.synl1netrical fractal patterns and eventually becomes to chaotic patterns. This observation also applied to the unit circles of original doma.ins at other function orders.

4.2

Normalized Phase factor ( parameter: exp(9i (z) )

In the following , we examine 4th order domains as shown in Figure 1 with parameter of normalized energy level a = OJn and normalized phase factor at 88 degrees when the original domain rema.ins a stable domain and at 89 degrees when the original domain beeomes 11 chaotic domain. These two specific degrees are defined as follows:

g(z)

= 1f X i x (degr-ce/180)

(5)

vVe exmIline the original domains based on the order of iterations as shown in Figures "1 (a) through (d) for the ca.'le of 88 degrees and in Figures 5 (a) through (f) for the case of 89 degrees.

220

D. C Ni and C. H. Chin

Figure 3 (a) a = 0.01

Figure 3 (b) a = 0.1

Figure 3 (c) a = 0.2

Figure 3 (d) a = 1

Fig. 3. Symmetry Broken and Chaos induced by NonnaIized Energy Level Comparing with Figure l(c), Figure 4 (a) shows that the low-it.erated portion of fraetals at center of unit circle is rotated at 88 degrees while the bulbs extended from individual fractal branches Jag an angle against the central portion of fractal pattern. This is another symmetry-broken mechanism by the normalized phase factor, C:Dp([li(Z)). In addition, a phenomenon similar to frequency-doubling is that the boundries of low-iterated portion of fractals at center is becoming a six-branched pattern instead of original 3branched pattern. \Ve call this phenomenon as branch-ramification. When the iternation number increases, we observe spirals formed to connect the extended bulbs back to the original branches as shown in Figure <1 (b) (iteranion number = 250). As the iternation numbers are further increased as Figure <1 (c) and Figure 4 (d), the central portion of fractal rema.ins in the same fractal boundaries and only the turns of connected spiral increases and more extended bulbs around the inner boundary of the unit circle (i.e., H = 1) appear. \Ve claim tha.t this set of parameters, namely, a = 0.01 and degTee "'.c 88, defines a st.able domain. vVhen the rotahon angle increases from 88 degrees to 89 degrees, as shown in Figure 5, we observe that the branch-ramification as shown in Figure 5 (b) and ·Figure 5 (c) are not stopping at particular iteration, these newly growing branches further extend and conned to the angle-lagged exteneded bulbs. As

Symmetry Broken in [,ott' Dimensional N-Body Chaos

Figure 4 (a) iteration = 50

Figure 4 (b) iteration = 250

Figure 4 (c) iteration = 1000

Figure 4 (d) iteration = 2000

221

Fig. 4. Stable Domains with different iteration at 88 degrees

the iteration number is equal to 100 shown in Figure 5 Cd), we observe the spirals, as shown in Figure 4, appear and compete for connecting to the extended bulbs. As the iteration number increases to 200, shown in Figure 5 (8), the branch-ramification and bulb-connecting competetion are further developed and the original domains eventually becomes a chaotic domain as shown in Figure 5 (f). This indicates that all points in the original domain C 1 :::; R ) will eventually diverge at a given number of iteration. As the normalized energy level increases, the chaotic transitions occur a.t lower rotation angles of the normalized phase factor, crp(g(z)) . Table 1 illustrates the eases for a = 0.1, a = 0.2, and a = 0.3. As normalized energy level increases to a crital value, the origianal domains become chaotic domains without any rotation applied as shown in Figure 3.

Table 1. Chaotic n·ansition in Parametric Space

222

D. C. Ni and C. H. Chill

Figure 5 (a) iteration

50

Figure 5 (c) iteration = 80

Figure 5 (e) iteration

=

200

Figure 5 (b) iteration = 70

Figure 5 (d) iteration = 100

Figure 5 (f) iteration = 1000

Fig. 5. Chaotic Domains with different ih:ration at 89 degrees

5

Remark

In this paper we examined the transiton to chaotic status of the original domains of fUllction f = hez) TIdca:p(g;(z))[(a; -z)/(1- aiz)]}. The normalilled energy level and the normalized phase factor arc two parmnters in the parametric space of this function induce symmtry broken in the fractal patterns. Accompanying with symmetry broken, we observed the trasitions from stable domains to chaotic domains under the iufiunee of combined ef-

Symmetry Broken in Low Dimensional N-Body Chaos

223

feet of these two parameters. Branch-ramification and bulb-connecting competion are the observed phenomena during the transitions. We will further investigate mathematical and physical implications of these computational observations.

References l.Benoit B. Mandelbrot, The Fractal Geometry of Nature, New York, W.H. Freeman and Company, 19S3. 2.K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, New York, John Wiley & Sons, 1990. 3.J. Milnor, Dynamics in One Complex Variable, Vieweg, 2000. 4.A. Fathi and J. -C. Yoccoz (Eds.), Dynamical Systems: Michael Herman Memorial Volume, Cambridge University Press, Feb. 2006. 5.David C. Ni and C. H. Chin, Z- l C 1 C2C3C4 System and Application, TIENCS workshop , Singapore, August 1-5, 2006. 6.David C. L. Ni and C. H. Chin, A Novel Approach for Designing Fractal Antennas, Proc. of International Conference on Wireless Information Networks and Systems, pp. 157-160, (Eds. S.O. Mohammand et al.), Barcelona, Spain, July 2007. 7.David C. Ni and C. H. Chin, Broadband Fractal Antennas , Proc. of International Symposium on Antennas and Propagation , Taipei, Taiwan, pp. 6S (Abstract), October 27-30, 200S S.C. H. Chin,Hypertranscendental Geometry , Preprint NCTU-IMS-OS01, Taiwan, R.O.C. 9.David C. Ni and C. H. Chin, Herman Ring Classification on Function f = h(z) Il{exp(gi(z)) [(ai - z)/(1 - a;z)]} , accepted for publication in the the 5th International Conference 2009 - Dynamical Systems and Applications , Constantza, Romania, June 15-1S, 2009

224

On Markov Processes: A Survey of the Transition Probabilities and Markov Property Gabriel V. Orman Department of Mathematical Analysis and Probability "Transilvania" University of Bnl~ov 500091 Brasov, Romania (e-mail: [email protected] )

Abstract: Some problems concerning the Markov processes are discussed shortly. Among other things, it is presented an extended Markov property. Also, two variants of the strong Markov property, as they are synthetized by Kiyosi Ito, are discussed. Keywords: random walk, transition probabilities, transition operators, Markov property. 2000 MS Classification: 60H25, 60H30, 60J1O, 60J27, 60J60.

1

Markov processes

Generally speaking, it can be considered a problem where comput ation is split among several processors , operating and transmitting data to one another asynchronously. Such algorithms are only being to come into prominence, due to both the developments of decentralized processing and applications where each of several locations migth control or adjusted local variable but the criterion of concern is global. For example a current decentralized application is in Q-learning where the component update at any time depends on the state of a Markov process. The usual class of Markov processes which we involv many times has some restrictions that it does not cover many interesting processes. This is the reason for which we try often to obtain some extensions of this notion. Some researches in such a direction are due, among others, to Kiyosi Ito. In this context we shall refer here to Markov processes and their impact for some practical problems (see for example [4], [5], [6]).

1.1

Transition probabilities

Let S be a state space and it is supposed that a particle moves in S. Also, it is supposed that the particle starting at x at the present moment will move into A C S with probability pt(x,A) after time t "irrespectively of its past motion"; that is to say this motion is considered to have a Markovian character. The transition probabilities of this motion a re {Pt(x , A)h ,x,A and it is considered that the time parameter moves in T = [0, +00). The state space S is considered to be a compact Hausdorff space with a countable open base so that it is homeomorphic with a compact separable metric space by the Urysohn's metrization theorem. The a-field generated by the open space (the topological a-fieldon S) is denoted by K(S). Therefore, a Borel set is a set in K(S).

A Survey of the Transition Probabilities and Markov Property The mean value m

=

M(/-I)

=

225

JRx /-I(dx)

is used for the center and the scattering degree of a one-dimensional probability measure /-I having the second order moment finite; and the variance of /-I is the following

On the other hand , from the Tchebichev's inequality we have (for any t

/-I(m - tu, m

> 0)

1

+ tu) < - t-2

so that several properties of I-dimensional probability measures can be derived. But in the case when the considered probability measure h as no finite second order moment, u becomes useless. In such a case one can introduce t he central value and the dispersion that will play similar roles as m and u for general I-dimensional probability measures . Note 1. W e remind that J.L. Doob defin ed the central value, = ,(/-I) as being the real number, which verifyes the following relation

JRr arctg(x -,) /-I(dx)) =

O.

Here, the existence and the uniqueness of 1 follow from the fact that arctg(x - I) decreases strictly and continuously from ~ to - ~, for x fixed and 1 moves from 2 2

-00

to

+00.

Then , the dispersion <5 is defined as follows

Regarding to the transition probabilities {Pt(x, A)}tET,XES,AE K (S) , it is supposed that they satisfy the following conditions: (1) for t and A fixed , a) the transition probabilities are Borel measurable in x; b) Pt(x, A) is a probability measure in A; (2) po(x, A) = 6x (A) (i.e. the 6-measure concentrated at x); (3) Pt(x,·) ---'weak Pt(xo , ·) as x [that is to say

->

Xo for any t fixed ;

xl!...rICioJ f(y)pt( x,dy) for a ll continuous functions

=./

f(y)pt( xo, dy)

f on Sj;

(4) Pt(x, U(x)) ---. 1 as t 1 0, for any neighborhood U(x) Of x; (5) the Chapman-Kolmogorov equation holds:

Ps+t(x,A) = }SPt(x, dy)ps(y, A).

226

G. V. Orman

In a similar manner the transition operators can be defined. It is considered that C = C(S) is the space of all continuous function s (it is a separable Banach space with a supremum norm). The following operators, denoted by Pt ,

(ptf)(x) =

is

Pt(x, dy)f(y),

f EC

are called transition operators. Now the conditions for the t ransition probabilities can b e adapted to the transition operators. We do not insist here on this case (for more details see [4], [6], [2]). Let us observe that the conditions (1) - (5) above are satisfied for Brownian transition probabilities. One can define 1

pt(x , dy)

=

pt(oo , A)

=

_

=e tv 27r o=A.

(,, _ x)2

dy

2,2

inR

Definition 1. A Markov process is a system of stochastic processes

{Xt(w),t E T,w E (O,K,Pa)}aES. [For each a E S, {Xt}tES is a stochastic process defined on the probability space (0 , K , Pall. The transition probabilities of a Markov process are {pet , a, Let us d enote by {Ht} the transition semigroup and let R ", b e the resolvent operator of {Ht}. Now pet, a, B), Ht and R", can be expressed in terms of the process as follows.

Bn.

Theorem 1. Let f be a fun ction in C(S). Then 10. p(t,a,B) = Pa(Xt E B). 20. For EaO = .f~ ·Pa(dw) one has Htf(a) = Ea(f(Xt)). 30. R ", f (a) = Ea e-"'t f(Xt)dt).

Uo=

Proof. One can observe that 1° and 20 are immediately. Now, for 30, it is considered the following equality:

R",f(a) =

1=

e - ",t Hd(a)dt =

1=

e-",t Ea(f(Ht))dt .

But f (X t (w)) is right continuous in t for w fixed and measurable in w for t fixed and therefore it is m easurable in the pair (t, w). Thus, the Fubini's theorem can be used and one gets

that is just 30..

Definition 2. The operator

et

:

° ° --t

defined as follows

Uhw)(s) = w(s

+ t)

for every s E T, is called the "shift operator". Evidently, (the semigroup property). Notation. For e a a-field on 0 , the space of all bounded e-measurable functions will be denoted by B(O, e), or simple B(e).

A Survey of the Transition Probabilities and Markov Property

2

227

Markov property

We shall develop below some aspects concerning the Markov property and related ideas in a vision of Kiyosi Ito. Firstly the classical and the extended Markov properties are presented. Then the strong Markov property will be considered. The Markov property is expressed in the theorem below. Theorem 2. Let be given r E K. The following is true

that is to say Note 2. The following notation can be used

Proof. It will be suffice to show that (1)

for r E K a nd DE K t . One can distinguish three cases. [I.] Let us consider rand D as follows:

and with

o :S Sl < S2 < ... < Sn o :S t 1 < t2 < ... < tm :S t and B i , Aj E K(S). Now it will be observed that the both sides in (1) are expressed as integra ls on srn+n in terms of transition probabilities. Thus, one can see that they are equal. [II.] Let now be r as in the case [I.] and let us denote by D a general member of Kt. For r fixed the family 'D of all D's satisfying (1) is a Dynkin class. If M is the family of all M's in the case [I.] then , this family is multiplicative and Me 'D. In this way it follows 'D(M) c 'D = K(M) = K t and one can conclude that, for r in the case [I.] and for D general in K t , the equality (1) holds. [III.] (General case.) This case can be obtained in a same manner from [II.] by fixing an arbitrary D E K t . It will be obtained that Pa(r) is Borel mea surable in a for any r E K. Corollary 1.

Ea(G

0

et , D)

Ea(EXt (G), D)

for G E B(K), DE Kt,

ell ) etlKtl

Ea(F. EXt (G))

for G E B(K), FE B(Kt) ,

E a(F · (G 0 Ea(G

0

EXt (G)

(a.s.)(Pa ) for G E B (K) .

228

G. V. Orman

2.1

The extended Markov property

It is interesting to see that the Markov property can be extended. The following theorem will refer to the extended Markov property.

Theorem 3. for

r

E K.

Proof. Let us come back to the equality (1) before. Now it will be proved for D E K t +. To this end the following equality will be shown:

for fi E C(S), DE Kt+ and 0 <:: 81 < 82 < ... < Sn· But D E K t+ h for h > 0, so that by Corollary 1 it results

Ea(h (XSI (et+l,w)) ... fn(Xsn (et+hw)), D) = Ea(Ext+,,(h (X S1

) •..

fn(Xsn)), D). (3)

Now one can observe that

is continuous in a, if it is considered that

and Hs : C ---+ C. But Xt(w) being right continuous in t, one gets

1 o.

as h

Now , the equality (2) will result by taking the limit in (3) as h 1 o. In this way, for G i open in S, the following equality will result from (2)

Ea(XSi(etw) E G 1 ,'" ,XSn(etw) E G n , D)

= Ea(PXt(XSl

E G 1 ,'"

(4)

,Xsn E Gn) , D),

and now the Dynkin's theorem can be used. 1

2.2

The strong Markov property

As it is known, the intuitive meaning of the Markov process (for example X(t)) is the fact that such processes "forget" the past, provided that t n -1 is regarded as the present. Now, the intuitive meaning of the Markov property is that under the condition that the path is known up to time t, the future motion would be as if it started at the point

Xt(w) E S. But a question arises: what will happen if "t" will be replaced by a random time

a(w)? 1Theorem (Dynkin's formula). Let us suppose that a is a stopping time with Ea(a) < 00. Then, for u E :D(A) it follows:

Ea

(~OO AU(Xtl dt )

=

Ea(u(Xcr)) - u(a).

A Survey of the Transition Probabilities and Markov Property Definition 3. A random time to {Kt} if

0" :

it

-->

229

[0, (Xl] is called a "stopping time" with respect (5)

for every t. But such a condition is equivalent to the condition

(6) for every t.

Observation 1. If the condition (5) holds, then it results

But if the condition (6) holds, then we have {O" <::: t} =

n

{O"

< t + ~}

E K

t+;k

n?:.m

for every m. Therefore {O"

< t}

En Kt+;k

=

Kt

m

by right continuity of K t. A trivial example is the deterministic time

0"

== t.

Now the strong property can be given (more details and proofs can be seen in [4],

[5], [6], [10], [2]). The strong Markov property is given in the theorem below

Theorem 4. The following equalities hold a.s. 10. P IL (8;11jKo-)=Px a (r)

2°. EIL(F(8o-wIKo-) K -measurable.

=

Ex" (F)

a.s. (PIL ) a.s.

on

< (Xl}, whererEK. {o- < (Xl}, where F is

{O"

(PIL ) on

bounded and

Note 3. The following conclusion are true. i.

8o-w = 8o-(w)w for any w with O"(w)

< (Xl, and

8;;lr = {w: O"(w) ii.

< (Xl

and 8o-w E

n.

It is observed that I" is arbitrary and for this reason the both F(8o-w) and Ex" (F) are K -measurable.

Finally, a variant of the strong Markov property is given below. It is referred to as the time-dependent strong Markov property.

Theorem 5. It is supposed that F(t, w) is bounded and K[O, (Xl] x K - measurable. Then, the following equality holds

(7) Proof. In view of that

0"

is

K (T-measurable we come to the conclusion that F(t, w) = f(t)G(w).

Now, by considering the aditivity of both sides of (7) in F, the general case results.

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G. V. Orman

3

Application

Absorbing and reflecing barriers Let us consider that a particle located on a straight line moves along the line via random impacts occurring at times tl, t2, t3, .. '. The particle can be at points with integral coordinates a, a + 1, a + 2, ' .. ,b. At points a and b there are absorbing barriers. Each impact displaces the particle to the right with probability p and to the left with probability q = 1 - P so long as the particle is not located at a barrier. If the particle is at a barrier then, it remains in the states Al and A n - I with probability l. A similar example can be considered for a particle being in a random walk, when at points a and b there are reflecting barriers. The conditions remain the same as in the former case, the only difference being that if the particle is at a barrier, any impact will transfer it one unit inside the ga p between the ba rriers. 1. Let now be the case of a Brownian motion with an absorbing barrier. The forward Kolmogorov's equation (9) for a Brownian motion on x > with an absorbing boundary at x = is given by ap 1 a 2p at 2 ay2 in y >

°

°

°

p(O,t , y) = 0, p(x , t , y)

--+

t>O, y > O

o(x - y)

as

t 1 0, x

> 0,

y

> 0.

The solution of such an initial boundary value problem is as follows 1

[(X _ y )2

p(x,t,y) = - - e -

tV21T

~

-e-

(x+y)2]

~

.

It can be seen that by symmetry, p(x , t , 0) = 0. Then , it can be shown that 1

j +oo

!
as t

1

°

if y

tv 211'

> 0.

C,,+ y) 2 e - ~ 'P(x)dx

--+

'1'( -y) =

-00

°

Therefore,

p(x, t, y)

--+

o(x - y) as t 10 for all x> 0, Y

> 0,

where 0 is the Dirac 's o-function. 2. Now let us consider the Brownian motion on x > but with a reflection barrier at the origin. The forward Kolmogorov 's equation for a Brownian motion on x > with an reflection boundary at x = is given by ap 1 a 2p y>o at 2 ay2'

°

°

°

ap(x , t , y) / =0 ay y=o p(x,t,y)

--+

o(x - y)

as

t

1 0, x > 0, y > 0,

with 0 - the Dirac's o-function. In this case the following solution is found

p(x,t,y)

= -1tV21T

[ ( X_ y ) 2

e-

~

and the condition

ap(x, t, y) -0 / ax x= o holds too.

( x +y) 2 ]

+e- ~

A

Survey of the Transition Probabilities and Markov Property

231

Note 4. It is known that in some conditions of existence, the transition density satisfies the folowing two Kolmogorov's equations (they will be referred to as the backward Kolmogorov 's equation and respective the forward Kolmogorov 's equation). o - .f~(c:..:t,-=-:.x,--;T-,--,y~) ( ) of(t,x;T,y) - -1 b( t x ) 0 2 f(t,x;T , y) = -a t x (8) ot 'ox 2' ox 2 and of(t , x; T, y) 0 1 02 OT = - oy [a(T,y)f(t,x;T,y)] + 2 oy2 [b(T,y)f(t,x;T,y)], (9)

where aCt , x), bet , x), aCT, y) , beT, y) are functions satisfying some conditions to assure the existence and the uniqueness of the solution of the equations. The forward Kolmogorov's equation is also referred to as the Fokker-Planck equation. We emphasize that the expression "the backward Kolmogorov equation" m eans: in the "backward" variables t and x. Similarly, "the forward Kolmogorov equation" must be understood that it is considered in the "forward" variables T and y. One can observe that this Application implyes stochastic differential equations in connection with the K olmogorov 's equations for a Markov process; more details and related topics can be found in [lS}, flO}, (13).

References [1] BERTOIN, J. , Levy Processes , Cambridge U niv. Press , Cambridge, 1996. [2] BHARUCHA-REID, A.T. , Elements Of The Theory Of Markov Processes And Their Applications, Dover Publications, Inc., Mineola, New York, 1997. [3] G NEDENKO, B.V., The Theory of Probability, Mir Publisher, Moscow , 1976. [4] ITO , K. , Selected Papers. Springer, 1987. [5] ITO , K. , McKEAN , H.P. , Diffusion Processes and their Sample Paths. Springer-Verlag Berlin Heidelberg, 1996. [6] ITO, K., Stochastic Processes. Eds. Ole E . Barndorff-Nielsen, Ken-iti Sato, Springer, 2004. [7] KALLENBERG , 0. , Foundation of Modern Probability, Springer , New York, 1997. [8] LOEVE, M., Probability Theory, vol. I , Springer-Verlag, New York , 1977. [9] LOEVE, M. , Probability Theory, vol. II, Springer-Verlag, New York , Heidelberg Berlin, 1978. [10] 0KSENDAL , B. , Stochastic Differential Equations: An Introduction with Applications, Sixth Edition, Springer-Verlag, 2003. [ll] 0KSENDAL, B. , SULEM A., Applied Stochastic Control of Jump Diffusions, Springer, 2007. [12] ORMAN , G . V., Capitole de matematici aplicate, Ed. Albastra, Microlnformatica, Cluj- Napoca, 1999. [13] ORMAN G. V., Lectures on Stochastic Approximation Methods and Related Topics , Preprint. " Gerhard Mercator" University, Duisburg, Germany, 2001. [14] ORMAN , G.V., Handbook Of Limit Theorems And Sochastic Approximation, Transilvania University Press , Brasov, 2003. [15] SCHUSS Z., Theory and Application of Stochastic Differential Equations, J ohn Wiley & Sons, New York, 1980. [16] STROOCK , D.W. , Markov Processes from K. Ito Perspective, Princeton Univ. Press, Princeton, 2003.

232

Qualitative Dynamics of Interacting Classical Spins Leonidas Pantelidis Department of Physics and Astronomy, Hanover College Hanover, IN 47243, USA (e-mail: [email protected]. edu)

Abstract: \Ve consider the classical Heisenberg model (Hl'vI) with z-axis anisotropy and external magnetic field. Its phase space is foliated into a family of invariant leaves diffeomorphic to (S2)A The flow on each leaf S is Hamiltonian. For the isotropic Hl'vI with zero field, the manifold :F of longitudinal fixed points (LFPs) intersects each leaf S orthogonally. In addition, we show that the ferromagnetic (FR) state and the antiferromagnetic (AF) state with non-zero total spin are both stable LFPs. This is a direct implication of a Lemma which extends Lyapunov stability from an invariant subspace to the whole leaf by exploiting the rotational symmetry. The lemma does not apply in the case of zero total spin, and indeed, the AF state on an equal-spins leaf is shown to be unstable. PACS numbers: 05.45.-a

L Introduction The Heisenberg model (HM) is a simplified model of magnetic ordering in materials [1, 2, 3, 4] which has drawn considerable attention for a long time. Theoretical and experimental studies on the quantum version of the HM are abundanq. The classical HM has also been of interest, both as a model of ferromagnetic and other phenomena and as a testground for methods of nonlinear dynamics. However, cla.'isical Heisenberg spin systems, especially chains, have been mostly studied in the continuum limit§. From the point of view of quantitative analytical dynamics, the discrete HM is barely tractable with even a simple Heisenberg ring (closed chain of spins with equal nearest-neighbor only interactions) being non-integrable. The only known exact solutions for the HM on a general periodic lattice arc the (propagating) ferromagnetic (FR) and antiferromagnetic (AF) nonlinear spin waves (NLSWs), and some special planar solutions [32, 33, 34]. These arc finite-amplitude generalizations of the smallamplitude linear spin waves, the latter being approximate periodic solutions of the nonlinear equations of motion in the vicinity of the FR and AF fixed points (FPs) II. :j: Some important work in the field can be found in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 251 and references therein, although this list is by no means comprehensive. § For instance, see [26,27, 28, 29, 30, 311 and references therein.

II

Also termed as critical points, equilibrium points, or steady states.

Qualitative Dynamics a/Interacting Classical SpillS

233

Certain linear combinations of NLSWs havc also been found to satisfy the equations of Illotion [35], although these solutions do not, in general, reside on the leaf (shell) of equal spins ("off-shell" NLSW s). In all the aforementioned solutions the spins have constant and equal (in each sublattice, in the AF case) z-components, that is, all known solutions are of a rather special type. This paper focuses on the qualitative dynamics of the classical HM. In section 2, we introduce the general classical HM with z-axis anisotropy and magnetic field. In section 3, we discuss its phase space structure and in particular the foliation of the phase space into invariant leaves with Hamiltonian flow. In section 4, we discuss the usual FPs of the flow and point out that they form manifolds which intersect the leaves orthogonally. Section 5 addresses the question of stability of FPs. This question is answered in the special but interesting cases of the FR and AF states on the basis of a general lemma which extends Lyapunov stability from an invariant subspace to the whole leaf by exploiting the rotational symmetry of the model. 2. The classical HM The standard quantum HM Hamiltonian reads A

A

(2.1 )

H = - LgjlSJ' S[- LBJ' Sj, j
where gjl are the exchange constants and the second term, the Zeeman term, arises in the presence of an external magnetic field B. We can also introduce a longitudinal anisotropy L'l. by setting Si' S[

==

S;-.

st + L'l.SjSF == (SjSF + sysn + L'l.SjS{ .

(2.2)

In a mean-field approach, in particular in a time-dependent Hartree- Fock approximation, the average spins Uj, j = 1 ... A interact according to the first order system of A coupled quadratic equations [34]

~~j

= Uj

x

(tgjl(ut +

L'l.l1tZ)

1=1

+

Bi) ,

j

=

1, ... ,A,

(2.3)

which constitute the classical HM. Taking the dot product of both sides of (2.3) with Uj, we find that the lengths of the individual spin vectors arc first integrals of the motion, that is

IlJ = 57.

j

(2.4)

= 1, ... ,A ,

where the 5j are arbitrary non-negative real constants. Additional constants of motion can be found by inspecting the sum of equations (2.3), d

A

dt L Uj = j=1

A

(L'l. - 1) L j.l=1

A

gj[(Uj x

I1tZ)

+L j=1

Uj x B j .

(2.5)

234

L. Pantelidis

One sees that, in the isotropic case L). = 1 and for a homogeneous magnetic field, the total spin v = L.~'=l Uj has constant length and precesses about the applied field. In the absence of an external field, both the direction and the magnitude of the total spin arc conserved. If L). # 1, only the z-component of the total spin is conserved, provided the magnetic field is 7:ero or parallel to the z-direction. In passing, note that (2.3) is invariant under the rescaling transformation

(lit, g, B)

-+

(2.6)

J(llt, g, B)

where J is a non-zero real constant. Hence, an overall renormalization of the exchange and ficld strengths does not affect the dynamics. Jdoreover, the dynamics essentially depcnd only on the rclative spin lengths since (2.3) is invariant under the rescaling

(lit, Uj, B)

-+

(2.7)

s(l/t, Uj, B)

for any non-zero real constant s.

3. Phase space structure of the HM Spins of zero length are zero vectors and, according to (2.3), they neither change themselves nor affect the non-zero spins. Therefore. A can be considered to be the number of non-7:ero spins and so the total phase space of the HM is the open submanifold

P

=

{(Uj)

E IR:JA

:uJ > 0, \lj E {I, ... , A}}

(3.1 )

of 1R3A If we consider the Coo surjection

its Jacobian matrix has non-zero orthogonal rows so rankn = A. Hence n is a submersion and therefore it induces the codimension-A, class Coo foliation {Sh) )EJP:~' where

hSj

and P

=

(~)S(sJ)' By definition of a foliation, the leaves S(8j) are disjoint and connected

(see Figure 1). In addition, as fibers of a submersion, they are 2A-dimensional closed regular submanifolds of P. Since they are also bounded, Sis}) are compact regular submanifolds of P. For A = 1 we recover the well-known result that the sphere 52 of radius s is a 2-dimensional compact regular submanifold of 1R3~. This immediately implies that (5 2 )A is a 2A-dimensional compact regular submanifold of 1R3A Because there is a unique 2A-dimensional regular submanifold structure on the set S, we conclude that S is diffeomorphic to (5 2)A

'If It is also well-known that all spheres of non-zero radii are diffeomorphic and that, since dim 52 2 < 4, there is a unique differentiable structure on 52.

=

Qualitative Dynamics of Interacting Classical Spins

235

It can be shown [36] that S is a symplectic manifold on which the spins satisfy the angular momcntum Lie algebra and their motion is a Hamiltonian flow with Hamiltonian h : S - t lR dcrived by replacing in (2.1) thc operators S with c:lassical spins u: A

h

=

A

~ """"' gzu . uz ~ """"' ~.1.1 ~ B J .uJ j
(3.2)

j=]

Moreover, since S is a connected manifold, any other Hamiltonian 9 generating the same flow, i.e., such that Vg = Vi" differs from h only by a constant [37]. Note that the classical HM is not a mechanical system, in the sense that the Lagrangian is not quadratic in velocities. In what follows, we will consider only the isotropic case with zero magnetic field, although our analysis could be extended to deal with more general cases. The Hamiltonian h is of course conserved as it does not depend on time explicitly. As we saw in section (2), the total spin v is also conserved. By comparison with the quantum case, we expect that the total spin v generates the rotations of the spins in lR 3 , which makes it the c:lassical total angular momentum of the system. Indeed, as one can verify,

where Ret is the infinitesimal rotation about the a-axis parametrized by f. The timetranslation and rotation arc continuous symmetries of the Hamiltonian and thus of the equations of motion. We can also see that directly from (2.3). These arc Niiether symmetries as they are generated by Hamiltonian vector fields, that is, vector fields which leave the 2-form invariant+: LvJ2 = Lv"" n = O. Notice that space inversion is a discrete symmetry of the Hamiltonian but does not leave the equations of motion invariant as the cross product in (2.3) is a pseudovector. A maximal set of known first integrals in invol'Ution (pairwise commuting) has only 3 elements, e.g., {h, u 2 , U Z } , and so the system is far from integrable in the general case. (For early numerical evidence of non-ingrability see [32].) The trivial case of 2 spins and the cases of the 3- and 4spin Heisenberg rings are exceptions. Note that the constants (2.4) cannot be related to Hamiltonian symmetries since they define the very phase space S on which the equations of motion (2.3) have Hamiltonian form. Each leaf S is covered by a 4-parameter family of invariant subspaces ~E,v' The generic ~ is a (2A ~ 4)-dimensional submanifold of S. However. since the parameters E and v are nonlinear functions of the intrinsic coordinates of S they are not always independent, and on the other hand additional conserved quantities may be present. Therefore, one may find ~s of lower or higher (than 2A ~ 4) dimensionality and even ~s that are not submanifolds. A simple example is the FR state (all A spins parallel to a given direction) which forms a O-dimensional submanifold by itself. + Non-Niiethcr symmetries arc automorphisms which preserve the equations of motion but not the 2-fonn. Both Nc)ether and nOll-Noether sYlnnretries are associated with constants of Illation.

236

L. Pantelidis

s

I

I

Figure 1. The phase space structure for the discrete classical HM. The 3A-dimensional phase space is foliat ed into 2A-dimensional, connected, compact, ,·eg·ular, invariant submanifolds S. The 2A - 1 (11. + 2)-dimensional submanifolds of LFPs, denoted by F, are connected, unbounded, and intersect every leaf S orthogonally along an 52

4. The manifold of LFPs From the equations of IlIotion (2.3), we can immediately see th at a configuration of spins is stationary iff (if and only if) the spin and the vector in parenthesis, which is the mean field in the Hartree-Fock approximation, are linearly dependent at every site. In the isotropic zero-field case that we examine, this condition is met when all spins are collinear. Therefore, on each S, there are 2A - 1 disjoint spheres 52 of fixed points of the flow. Each sphere is generated by rotations of a configuration with one spin in a given direction and the rest parallel or anti-parallel to that direction. We will refer to these points as longitudinal fixed points (LFPs). These are the only equilibria for

Qualitative Dynamics of Interacting Classical Spins

237

generic exchange constants. However , additional stationary states of non-collinear spins can exist, depending on the exchange constants (e.g., planar fixed points for closed Heisenberg chains etc.) By varying the lengths of the spins, we see that each LFP belongs to an (A + 2)-dimensional submanifold F of LFPs within the 3A-dimensional phase space. Clearly, there are 2A - 1 such submanifolds and they are disjoint. Moreover , every F is connected, unbouncled , and intersects every leaf S along an 52 Since the lengths of the spins are constant on a shell S we expect that F and S are orthogonal. This orthogonality can be form ally demonstrated [36].

5. Stability of LFPs To investigate the stability of LFPs, we can use the following lemma which is implied by Lyapunov's direct method: Lemma 5.1. Consider the Hamiltonian system (M, 0, h). If x is a strict local extremum point of the Hamiltonian h, then x is a stable FP of the flow. Incleed, there are a few interest.ing cases where LFPs do ext.remize the Hamiltonian. The problem though is that, due to the 50(3) symmet.ry of the model, these extrema are not strict on S and therefore Lemma 5.1 does not apply. Luckily, it. is possible to overcome t.his difficulty by finding an invariant subspace r of S in which the above extrema of the Hamiltonian are strict. and thus stable FPs, and then extend t.his stability to S by using the symmetry of the model. This idea is formalized in a general theorem [36]. In this paper, we will tailor t.his strategy directly to t.he model at hand: Lemma 5.2. If a LFP x is a strict local extremum of the Hamiltonian h in

where n is some unit 'Vector, then it is Lya.punov sta.ble in

S(s}J'

Proof. Let x be the LFP, r.r the r-manifold through x, B(x, f) the ball in S cent.ered at with radius f > 0, and Ft the flow on S. According to Lemma 5.1, since :r is a strict local extremum in r x , x is st.able in r x ' Thus, 3E],E2,E3 : E/2 > E] > 2E2 > 4E3 > 0,

.T

and (5.1)

Since the total spin v is a continuous function on Sand Vx # 0, 3151 E (0, (3) : Vy E B(x, bd. Then, there exists a continuous function R : B(x, 15]) --750(3), where Ry is a rotation on the plane of the unit vectors Tt x , Tty such that Tty = RTt x · This implies that the function B(x, 151 ) 3 y --) Z = (EBJ=I Ry)x E ry is also continuous. Therefore, 315 E (O,bd : z(B(x, b)) c B(z(x), (3) = B(.T, (3). Now, if y E B(:r, b), then d(y, z) :::; d(y , x) + d( z, x) < 15 + E3 < 15 1 + fa < 2E3 < E2 , thus y E B(z, (2) n r z · Since Ry == EB.7=1 Ry is a symmetry transformat.ion, [Ry, Ft ] = 0, and so, acting wit.h Ry on (5 .1) yiclds: B(Z,EI) nr z :) Ft (B(z,E2) nrz) 3 Ft(y) = y(t). As a result, Vy

# 0,

238

L. Pantelidis

"It E lR, d(.r,y(t)) :::; d(1:,z) + d(z,y(t)) < E3 + E1 < 2E1 < E. We have shown that "IE> O,:J5 > 0: Ft (B(.T,5)) C B(1:, E), "It E lR, therefore 1: is Lyapunov stable on

S.

0

One may want to include the case that :r has zero spin by replacing the semi closed interval in the definition of fh).n with a closed interval. However, if v'" = 0 the proof of the Lemma breaks down for the map R cannot be defined in that ca.se. Now a.s long as t.he total spin docs not vanish, Lemma 5.2 immediately applies t.o t.he following four cases: • Ferromagnet. minimum h.

(gjl

:::> 0 for all j, I). The FR stat.e (all spins parallel) corresponds to

• Ant.iferromagnet without intra-sublattice interaction (bipart.it.e lattice, sublattices A and B, gj(A)I(A) = gj(B)I(B) = 0, gj(A)I(B) :::; 0). The FR statc corresponds t.o maximulIl h. • Ferromagnet wit.hout. intra-sublattice interaction (bipartitc lattice, sublattices A and B, gj(A)I(A) = gj(B)I(B) = 0, gj(A)I(B) :::> 0). The AF state (all spins parallel in a sublat.tice, sublattices anti-parallel) corresponds to maxilIlUIn h. • Antiferromagnet (bipartite lattice, sublattices A and B, gj(A)I(A) :::> 0, 9j(A)I(B) :::; 0). The AF stat.e corresponds to minimum h.

gj(B)I(B)

:::> 0,

Note that in all four cases the extrema arc global. Moreover, they arc extrema on every leaf S, i.e., for random spin lengths (ferrimagnetie case). On the other hand, a LFP 1:, which is a strict local extremum of the Hamiltonian, may well be unstable if its spin is zero. To show this, consider an antiferromagnet or ferromagnet with a finite bipartite lattice without int.ra-sublattice exchange and with periodic boundary conditions. In this ca.se, there exists a solution such that all the spins in a sublattice are equal [36]. The sublattice total spins VA and VB precess around the direction of V = VA + VB (Figure 2) with frequency

Now notice that if VA f 1)B, then as the geometric angle between VA and VB tends to 7r, the angle between V and VA (or VB) tends to either 0 or 7r. In either case the amplitUde (i.e., the radius) of revolution of each spin t.ends to zero in agreement with the fact that the AF state with V f 0 must be a stable LFP. However, if VA = VB, t.he angle between V and VA (or VB) tends to 7r /2 which corresponds to maximum amplitude of revolution, i.e., the spins wander away from their nearly equilibrium position. Of course, at this limit the frequency tends to zero. Nevert.heless, we have found periodic orbits passing arbitrarily close to t.he AF st.at.e on the equal-spins leaf (which has V = 0), while also moving far from this state after a long enough time. We conclude that the AF state on the equal-spins leaf is unstable.

Qualitative Dynamics of Interacting Classical Spins

r: /' -

j j

I v iO

-I-

'~

- r- - ~ I I I I

~v

239

II lJ v=O

\ \ V \ \ \ \ 1\ 1\ \

I

1\ I I II

Figure 2. Solutions for a 4-spin ring with the two spins in each subiatbee equal. If the total spin v of the AF state is no t zem (left), the AF state is stable. If v vanishes (right), the AF state is unstable.

Acknow ledgments The author is indebted to Dr. Harilaos Skiadas for helpful discussions.

References [lJ P. \V. Anderson, New approach to the theory of svperexchange interactions, Physical R eview 115 (1959), no. 1 2-1 3. [2J R. lvI. White, Quantum Theory of Magnetism. IvlcGraw-Hill , Inc ., 1970. [3J K. Yosida, Theory of Magnetism. Springer-Verlag, 1996. [4J L. P. Levy, Magnetism and Superconductivity. Texts and Monographs in Physics. Springer-Verlag, 2000. [5J E. H. Lieb and D. Mattis, Ordering energy levels of intemcting spin systems, Journal of Mathematical Physics 3 (1962) 749-75l. [6J .J. des Cloizeaux and J . .J. Pearson, Spin-wave spectmm of the antiferromagnetic linear chain, Physical R eview 128 (1962) , no. 5 2131- 21 35. [7J J. C. Bonncr and M. E. Fisher, Linear' magnetic chains with anisotropic coupling, Physical Review 135 (1964), no. 3a 640A 658A. [8J .J. Frohlich and E. H. Lieb, Phase tmnsitions in anisotropic latNee spin systems, Comrrmnicah.ons in Mathematical Physics 60 (1978) 233-267. [9J F. Haldane, Continuum dynamics of the 1 - D Heisenberg antiferrvmagnet: Identification with the 0(3) nonlineaT sigma model, Physics Letters 93A (1982), no. 9 464 468. [lOJ R. Botet, R. Julien and M. Kolb , Finite-size scaling study of the spin-l Heisenberg-Ising chain with uniaxial anisotropy, Physical R eview B 28 (1983) 3914-392 1. [l1 J J. B. Parkinson and J. C. Bonner, Spin chain8 in a field: CTOssoveT fTOm quantum to classical behavior, Physical Review B 32 (1985) , no. 74703- 4724. [12J H. J. Schulz, Phase diagr'ams and correlation exponents for quantum spin chains of arbitrary spin quantum number, Physical Review B 34 (1986), no. 9 6372- 6385. [13J 1. Affleck and E . H. Lieb , A proof of part of Haldane 's conjecture on spin rha'ins, LetteT8 in Mathematical Physics 12 (1986) 57-G9.

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[14] I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki, Rigorous resnits on valence-bond ground states in antiferromagnets, Physical Review Letters 59 (1987), no. 7799-802. [IS] P. Horsch and W. von der Linden, Spin-correlations and low lying excited states of the spin-l/2 He'isenberg antifer-romagnet on a square lattice, Z. Phys. B 72 (1988) 181-193. [16] D. R. N. S. Chakravarty, B. I. Halperin, Two-dimensional quantum Heisenberg antiferromagnet at low temperatures, Physical Review B 39 (1988), no. 42344-237l. [17] H. Neuberger and T. Ziman, Finite-size effects in Heisenbery antiferromagnets, Physical Review B 39 (1989), no. 4 2608-2618. [18] E. Manousakis, The spin-l/2 Heisenbery antiferromagnet on a square lattice and its application to the caprous oxides, Review of Modern Physics 63 (1991), no. 1 1-62. [19] T. Kennedy ane! H. Tasaki, Hidden symmetry break'ing and the Haldane phase in S = 1 quantum spin chains, Communications in Mathematical Physics 147 (1992) 431--484. [20] S. R. \Vhite and D. A. Huse, Numerical rerwrmalization-group study of low-lying eigenstates of the antiferromagnetic s = 1 Heisenberg chain, Physical Review B 48 (1993), no. 6 3844-3852. [21] O. Golinelli, T. Jolicoeur and R.. Lacaze, Finite-lattice extmpolat'ions for a Haldane-gap antiferromagnet, Physical Review B 50 (1994), no. 5 3037-3044. [22] S. Yamamoto, Qnantum Monte Cado approach to elementary cuitations of antiferromagnetic Heisenbery chains, Physical Review Letters 75 (1995), no. 18 3348 -335l. [23] E. Dagotto ane! T. M. Rice, Surprises on the way from ld to 2d quantum magnets: The novel ladder materials, Science 271 (1996) 218-235. [24] B. B. Beare!, R. J. Birgeneau, M. Greven and U. J. Wiese, The square-lattice Heisenbery antiferromagnet at very large correlation lengths, Physical Review Letters 80 (1998) 1742-1745. [25] X. Wang, S. Qin ane! L. Yu, Haldane gap for the s=2 antifermmagnetic Heisenberg chain revisited, Physical Review B 60 (1999), no. 21 14529-14532. [26] K. M. Leung, D. W. Hone, D. L. Mills, P. S. Riseborough and S. E. Trullinger, Solitons in the linear chain antiferromagnet, Physical Review B 21 (1980), no. 9 4017-4026. [27] H. J. Mikeska and Iv!. Steiner, Solitary excitations in one-dimensional magnets, Advances in Physics 40 (1991), no. 3 191356. [28] Iv!. Daniel and R. Amuda, Nonl'inear dynamics of weak fermmagnetic spin chains, Journal of Physics A: Mathematical and Geneml28 (1995) 5529-5537. [29] Iv!. DanieL K. Porsezian and Iv!. Lakshmanan, On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in ar'bitmTY d'irnensions, Journal of Mathematical Physics 35 (1994), no. 126498 6510. [30] K. Porsezian, Nonlinear' dynamics of the radially symmetric and site dependent anisotropic Heisenberg spin chain, Chaos, Solitons and Fractals 8 (1997), no. 1 27-3l. [31] Iv!. Daniel and E. Gutkin, The dynamics of a generalized Heisenberg fermrnagnetic spin chain, Chaos 5 (1994), no. 2 439-442. [32] .1. A. G. Roberts and C. J. Thompson, Dynamics of the classical Heisenberg spin chain, Journal of Physics A: Mathematical and General 21 (1988) 1769-1780. [33] R. Balakrishnan and R. Dhamankar, E1:act nonlinear spin waves in some models of interacting classical spins on a one-dimensional lattice, Physics Letters A 237 (1997) 73-79. [:l4] L. Pantelidis, Nonl'inear spin waves for' the He-isenberg model and the ferTOmagnet-ic-antiferromagnetic bifurcations, JOTJ.rnal of Physics A: Mathematical and General 37 (2004) 8835-8852. [:J5] L. Pantelidis, Ojj~8hell nonlinear spin waves for the Heisenbery model, Journal of Physics A: Mathematical and TheoTetical41 (2008) 1-21. [36] L. Pantelidis, Dynamics of the Heisenbery model a.nd a theorem on stability, .mbmitted to the Journal of Physics A: Matherrwtical and Theoretical. [37] R. Abraham and J. E. Marsden, FO"UTuiatioTl.8 of Mechanics. The benjamin/cummings publishing company, 1978.

241

A Transmission Line Model for the Spherical Beltrami Problem C. D. Papageorgiou I and T. E. Raptis 2 Dept. of Electrical & Electronic Engineering National Technical University of Athens, Greece (e-mail: [email protected]) 2 Division of Applied Technologies National Centre for Science and Research "Demokritos" Patriarchou Grigoriou & Neapoleos, Athens, Greece (e-mail: [email protected]) I

Abstract: We extend a previously introduced model for finding eigenvalues and eigenfunctions of PDEs with a certain natural symmetry set based on an analysis of an equivalent transmission line circuit. This was previously applied with success in the case of optical fibers [8], [9] as well as in the case of a linear Schroedinger equation [10], [II] and recently in the case of spherical symmetry (Ball Lightning) [12]. We explore the interpretation of eigenvalues as resonances of the corresponding transmission line model. We use the generic Beltrami problem of non-constant eigen-vorticity in spherical coordinates as a test bed and we locate the bound states and the eigen-vorticity functions.

1. Introduction

The notion of a Beltrami field appears first in hydrodynamics in the studies of Eugenio Beltrami [I] at the end of the 19th century where it has found widespread application in vortical fluid states with a characteristic helical geometry. Its rediscovery in modern electromagnetism took place at the middle of the 20th century when Lundquist [2] proposed the same model for galactic magnetic fields where the field appears to be parallel to its own rotation (neglecting displacement current). Later, Lust and Schluter [3] and Chandrasekhar and Woltjier [4] proposed a theoretical justification of these states in terms of magnetic plasma equilibrium. Several types of solutions are reviewed in [6] and [7]. Nowadays, the Beltrami equation serves as a model of a) "Force-Free Fields" for which the magnetic part of the Lorentz force disappears (J II B) in equilibrated plasma and b) for parallel Ell B components when applied to the vector potential directly (in which case we get E

~

-{()A, B

~

A(r ,t)A).

The Beltrami problem is defined by at least two and at most three PDEs given by V xA

~

A(r,t)A

(J a) (J b)

The second condition is usefull in applications in electromagnetism when A stands either for the vector potential or the magnetic field. Applying twice the rotation operator in the first of (I) gives the most general representation of the non-linear inhomogeneous Helmholtz operator for Beltrami flows as

242

C. D. Papageorgiou and T. E. Raptis V'A-l'(r,t)A=VAXA

(2)

In the case of constant A the inhomogeneous part is of course zero. Solutions of this class are usually termed linear "Trkalian" fields as they were first studied extensively by Viktor Trkal [5]. In the next paragraph we turn to equation (I) and attempt to locate eigen-modes in terms of the orthogonal basis of vector spherical harmonics. In section 2, we perform the expansion and derive the associated ODEs and in section 3, we introduce the transmission line model for this system. In section 4, we solve the model for the associated angular momentum eigenvalue and we approximate the specific eigen-mode for a "bound state".

2. Spherical expansion of the Beltrami equation In what follows we restrict attention to purely radial functions for the eigen-vorticity ACr). We introduce the usual basis of Vector Spherical Harmonics as CY = r'l"m' tft = rV'l',m. = r X V'l"m } and we seek an appropriate decomposition of the field A and its eigen-vorticity A(r). Then there must be a unique reference frame for which both equations (I a-b) can be satisfied by taking

,

A=

I I 10",0

(3)

[a'm(r)Y"'m + b'm(r)tft'm +c'm(r),J

1I1'=-{

Then condition VA = 0 is satisfied if VA -- L..J ~ ~[(a + 2a'm L...J 1m !=() 111=_/ r

J- 1(1 + I) b r

/m

( r)

J'l' /111-

0

(4)

while the choice of the A function is restricted by VA. A = O. For a purely radial A function it suffices to take the additional condition that the radial part of A is set to zero which is a very stringent condition on the coefficients of (3). We note however that in the more general case we may allow for violations of condition (I b). In electromagnetic applications we may instead assume the presence of static charges satisfying the Lorenz gauge ,t/J = -cV A = cxlv A. A .

a

From now on we will deal solely with the first condition (I a) for the rotation of A. We now turn back to the first of (I) for the rotation operator which obtains

VxA =

I~ I' {- -1(1-r+-I)c !=() 111=-1

1m

c)'-l' + (b.

- - ( i: +-""Y fill 1m r

/111

/111

a J- } (5)

b +-""--~

r

r

/111

For the rotation to be parallel to AA we should have 1(1 + I)

----e,1II = r

C/I/!

All/III

+~+AbIIl1 r

b1m + him - ~ r

r

=0

AC fill = 0

(6)

A Transmission Line Modelfor Spherical Beltrami Problem

243

Using the first of (6) to the rest two equations and replacing elm with e and blm with b results in the ODE system (rb)'

=[A - 1(1r2+Al)](rc)

(7)

(re)' = -A(rb)

3. The Transmission Line model At this section we introduce an isomorphism between the system (7) and a lossless transmission line via the mapping

v = re ---> dV = -jAI dr

. dl y' JI =rb --->-=--V dr jA

(8)

Where we have introduced the propagation constant defined by the relation Equivalently we may define the transmittances to be Z, = -jA

Y,

=-j~

Then the characteristic impedance are defined through the relations: -Z

(9)

Z and the propagation factor y of the equivalent transmission line

rz; ..,1, '~l =Vy:=Jy=V;:y--l

(10)

y=~Z,y, From standard transmission line theory we have that propagation is defined for IAI

2

Y <0 or r>

~

(11)

-yl(l+l)

While damping (evanescent modes) for ,

1.,1,1

r>O or r< ~ -yl(l+l)

(12)

An element of such a transmission line corresponds to a lumped-element model of the initial domain [0, ooj of the radius r into a set of mesh points ro' r, "'" r" with each element interpolating into the arbitrary intervals [r",r,,+,j,We then take each element to correspond to a segment of infinitesimal width

LY"

= ':,+1 - ,:, for which the propagation factor y and the characteristic impedance Z can be

244

C. D. Papageorgiou and T. E. Raptis

taken approximately constant (defined as Yn and Z,, ) . For such an element we may take the equivalent relations for a homogeneous transmission line as follows.

[ cosh(q,,) [ V(~,ll I (~,l = sinh(q,,) / Z"

(13)

where we have put q" = r" . /';.~, . Thus the value of the input impedance at each point of the line will be given by the recursive formula Z(r ) = V(r,,) = " I (r,,)

Z

Z(r,,+l) + Z" tanh(q,,) " Z(r,,+l) tanh(q,,) + Z"

(14)

The boundary conditions are defined through the respective limitsZ(r ~ O),Z(r ~ 00) based on the assumption that the asymptotic limit of a lossless transmission line corresponds to the fixed value of the characteristic impedance given by Z" according to (14) for r ~ 0 and r ~ 00.

4. Trapped modes and bound states The form of the propagation constant leads to the possibility of "trapped" waves and their respective eigen-modes. A necessary condition for trapping is to have at least one region r1 ::; r ::; r, where y2 (r) < oallowing propagation and two disparate regions r < r" r > r 2 with y2 (r) > 0 where only evanescent modes exist. This situation is reminiscent of the bound states of quantum mechanical systems with the function 112 (r) playing a role analogous to that of the potential in the radial Schroedinger equation. Evanescent modes are then analogous to the scattered wave functions above a threshold. On the other hand, a constant value of A allows only a simple root r3 = ~l(l + I) / Au beyond which propagation is allowed up to the infinity. We will next evaluate one such example of an eigenvorticity function supporting a trapped mode. According to a standard technique introduced in [11] and [12] , a method for locating the eigenvalues and compute the associated eigen-mode is based on a simple resonance theory for transmission lines. In this viewpoint, we may take the bound state to correspond to an area where ordinary resonance takes place through the condition Z L = Zc between the "capacitive" and "inductive" elements in the imaginary part of the respective complex impedance. This can effectively be restated as (15) Where Z + (~) and Z - (~.) are the input impedances of the equivalent transmission line as shown from the point rc (right and left of it). Evolution of the above quantity in the region [0,00] prescribes also the associated eigen-mode. In order to locate a certain mode for a given value of angular momentum we iterate the mapping (14) forward (0 ~ r, ) and backward (00 ~ 1',. ) and expand the interval until we find a root of (15). As an example let us make a simple choice of the propagation factor given below

r'= L2(r-~)(r-r2) r + fir2r 4

2'

h

were,

L=l(l

+ 1)

(16)

A Transmission Line Modelfor Spherical Beltrami Problem The propagating region is 'i

:0;

245

r :0; r, and the evanescent regions are r < ~, r > r,

· ,.2 ,'.It can be proved that: · th Usmg ereIatlOn r = 2L' - /L. r

,12 = L2 .(~ +r2)

(17)

r·(r' +~r,)

For given integer values of I and real positive values of rl using the equation (15) for rl and variable values of r2>rl i.e. (18) Quantities Z+(~) and Z·(~)are calculated through the recursive relation (14). Then the value of r2 for which the equation (18) holds can be defined and though it the proper function A(r) by (17) for which the Beltrami equation (1 a) is fulfilled. In Fig. 1 we show fori = I rl=1 and r2=8.12 the exact form of the propagation factor y' and the eigen·vorticity ,12 that is positive definite everywhere. 1=1 ;r1=1 ;r2=B.12 3,-----~-----,------._----_,------,_----~

.£ " t o > C


.ijj

~

o

I I

I

I

j

--I----~------T-------I-------I-------~------

2

--1---r------1------~-------r------r------

I

I

I

\

"ffi

~c

I

2.5

I

I

I

I

I

I

I

I

I

I

\

1.5

I

I

I

\

- - - --\. - - T - - - - - - I - - - - - - -1- - - - - - -

r - - - - - -

j

T - - - - --

o

.~

g>

e"'a..

0.5

·0.5'----~-----'------'-----'----~---~

o

2

4

6 radial distance r

B

10

12

Fig.! In Fig. 2 we show the evolution of the Function (25) of the sum of the imaginary parts for 1= 1 and rl=l.The boundary values of the impedances are defined for the terminal points left and right + 2d I·Z',1· Z·,"",,,/ =Z(drI2),Z '"'Ii,,/ =Z(R+drl ) uetot hereatlOn =J._=J.

r

r· (~

+ r,)

. The roots

(r-~)·(r-r,)

are between the poles of the Function (18) and can be easily calculated. The first three values of r2 are 8.12, 33.5428, 86.8528.

246

C. D. Papageorgiou and T. E. Raptis 1~1 ,rl~1

,the roots of the Function are between the poles of the Function

300

~ ~oo

---I----~----~-----I----~----~----~---

-300

---I-------j----,-----I-------,-------j----.,---

'c::" "g>

.~

"

'0

c:: -400 0

,

,

---I----'----~-----I----~----~----~---

,

"-B c::

::>

LL

-500

I

I

I

, ,----

" I

I

I

I

I

--~----~----~----~---------~----~---

~----

-600 20

40

60 80 100 120 140 eigen values of r2 as roots of the Function

160

180

Fig. 2 In appendix I a matlab program of the Function( IS) is shown.

5. Beltrami solutions through the Lorentz group From the previous analysis a fundamental observation occurs based on the hyperbolic relation connecting the triplet {y, It, LI r} which can be written as

(26)

Going to the transformed variables

x

= y'(r)

(27)

y =lt2(r) T

= j/ r

leads to the Lorentzian manifold M(2,1) with the usual Minkowski metric of signature (1,1,-1) that has an invariant "null" length (virtual photon path) s2=X'+Y'-(LT)2=O

(2S)

and a "speed-of-light" c = L. This means that given a set of "core" functions (Yo,lto ) and an angular momentum value L we can produce an infinite family of deformed solutions which are connected by both proper and improper Lorentz transforms. Thus, for the choice represented by (17) we may find a continuous

A Transmission Line Modelfor Spherical Beltrami Problem

247

trajectory passing through different families of functions {r", A" ) by a direct application of the generic linear transformation (X ,T)~(X',T'): (rv,A,,)~(rv+' ,Av +,) where the matrix A(L) is parametrized according to the four conjugacy classes of the full Lorentz group (elliptic, hyperbolic, parabolic and Loxodromic) thus leading to different families of solutions.

6. Conclusions We showed that a general class of solutions of the spherical Beltrami problem can be found in terms of vector spherical harmonics of which the resulting system is equivalent with a generic loss less transmission line. When applied to a specific choice of eigen-vorticity function it allows finding the set of parameters necessary for each mode to exist. We have given explicitly an example of such a function for which trapped modes can exist in a region. We believe that this phenomenon deserves further examination especially with recourse to electromagnetic applications.

References [1] E. Beltrami, Rend. Inst. Lombardo Acad. Sci. Lett. 22 (1889) 121-131 [2] S. Lundquist, Ark. Fys. 2 (1951) 361 [3] R. Lust, A. Schluter, Z. Astrophys. 34 (1954) 263 [4] S. Chandrasekhar, P. C. Kendall, Astrophys. J. 126 (1957) 126 [5] A. Lakhtakia, Czechoslovak J. Phys. 44(2) (1994) 89 - 96 [6] H. Zaghloul, O. Barajas, Am. J. Phys. 58(8) (1990) 783-788 [7] G. E. Marsh, "Force-Free Magnetic Fields", World Sci. (1996) [8] C.D. Papageorgiou, 10 Kanellopoulos, J. Phys. A: Math. Gen. 15 (1982) 2569-2580. [9] c. D. Papageorgiou, A. D. Raptis, Compo Phys. Com. 43 (1987) 325 - 328 [10] c. D. Papageorgiou et aI., J. Compo App. Math. 29 (1990) 61-67 [II] C. D. Papageorgiou et al.,J. Compo Phys. 88(2) (1990) 477-483. [12] C. D. Papageorgiou, T. E. Raptis, under preparation.

248

C. D. Papageorgiou and T. E. Raptis

Appendix I function y=beltram(r2) global 1 rl l,rl are defined externally dr=rl/M, M=number of elements left of rl,N=total number of elements range dr/2
;

N~200*M; LL~l*ll+l);

dr=rl/Mi R=dr*Ni % defining the terminal impedance in the outer terminal point R+dr/2 r=R+dr/2i L~sqrtILL*lrl+r2)/r/lrA2+rl*r2));

cn=sqrt(LL/r A 2-L A 2+10 A -IO);zn=j*L/cn; zl= zn i recursive formula for the calculation of "right of rl" impedance for n=l:N-M r~R-dr*n+dr/2;L~sqrtILL*lrl+r2)/r/lrA2+rl*r2));

cn=sqrt(LL/r A 2-L A 2+10 A -IO);zn=j*L/cn; z1=zn* (z 1+zn *tanh (cn*dr) ) / (z 1 *tanh (en *dr) + zn) ; end

% defining the terminal impedance in the inner terminal point dr/2 r~dr/2; L~sqrt ILL* (rl+r2) Ir I IrA2+rl *r2) ) ; cn=sqrt(LL/r A 2-L A 2+10 A -IO);zn=j*L/cn; z2=zn;

recursive formula for the calculation of "left of rI" impedance for n=I:M r~n*dr-dr/2;L~sqrt(LL*(rl+r2)/r/lrA2+rl*r2)) ; cn=sqrt(LL/r A 2-L A 2+10 A -IO);zn=j*L/cn; z2=zn* (z2+zn *tanh (cn *dr) ) / (z2*tanh (cn*dr) + zn) ; end y=imag(z2+z1);

249

Two Aspects of Optimum CSK Communication: Spreading and Decoding Theodore Papamarkou University of Warwick Department of Statistics Coventry, CV 4 7AL, UK (e-mail: [email protected] . uk)

Abstract: This paper focus es on the optimization of performance of single-user chaos shift-keying (CSK). More efficient signal transmission is achieved in the coherent case by introducing the class of the so-called deformed circular maps for the generation of spreading. Also, the paired Bernoulli circular spreading (PBCS) is introduced as an optimal choice, which attains the lower bound of bit error rate (BER). As interest shifts to the non-coherent version of the system, attent ion moves to the receiver end. Maximum likelihood (ML) decoding is utilized serving as an improvement over the correlation decoder. To make the methodology numerically realizable, a Monte Carlo likelihood approach is employed. Keywords: communication systems, chaos shift-keying, optimal spreading, correlation decoding, likelihood decoding, Monte Carlo likelihood.

1

Single-user coherent CSK: The model

A brief description of single-user coherent CSK, based on [4] , is provided. Assume that the information source comprises of K binary bits, each of which is transmitted N times for reliability purposes. Let b = ±1 denote any single binary bit out of the K, which is replicated N times. A stationary process X := (Xo, Xl ' . .. ' X N-d of mean J-Lx and variance O"~ is involved in the modulation process. X is called the spreading and N the spreading factor. The transmitter emits the scalar product T:= b(X - J-Lxl). The N-length transmitted signal T is degraded as it passes through the channel. Stochastic channel noise E := (EO , EI , . . . , EN- d models the corruption of T. It is assumed that the system is affected by white channel noise, which means that E ~ N(O, 0";1), where 0 and 1 stand for the N -length null vector and the N x N identity matrix respectively. A first modelling approach would be to set the received signal R to be the transmitted signal T distorted by the additive channel noise E , that is

R:= T

+ E = b(X -

J-Lxl)

+ E.

(1)

250

2

T. Papal7larkou

Criteria for optimal spreading

When evaluating the reliability of the system, simulations of bit error rate (BER) against signal-to-noise ration (SNR) are run. BER is defined as the probability of erroneously decoding a single bit b. A lower bound has been found for BER. For more details, see [4]. Vital for the outcome of BER simulations is the bit energy S(X) of spreading X, which is defined to be the function S(X):= I X ~ P,xlI12/(J"~. The choice of X affects BER. There thus arises the question how could X be selected in order to minimize BER. In an attempt to offer an answer, the concept of bit energy helped to form three criteria for optimally choosing the spreading. They are summarized here hierarchically from the most to the least stringent. As the firmness of the condition imposed by each of the three successive criteria loosens, the possibility of attaining the condition increases with a relative payoff in the system's performance. Optimality Criterion I: Define the spreading X in a way that its bit energy S(X) becomes a constant equal to the spreading length N. X is then optimal. Optimality Criterion II: Set X such that S(X) has zero variance. Optimality Criterion III: Choose X whose mean-adjusted quadratic autocorrelation has lag(l) which is as close as possible to ~l, that is

Corr[(Xt~p,X)2,(Xt+l~P,X)2] =~1.

(2)

[8] and [5] elaborate on how these criteria have been derived.

3

Family of deformed circular maps

One way of generating the stationary spreading X would be to start from a random variable Xo and iteratively produce the rest of the N ~ 1 members of X by means of an one-dimensional map T : [c, d] ----) [c, d], c, dE R Then,

Xn=T(Xn-d, n=1,2, ... ,N~1.

(3)

Widely used choices of T are, as instances, the tent, Bernoulli and logistic maps. However, T can be defined in ways that better conform with the third optimality criterion. In trying to reduce the lag(l) of mean-adjusted quadratic autocorrelation function (ACF), the class of deformed circular maps has been defined as

(4)

Two Aspects of Optimum CSK Communication

251

An explanation on how the deformed circular family has been established is available in [5]. r is called the deforming parameter and takes values in (0, 1). Figure 1 demonstrates that values of r < 0.5 squeeze the central branch of the map, while values of r > 0.5 stretch it.

---l

1

r-- ---,

1r------

I

\

-1

-1

-1

Li. ____--'

-1

-1

(a) r = 0.1.

(c) r=0.8.

(b) r = 0.42.

1\\

\

L i

o -1

-1

(d) r=O.1.

/ \

/

·/ I

---~

0

(e) r=0.42.

I I

o ~~_.---.J -1

0

(f) r = 0.8.

Fig. 1. Three examples of deformed circular maps and their invariant densities. (a): Map for r = 0.1, (b): map for r = 0.42, (c): map for r = 0.8, (d): density for r = 0.1, (e): density for r = 0.42, (f): density for r = 0.8.

It can be confirmed that the function f(x), given by f(x) '= {-2(1 - r)x, -1::; x ::; 0 .' 2rx, 0 < x ::; 1 '

(5)

satisfies the Perron-Frobenius equation. Thus, f(x) can be seen as the invariant distribution of spreading iteratively produced by deformed circular maps. Figure 1 displays the plot of f(x) for three values of r. For r = 0.5, the so called circular map has the "V-shaped" invariant density f(x) = lxi, x E [-1,1], which results in zero mean. The meanadjusted quadratic ACF is known for any lag (see [6]):

(6) lag(l) = -0.5 according to (6) and therefore the circular map has smaller first lag than the tent and Bernoulli maps, which have lag(l) = 0.25.

252

T. Papamarkou

The lo,g(l) value of the mean-adjusted quadratic ACF has been calculated as a function of deforming para~eter r, see Figure 2. With the help of numerical minimization, it has been found that the minimallo,g(l) is approximately equal to -0.722 and is achieved for r c:::' 0.42. So, the optimal deformed circular map, with respect to the third optimality criterion, is the one with deforming parameter r c:::' 0.42.

Fig. 2. Dashed line: Plot of lag(l) of mean-adjusted quadratic ACF of deformed circular family versus its deforming parameter r . Solid line: Fnkhet lower bound of lag(l) of mean-adjusted quadratic ACF for the class of spreading sequences with invariant density given by (5).

Jag(l ) 0.4 0.2 - 0.2 -0.4 - 0.6

- 0.8 -1.0

Also, the Fnkhet lower bound of lo,g(l) has been computed, see [7]. Such a lower bound gives the minimal attainable lo,g(l) mean-adjusted quadratic autocorrelation among all stationary processes with invariant distribution described by (5), see Figure 2. Despite having not reached the lo,g(l) lower bound, spreading produced by the deformed circular maps outperforms that generated by the logistic, Bernoulli, tent and other conventionally used maps. BER simulations confirm the superiority of deformed circular family, see Figure 3.

10-2 +------------ , Tent Logistic Del Circ, r=0.42

10-3

10-3 ~---L-B~------~----~ o 5 10 -5

10-4

(a) N

=

2.

Tent Logistic Del Circ, r=O.42 LB

-5

0

5

10

(b) N=5.

Fig. 3. Simulated BER of tent, logistic, circular and deformed circular (r = 0.42) spreading for two values of spreading factor N.

Two Aspects of Optimum CSK Communication

4

253

Paired Bernoulli circular spreading (PBCS)

When r = 0.5, the Frechet lower bound of the first lag equals - 1. This implies that there exists a stationary process with the "V-shaped" invariant density f( x ) = lxi, x E [- 1, 1] and lag(l) mean-adjusted autocorrelation equal to -1. Starting from that knowledge of the distribution of spreading which fully meets the third optimality criterion, the Bernoulli circular spreading (BCS) has been defined, see [7]. Although BER simulations showed that BCS is a relatively efficient type of spreading, it has a main drawback; BCS can take only four values. As an improvement over BCS , paired B ernoulli circular spreading (PBCS) has been suggested. To introduce PBCS, let X := (Xo, Xl,'" , XN - d denote the N - length spreading sequence, where N is any even natural number. Also , consider the function f

k lR. j '( x) '= { Ix - kl, x E [k - 1, k + 1] . 0, x E lR. \ [k - 1, k + 1]' E ,

(7)

and the random vector (AI , A3 , . .. , AN-d given by A 2i -

l

:=

(_1)1-<1'2,- 1,

l

~ B er(p),

i E { 1, 2, ··· ,

~} .

(8)

The even subscripted members of PBCS are defined as X 2· 1.

~ .

-2}

N fiE { 1 2 ... , '" 2

(9)

'

while the odd subscripted ones are

X 2i- l = k

+ A 2i- l VI -

(X2i - 2

-

kF, i E

{I,

2, ··· ,

~} .

(10)

It has been shown that (7) satisfies the P erron-Frobenius equation and can thus be accepted as the invariant distribution of PBCS. To further explain how PBCS is constructed, it could be said that the even subscripted members X o, X 2 , X N - 2 of X are sampled from (7). To obtain the odd subscripted members X 2i - 1 , a Bernoulli trial is performed to specify A 2i - l and then (10) is invoked in order to calculate X 2i - l by means of the previous value X2 i-2, which has been sampled from (7). Thinking of PBCS pairwise allows for its geometrical interpretation. Each ordered pa ir (X 2i , X2 i+l), i E {O , 1, · .. , (N - 2) / 2} , represents a point in a circle of centre (k , k) and radius 1, see Figure 4. PBCS can take an infinite number of values. Geometrically speaking, any point of the unit circle with centre (k, k) could be included in PBCS. On the contrary, BCS would only allow four possible points of the circle to appear

254

T. Papamarkou 1

~O I"..

:.

-1L~ -1

0

-1

\

\

)

, - . -.•.. -~1--

6

~/! 1

X2i

(c) N = 1000.

(b) N = 50.

(a) N = 20.

I

I'

1

X2i

/ _ ••• _ ' "

Fig. 4. Simulation of PBeS from the unit circle of centre (0,0) for spreading length N = 20, N = 50 and N = 1000. in the spreading. So, one of the goals of PBCS has been achieved, namely to heal the weakness of BCS by allowing an infinite number of real-valued numbers in [k - 1, k + 1] to appear in the spreading. Another question to pose is how well PBCS satisfies the third optimality criterion. The lag(l) of quadratic ACF is given by

[

2

CarT (Xi - k) ,(Xi+l - k)

2]

=

{ -1,O'fif i. is. odd ,1

1. IS

even

.

(11)

So, the first lag equals -1 as intended. However, this is the case only for even subscripted members of the spreading and from that point of view the third optimality criterion is not fully met. It is interesting to point out that the invariant distribution of PBCS for k = 0 coincides with the "V-shaped" density of circular map, which has deforming parameter r = 0.5. Recall that the Frechet lower bound for the first lag has been calculated to be -1 when r = 0.5. So, there has been found a stationary process whose odd subscripted members attain the Frechet lower bound of lag(l) mean-adjusted quadratic autocorrelation. More importantly, it has been proven that the bit energy of PBCS is constant and equal to the spreading length N. PBCS therefore meets the first optimality criterion. It has also been proven that any stationary process which meet the first optimality criterion has BER equal to the BER lower bound given in [4]. BER simulations have been run too and they verified the theory. So, one could safely state the conclusion that PBCS is optimal without leaving room for further reduction of BER.

5

Single-user non-coherent CSK: The model

So far optimization has focused on the transmitter and more specifically on the choice of spreading. In the coherent version of single-user CSK, there is not much to be done at the receiver end. The traditional engineering tool for decoding is the correlation decoder, which coincides with the likelihood decoder in the coherent case.

Two Aspects of Optimum CSK Communication

255

However, one may assume that the spreading X is not known at the receiver, as opposed to the coherent system. Then not only T = b(X - /LxI), but also X has to be transmitted. Both N-Iength signals T and X are corrupted by noise when passing through the channel. Eventually, two vectors Rand Y received:

R

:=

b(X - /LxI)

Y := X - /Lx 1

+ E,

(12)

+ Tl,

(13)

where it is assumed that E ~ N(O, (]"21) and Tl ~ N(O, (]"21). Since T and X pass through the same channel, it is plausible to think of E and Tl having common error variance (]"2. E and Tl are also assumed to be independent of each other.

6

Monte Carlo Maximum Likelihood Decoding

For the non-coherent system described by (12) and (13), the correlation decoder C(y, r) computes as the inner product C(y, r) := (y, r). The decoding rule based on the correlation decoder is

C(y,r) ~ 0 =:?

b=

+1, C(y,r) < 0 =:?

b =-1.

(14)

Motivated by [3], there was the thought that maximum likelihood (ML) decoding might be a more efficient alternative. Firstly, the likelihood function £(b, (]"2Iy, r) has been expressed as

£(b,(]"2Iy,r)

=

(27f(]"2)-N

J... J

e:rp [-2!2Ar.y,r(b)] ix(x)dx,

(15)

DUx)

where

A""y,r(b)

:=

Iy-

(x - /LxI)

112 + I r

- b(x - /LxI)

112.

(16)

Although the parameter of interest is b, the maximization of likelihood

£(b, (]"2Iy, r) requires the estimation of one more parameter, namely of channel noise (]"2. Notice that £(b, (]"2Iy, r) is a function of the discrete variable band of the continuous one (]"2. The ML decoder L (y, r) is defined as

and likelihood estimation imposes the ML decoding rule

L(y,r) ~ 0 =:?

b=

+1, L(y,r) < 0

=:?

b=-1.

(18)

256

T. Papamarkou

Direct maximization of likelihood (15) is probably intractable and therefore (17) is unlikely to be calculated. A computational solution is nevertheless realizable once (15) is rewritten as

R(b, (T21y, r) = Ex {(27W 2 )-N exp [- 2!2 Ax,y,r(b)] } .

(19)

It was the expectation appearing in (19) which gave the idea of a Monte Carlo maximum likelihood (MCML) approach, see [2] and [1]. For the estimation of the expectation one would have to sample rn spreading sequences Xi, 'i E {I, 2,'" ,rn}, each of length N. Then the Monte Carlo likelihood C(b, (T21y , r) calculates as

C(b, (T21y, r)

=

T~! ~ {(27W 2 )-N exp [- 2!2 AXi,y,r(b)] } .

(20)

BER. simulations suggest that MCML decoder L(y, r) outperforms correlation decoder C(y, r). MCML decoding allows for exact results to be obtained. It is hoped that the idea could be implemented in more complex communication settings.

References 1. Gelfand, A.E. and Carlin , B.P. Maximum-likelihood estimation for constrained

or missing data models. The Canadian Journal of Statistics, 21(3):303- 311, September 1993. 2.Geyer, C ..J. and Thompson, E.A. Constrained monte carlo maximum likelihood for dependent data. Journal of the Royal Statistical Society, 54(3):657- 699, 1992. 3.Hasler, M. and Schimming, T. Chaos communication over noisy channels. International Journal of Bifurcation and Chaos, 10(4):719- 735, April 2000. 4.Lawrance, A ..J. and Ohama, G. Exact calculation of bit error rates in communication systems with chaotic modulation. IEEE Transactions on Circuits and Systems, 50(11):1391 - 1400, November 2003. 5.Lawrance, A ..J. and Papamarkou, T. Higher order dependency of chaotic maps . In 2006 International Symposium on Nonlinear Theory and its Applications (NOLTA) , pages 695- 698 , Bologna, Italy, September 2006. Research Society of Nonlinear Theory and its Applications, IEICE. 6.Papamarkou, T. Statistical dependence of chaotic maps. Master's thesis, University of Warwick, Department of Statistics, September 2005. 7.Papamarkou, T. and Lawrance, A ..J. Optimal spreading sequences for chaosbased communication systems. In 2007 International Symposium on Nonlinear Theory and its Applications (NOLTA) , pages 208 -·211, Vancouver, Canada, September 2007. Research Society of Nonlinear Theory and its Applications, IEICE. 8.Yao,.J. Optimal chaos shift keying communications with correlation decoding. Proceedings of the 2004 International Symposium on Circuits and Systems, ISCAS, 4:lV - 593-596 , May 2004.

257

A Numerical Exploration of the Dynamical Behaviour of q-Deformed Nonlinear Maps Vinod Patidar l , G. Purohit, and K. K. Sud Department of Basic Sciences, School of Engineering Sir Padampat Singhania University, Bhatewar, Udaipur - 313601, India I (e-mail: [email protected]@spsu.aC.in) Abstract: In this paper we explore the dynamical behaviour of the q-deformed versions of widely studied ID nonlinear map-the Gaussian map and another famous 2D nonlinear map-the Henon map. The Gaussian map is perhaps the only I D nonlinear map which exhibits the co-existing attractors. In this study we particularly, compare the dynamical behaviour of the Gaussian map and q-deformed Gaussian map with a special attention on the regions of the parameter space, where these maps exhibit co-existing attarctors. We also generalize the q-deformation scheme of 1D nonlinear map to the 2D case and apply it to the widely studied 2D quadratic map-the Henon map which is the simplest nonlinear model exhibiting strange attractor. Keywords: q-deformation, Gaussian map, Henon map, Lyapunov exponent, chaos, co-existing attractors.

1. Introduction The fascinating theory of quantum groups has attracted considerable interest of physicists and mathematicians towards the special branch of mathematics dealing with q-deformed versions of numbers, series, functions, exponentials, differentials etc. (i.e. the q-mathematics) [1]. The q-deformation of any function is to introduce an additional parameter (q) in the definition of function in such a way that under the limit q ~ 1, the original function is recovered. Hence there exist several deformations of the same function. A recent study [2] induced the study of q-deformation of nonlinear dynamical system, where a scheme for the q-deformation of nonlinear maps (in analogy to the q-deformation of numbers, functions, series etc.) has been suggested. In this study authors have shown that the q-deformed version of logistic map (q-Iogistic map) exhibits a variety of interesting dynamical behaviours (which also exist in the canonical logistic map) including the co-existing attractors, which are not present in canonical logistic map. Further Patidar [3] and Patidar et al. [4] analyzed the dynamical behaviour of q-deformed version of another famous ID map - the Gaussian map, which is perhaps the only known I D map, exhibiting co-existing attractors. In this paper, we present the results of our recent analysis of the dynamical behaviour of various q-deformed maps. Particularly, we report the results of numerical exploration of the dynamical behaviour of q-deformed versions of widely studied ID nonlinear map-the Gaussian map and another famous 2D nonlinear map-the Henon map.

258

V. Patidw; G. Purohit. alld K. K. Sud

2. The q-deformation of nonlinear maps In this section, we briefly introduce the q-deformation scheme for the nonlinear maps proposed in [2], however for a thorough discussion on qdeformation, we refer the readers to [2] and references cited therein. In general a q-exponential function is given by =

11

[e X]q = I~ , 11 =0 [n]q'

(1)

where [n]q!= [l]q [2]q [3]q ... [n -1]q [n]q' [n]q

= (1- ql1) /(1- q) and

the subscript outside the square bracket denotes the q-deformed version of the argument inside the square bracket. It is clear from the above expressions that

under

[e X]q -7e x

the

q -71,

limit

[n] q -7 n ,

[n]q!-7 n! and hence

.

Another definition of q-exponential function proposed by Borges [5] is

[e X] q

=1+~QIl_IXIl L..J , n.

11=1

(2)

'

where QI1 = 1 (q)(2q -1) ... (nq - (n -1)). With the substitution 1 - q

= £ , Eq. (2) becomes

I4, n.

[e X]E =

T

=

11

(3)

11 =0

where

T" = 1

n = 0)

(for

T" = 1(1- £)(1- 2£) ... (1- (n -1)£) (for

and

n 2 1). It is clear that under

the limit q -71 i.e. £ -70, [eX]E -7 eX. If we compare Eqs. (1) and (3), we obtain a new definition for the deformation of numbers as

[n] E =

n 1-(n-l)£

,

(4)

Under the limit £ -70, [n] E -7 n . If we extend the above definition of deformation of numbers to any real number X as

[X] E =

X

1- £(x-l)

,

such that

lim[x] E = X . Equivalently, Eq. (5) can be written as E-->O

(5)

Numerical Exploration of Dynamical Behaviour of q-Deformed Nonlinear Maps

x 1- (1- q)(x -1)

259

(6)

such that

lim[x]q q--.l

= x.

form xn+l

=f

The q-deformation scheme for ID nonlinear map [2] of the

=f

(X n) is given by xn+l

([Xn]q) .

In this paper we are aimed to numerically analyze the dynamical behaviour of the q-deformed versions of the widely studied 1D nonlinear map-the Gaussian map and another famous 2D nonlinear map-the Henon map. In the next subsections, we briefly introduce the Gaussian and Henon maps and their q-deformed versions. 2.1. The q-dejormed Gaussian map The Gaussian map [6] is based on the Gaussian exponential function. It is characterized by two parameters band c as follows:

x n +J

= e-bx,~

+C

(Gaussian map)

(7)

The q-deformed version (i.e. q-Gaussian map) of the Gaussian map by following the above definition of the q-deformation of ID nonlinear map is given as:

xn+1

=e -b {{x"lql

2

+c

(q-Gaussian map)

(8)

or

b{ xn+1 = e

}2

x" J-(J -q)(x -1)

"

+c (q-Gaussian map)

(9)

Under the limit q -7 1 , the q-Gaussian map becomes the original Gaussian map (Eq. (7»). In this paper, throughout the discussion on q-Gaussian map, we prefer to use deformation parameter [; instead of q (which are related by the relation 1 - q = [; ). Hence the explicit form of q-Gaussian map, which we use, is given as:

xn+1 = e

-b{ 1-"'(;"-J)}' +c, (q-Gaussian map)

(10)

which under the limit [; -70, becomes the original Gaussian map (Eq. (7». 2.2. The q-dejormed Henon map One of the simplest mathematical models, which exhibits strange attractor is the quadratic map introduced by Henon in 1976 [7]. The mathematical form of the Henan map is given by

260

V. Patidar. G. Purohit, and K. K. Sud

X,,+l =f(x",y,,)=I-ax,~+ y"

(Henan map)

(11)

Y,,+l =g(X 'YIl)=j3x" Il

If we generalize the above definition of q-deformation of ID nonlinear map to the 2D case then we may introduce different deformation parameters for different state variables. Such generalized form of a 2D q-deformed map is given by

X,, +l = f([xJ qx,[Y,,]q , ) (2D q-defarmed map),

(12)

Y,, +l = g([xJ qx,[yJ q , ) X

"

1- (1- qx )( x" -1) an Clearly under the limits

~ land

qx

reduces to original map.

d[

y"

]

q ,-

1- (1- q y)(y" -1)

qy ~ 1the q-deformed 2D map

The explicit form of the q-Henon map (by

introducing 1- qx = Cxand 1- qx =

cx )' which we use throughout this

study, is given by

x

,,+1

=1-a(

x" l-c ,(x -1)

J2

ll

-

y".,

=

pC -£,~:"

which under the limits

cx

~

-I)

0 and

+[

y" l-c y (y,,-1)

J (13)

J

cy

~

0, becomes the canonical

Henan map (Eq. (II )). In the next section, we present results of our numerical exploration of the dynamical behaviour of above described qdeformed nonlinear maps.

3. Results and Discussion A recent study [2] on q-deformation of nonlinear dynamical systems has revealed that the q-deformed logistic map exhibits a variety of dynamical behaviours: fixed point, periodic, chaotic and more interestingly the coexistence of attractors, which is a rare phenomenon in the 1D nonlinear maps. In the first part of our study we are interested in analyzing another ID mapthe Gaussian map, which is known to exhibit period doubling route to chaos and co-existing attractors [3,6] under the same q-deformation scheme. Recently, Patidar [6] has done a detailed study on the regions of the parameter space of Gaussian map where co-existing attractors exist. Here we embark to analyze the effect of q-deformation on the regions of the parameter space of Gaussian map where co-existing attractor exist. In Figure 1, we have shown the results of the analysis on the Gaussian map [3]. Particularly in Figure 1(a), we have identified different regions of the parameter space.

Numerical Exploration o/Dynamical Behaviour ()f q-De{ormed Nonlinear Maps

261

I)

+I).,w

&. ~ ~~".,..,..,,~,...j..,~=,-~-+~""""_-i,",=-~_ _

(a)

4::14

1-·-~·--~"""'T~·"""·--'-"!""",-·'~""",,,-~~,,,,'~....,.----+·1.1(i I)

11,. P,"';lM__~

:; 0.00

un

HI

. , . ' ' - - _ - - ' -_ _~_ _ _L __

1~

~\J

20

'15

__'__ __'__ _~_,__+

0 .00

(b) .0.29

·0.29

ill iil .0.50

.1l.58

~g'

,.~

IL

.lUll

·1.1&

-0.81

-"-~--

...---~--.--.-~--,--~--+

·1.16

P Ilr ameter (b)

Figurc 1. Paramcter space (b, c) of the Gaussian map (a) showing the regions, where chaotic (black shade) and regular (white shade) motions appeaL (b) showing the regions where a period-] attractor co-exists with some other periodic attractor (grey shade) and the regions where a petiod-1 atlmctor co-exists with a chaotic attractor (black shade),

(b ,c) where the dynamics of Gaussian map is chaotic and regular (black and

white shades correspond to chaotic and periodic attractors respectively). The results shown in Figure lea) are based on the extensive numerical calculation of Lyapunov exponent by iterating the Gaussian map 1000 times (after neglecting initial transient behaviour up to 400 iterations) at 2 X 106 different points of the pammeter space defined by 0::;: b S 20 and -1.0 S c sO,

262

V Patidar, G. PUlVhit. and K. K. Sud

Chaotic situation has been recorded, whenever the Lyapunov exponent becomes greater than 1 X 10-4 • The calculation has been repeated for several sets of initial conditions to include all the co-existing attractors. In Figure 1 (b), we have shown the complete region of parameter space where Gaussian map exhibits co-existing attractors. -Q..&l O.Q

-i"-----,.,...·--""----·--'----'-------+t,.J(.I (a)

·0.50

-0.25

.\).!i411

,1).2:/j

O,;jfofrf,lIith';,r.

fI.O

Pi\'it'~ITWi'Iw !Ii:~

o.~

.~U

r -.......- . . . . . L -.......--iL....~........-..L--.-........-~-{M. b:5.0

(b)

Figure 2. Parameter space (E, c) of the q-Gaussian map for b=5.0 (a) showing the regions, where chaotic (black shade) and regular (white shade) motions appear, (b) showing the regions where a period· 1 attractor coexists with some other periodic attractor (grey shade) and the regions where a period-] attractor co-exists with a chaotic attractor (black shade).

Numerical Exploration of Dynamical Behaviour ofq-Deformed Nonlinear Maps

263

Particularly, the grey shade shows the regions of the parameter, where a period-l attractor co-exists with some other periodic attractor (i.e., period-I, period-2, period-4, ... , period-3, etc.), however the black shade shows the regions of the parameter space where a period-l solution co-exists with a chaotic solution . • (1.00 OO·i-~~--~~~--~~~--~--~--~~-~+~'.O

(a)

.... liO (1,(1

+------'--'---......- .......- ........- ......--~+Q.1i' b=7.5

(b)

Figure 3. Parameter space ( c , c) of the q-Gaussian map for b==7.5 (a) showing the regions, where chaotic (black shade) and regular (white shade) motions appear. (b) showing the regions where a period-l aUmctor co-exists with some other periodic attractor (grey shade) and the regions where a period-l attractor co-exists with a chaotic attJactor (black shade).

264

V. Patidar, G. Purohit. and K. K. Sud

As explained in Section 2 that the q-deformation of any function/map is to introduce an additional parameter (£ ) in the definition of that function/map in such a way that under the limit £ --7 0, the original function/map is recovered. The q-deformation of the Gaussian map (Eq. (10)) leads to a three parameter one dimensional nonlinear map. Now it becomes more complicated to analyze the dynamical behaviour of the q-Gaussian map in a three dimensional parameter space (b, £ , c). The effect of parameter b on the dynamical behaviour of Gaussian map has been analyzed [4], so we prefer to work in the two dimensional parameter space (£ , c) for some fixed values of parameter b. The results of our analysis for the co-existing attractors in qdeformed Gaussian map (the analysis similar to the Gaussian map reported in Fig. I) have been depicted in Figures 2 and 3 for b=5.0 and b=7.5 respectively. It can be easily observed from Figure 2 that for b=0.5, the nondeformed Gaussian map ( £ = 0 ) there is a range of c for which co-existing attractors exist. For all the values of c which belong to this range, both the co-existent attractors are periodic i.e., chaotic solution does not co-exist with a period-l attractor for any value of c. Now if we deform the Gaussian map by changing the value of deformation parameter £, then for all positive values of £ both the co-existent attractors are periodic and the range, for which co-existing attractors exist, is decreasing with the increase in the value of £ in positive direction. However, if we increase the value of deformation parameter £ in the negative direction, initially the range of c, for which coexisting attractors exist, increases and then after a particular value of £ , it starts decreasing. We also notice an important feature that for some negative values of deformation parameter £, one of the co-existent attractors is chaotic. A similar feature we observe for b=7.5, the only difference is that the whole pattern is shifting towards the higher values of deformation parameters. In conclusion to the study on q-deformation of Gaussian map, we may infer that the q-deformation of the Gaussian does not lead to a drastic change in the dynamical behaviour (i.e., the qualitative behaviour is similar), however it introduce an additional parameter in the definition of the map, which sometime may be useful for making a choice of the desired dynamical behaviour required for some specific purposes (in case if we do not have direct access to change the parameters of the Gaussian map i.e. b and c). In the second part of our study, we analyze the dynamical behaviour of Henon map under the same q-deformation scheme. Since the Henon map is a 2D map and to consider the most general case, we introduce the two different deformation parameters corresponding to the deformation of two different state variables as explained inSection 2.2. Moreover in the canonical Henon map two system parameters are present hence after introducing the qdeformation, it becomes a four parameter system. To analyze the dynamical behaviour of this four parameter system, we choose fixed values for the parameter of canonical henon map i.e. a and f3 and then analyze the the dynamical behaviour of the q-deformed system in the space of deformation

Numerical Erploration of Dynamical Behaviour of !I-Deformed Nonlinear Maps

-2

~

Henon Map Parameter Space a 1

2

3

1.0

1.0

0.5

0.5

is 0.0

0.0

...

-... Q)

~0.5

·0.5

(!J

(tf

0.

-1.ll

-1.0

-1.5

-1.5

-2.0

-2.0 -2

-,

0

1

2

3

Parameter (0:)

Figure 4, Pm'ameter space of the Henon map showing the regions correspond to variety of dynamical behaviours: fixed point solution (Black shade), pCl10dic solutions (blue shade). chaotic solutions (red shade) and unbounded ~olUl.ions (white shade).

·1.0 1.0

q-Hel1on Map Parameter Space for II ::: 1.4 & j3 ::: 0.3 -0.5 0.0 0.5

1.0 1.0

£05 ... .

0.5

(U

Gi E I!:! \'II

0.0

0. 0.0

c:

10

:; E ... o

-0.5

1il -0.5

o

-1.0 -'1.0

-1.0

-o.S

0.0

0.5

i .0

Deformation Parameter (E.)

Figure 5. Parameter space of the q-Henol1 map showing the regions correspond to variety of dynamical behaviours: fixed point solution (Black shade), periodic solutions (blue shade), chaotic solutions (red sbade) and unbounded solutions (white shade).

265

266

V. Patidar, G. Purohit, and K. K. Sud

parameters C x and

cy .

In this way, we will be able to see the effect of q-

deformation on the dynamical behaviour of henon map for a particular choice of parameters a and f3. In Figures 4 and 5, we have shown the results of one such analysis. Particularly in Figure 4, we have shown the regions of parameter space (a, f3) of non-deformed Henon map (i.e., c x = c y =0), where different different types of dynamics occur i.e., fixed point solutions, periodic solutions, chaotic solutions and unbounded solutions (diverges to infinity), etc. These results are based on an extensive calculation of Lyapunov exponent at 15 X 104 different points of the parameter space defined by - 2 a 3 and - 2 S f3 1. The black shade in Figure 4 represents the region of parameter space with fixed point solutions, blue and red shades represent periodic and chaotic solutions respectively and white shade represents the regions where dynamics diverges to infinity i.e., unbounded solutions. Now we choose a particular set of parameters a =1.4 and f3 =0.3 for which a chaotic strange attractor exists and see the effect of q-deformation on the chaotic dynamics of the Henon map. The results of this analysis are shown in Figure 5, which is again based on extensive Lyapunov calculation at 4 X 104 points of the deformation parameter space defined by -1 S C x S 1 and -1 S c y 1. The same shading scheme has been used

s s

s

s

as in Figure 4. It is clear from Figure 5 that the q-deformation leads to the suppression of chaos in the Henon map as in the large part of the deformation parameter space the dynamics becomes periodic. Now several questions arise - (i) How this conversion from chotic to periodic solution occurs in this case of q-deformation or in other words what's the route to chaos in qdeformed Henon map?, Whether the topological shapes of the strange attractor of canonical Henon and q-deformed Henon maps are same or different?, Is it a universal feature in the all 2D q-deformed nonlinear map?, etc. Work in this direction is in progress and will be reported elsewhere.

4. Conclusions In this paper we investigated the dynamical behaviour of q-deformed Gaussian and Henon maps. We numerically explored the dynamics of these maps in the complete parameter space including the deformation parameters using extensive computation of the Lyapunov exponent. We observed a variety of dynamical behaviours and we are able to produce the desired behaviour by slightly changing the deformation parameter without disturbing the canonical system parameters of the system, which are not accessible in some practical situations. A more detailed analysis on the q-deformed Henon map is in progress to clarify the routes to the suppression of chaos due to the deformation and some universal features (if exist!) in such q-deformation of nonlinear maps. We strongly believe that such studies of q-deformation of nonlinear maps can be used beneficially in modelling of several phenomena,

Numerical Exploration of Dynamical Behaviour of q-Deformed Nonlinear Maps

267

which are not modeled exactly with the standard maps but their q-deformed versions could serve the purpose.

References [I] C. Chaichian and A. Demichev.lntroduction to quantum groups. World Scientific, Singapore, 1996. [2] R. Jaganathan and S. Sinha. A q-deformed nonlinear map. Phys Lett A, 338:27787,2005. [3] Vinod Patidar.Co-existence of regular and chaotic motions in the Gaussian map, Electronic Journal of Theoretical physics, 3(13): 29-40, (2006). [4] Vinod Patidar and K. K. Sud. A comparative study on the co-existing attractors in the Gaussian map and its q-deformed version, Communications in Nonlinear Science and Numerical Simulation, 14: 827-838,2009. [5] E. P. Borges. On a q-generalization of circular and hyperbolic functions. J Phys A, 31 :5281-8, 1998. [6] R. C. Hillborn. Chaos and nonlinear dynamics. Oxford University Press, Oxford, 2000. [7] M. Henon. A two-dimensional mapping with a strange attractor. Comm. Math. Phys., 50: 69-77, 1976.

268

Spatiotemporal Chaos in Distributed Systems: Theory and Practice George P. Pavlos, A. C. Iliopoulos, V. G. Tsoutsouras, L. P. Karakatsanis, and E. G. Pavlos Department of Electrical and Computer Engineering Democritus University of Thrace, Xanthi, Greece (e-mail: [email protected])

Abstract:

This paper presents theoretical and experimental results concerning the hypothesis of spatiotemporal chaos in distributed physical systems far from equilibrium. Modern tools of nonlinear time series analysis, such as the correlation dimension and the maximum Lyapunov exponent, were applied to various time series, corresponding to different physical systems such as space plasmas (solar flares, magnetic-electric field components) lithosphere-faults system (earthquakes) brain and cardiac dynamics during or without epileptic episodes. Futhermore, the method of surrogate data was used for the exclusion of 'pseudo chaos' caused by the nonlinear distortion of a purely stochastic process. The results of the nonlinear analysis presented in this study constitute experimental evidence for significant phenomena indicated by the theory of nonequilibrium dynamics such as nonequilibrium phase transItIOn, chaotic synchronization, chaotic intermittency, directed percolation, defect turbulence, spinodal nucleation and clustering. Keywords: nonlinear time series analysis, spatiotemporal chaos, Self Organized Criticality (SOC), distributed systems, nonequilibrium dynamics.

1. Introduction In this study we are interested in distributed nonlinear physical systems such as space plasmas, earthquakes, brain and cardiac dynamics which include practically infinite degrees of freedom, as well as the theoretical presuppositions for understanding their complex dynamics. At the meso scopic level, these systems are described by variables which are functions of space and time as coarse grained fields of microscopic variables. Also, the spontaneous formation of the spatiotemporal structures is the result of sudden qualitative changes known as bifurcations, which occur if an externally controllable driving passes through critical values. At bifurcation points the system can perceive its environment and adapt to it by forming preferred patterns of modes of behaviour. According to Wilson, 1979 [15], precisely at the critical point, the scale of the largest fluctuations becomes infinite, but the smallest fluctuations are in no way diminished. The theory that

Spatiotemporal Chaos ill Distributed Systems

269

describes a physical system near its critical point must take in account the entire spectrum of length scale. Recently, for the understanding of such distributed physical systems two theoretical concepts, contradicting for many scientists, have been introduced: the self organized criticality (SOC) and the low dimensional chaos. In a series of papers we have supported experimentally and theoretically the broad universality of SOC and low dimensional chaotic processes for distributed physical systems. In this study, we present new experimental results and theoretical concepts that reveal the deeper and universal character of the nonequilibrium phase transition and the critical point theory characteristics, applied to distributed physical systems. Particularly, in section 2 we introduce significant theoretical concepts concerning the far from equilibrium phase transition process. In section 3, we present new results obtained by modem nonlinear time series analysis applied to data concerning five different physical systems, such as the solar system, the earth's magnetosphere and crust, the human brain and cardiac dynamics. In section 4 we summarize crucial results obtained by the data analysis and in section 5, we give theoretical explanations of our results embedded in the modem theory of spatiotemporal chaos.

2. Far from Equilibrium Physical Processes The far from equilibrium nonlinear dynamics includes significant collective phenomena such as: power law distribution and critical scale invariance, nonequilibrium fluctuations causing spontaneous nucleation and evolution of turbulent motion from metastable states, defect mediated turbulence and localized defects changing chaotically in time and moving randomly in space, spatiotemporal intermittency, chaotic synchronization, directed percolation causing levy-flight spreading processes, turbulent patches and percolation structures, threshold dynamics and avalanches, chaotic itinerancy, stochastic motion of vortex like objects. In particular, in the phase space of a distributed system various finite - dimensional attractors can exist such as fixed points, limit cycles or more complicated finite dimensional strange attractors, which correspond to chaotic dynamics as well as states known as self organized critical states (SOC). Generally, spatiotemporal chaos includes early turbulence with low effective dimensionality and few coherent spatial patterns or states of fully developed turbulence. These states are out of equilibrium steady states related to the bifurcation

270

G. P. Pavios et ai.

points of the nonlinear distributed dynamics, as well as to out of equilibrium second order phase transition processes. Also, spatial and temporal patterns as well as spatially localized structures are possible solutions of the nonlinear spatially extended dynamics. Solar flares, magnetospheric super-substroms, earthquakes and epileptic seizures could be some paradigms of physical systems which include dynamical coherent localized structures underlying these physical events.

2.1. The modern unified theory of the far from equilibrium physical processes A distributed system is governed by a generalized Langevin stochastic equation

a;; =.t:

(
(1)

Typical examples of such an equation are: the generalized GinrburgLandau equation, the Maxwell-Boltzman plasma equations, the Navier-Stokes equations, or the type of Kardak - Parisi - Zhang stochastic partial differential equations. The continuous stochastic processes of Eq. (1) can also be the field theoretical approach of coarsegraining for (contact) processes such as cellular automata (CA) models including threshold systems and avalanche type processes or coupled Map Latticies(CML). In this coarse-grained approach the threshold type coupling corresponds to continuous diffusion process. The probabilistic description of the stochastic Langevin equation (1) is given by the solution of the corresponding master equation or FokkerPlank equation for which the solution is as follows

p(,
(2)

obtained after path integral intergration. In Eq. (2), P (

,
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macroscopically dynamical parameters (order parameters) of the system, that drives its dynamics near the fixed point. Also, scaling (power laws) near criticality are caused by the combined effects and long correlations of the fluctuations (Chang, 1992[3]). The dynamic renormalization group shows that nonlinear stochastic systems can exist near critical states and can exhibit low dimensional chaotic or high dimensional self organized criticality (SOC) and fractal behaviour with intermittent turbulent character. The intermittent turbulence may grow producing more and larger coherent structures as well as fluctuations and new resonance sites, where random dynamic fields are extremely long ranged and there exist many correlation scales. According to these theoretical concepts, the solar flares, the magnetospheric superstorm-substorm events, the earthquakes in earth's crust, the epileptic seizures in the human brain or the cardiac arrhythmia can be related to the stochastic behaviour of nonlinear distributed dynamical processes.

3. Results In this section, we present significant results obtained by the nonlinear time series analysis apllied to signals concerning different physical systems such as solar flares, magnetospheric plasma, earthquakes, epileptic human brain and cardio rhythm. For an extended description of the algorithms of nonlinear time series analysis (geometrical and dynamical characteristics) see Pavlos et al. 2003[9],2007[10].

3.1. Solar flares Figure l(a) represents the daily Flare Index of the Solar activity that was determined using the final grouped solar flares obtained by NGDC. It is calculated for each flare using the formula: Q = (i * t) , where "i" is the importance coefficient of the flare and "t" is the duration of the flare in minutes. To obtain final daily values, the daily sums of the index for the total surface are divided by the total tome of observation of that day. The data covers time period from 1/1/1996 to 31112/2007.

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Figure 1(b) illustrates the slopes of the correlation integral estimated for the Solar Flares time series and its 30 surrogates using delay time r= 500 and embedding dimension m=lO, as a function of Ln(r). As it can be seen from this figure, there is no significant difference between the original series and the surrogate data. Similar results are obtained estimating the Maximum Lyapunov exponent

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(Lmax), shown in Fig. l(c), for the Solar Flares time series and its 30 surrogates using delay time r= 500 and embedding dimension m= 10, as a function of time (days). These results support the stochastic character of the original signal and indicate the possibility of correlated noise. Fig. led) represents the First Difference filtered Solar Flare time series. We used the high pass first difference filter in order to exclude the low frequency component. Figures l(e-f) show the slopes of the correlation integral and the Lmax using delay time r=20 and embedding dimension m=7, for the First Difference Solar Flares time series and its 30 surrogates, correspondingly. As it can be seen in Fig. lee) there is a low value saturation D of the slopes D "" 2 for a sufficient large embedding dimension m=7, revealing low dimensional profile of the underlying dynamics. Moreover, Fig. l(f) shows the existence of a positive Lmax, indicating sensitivity to initial conditions for the underlying system. In Figs. 1(e-f) we also show the clear and significant difference between the First Difference Solar Flares time series and the surrogate data. The above results support the hypothesis of the existence of a strange attractor underlying the Solar flare dynamics, revealed after using the first difference filter.

3.2. Magnetosphere In this section we present results for three time series concerning the Earth's Magnetosphere plasma sheet obtained by the spacecraft GEOTAIL. The data period corresponds to a Superstorm which took place at 27 July of 2004. In particular, we estimate geometrical (correlation dimension) and dynamical (maximum Lyapunov exponent) characteristics in the reconstructed state space of the Magnetic field (Bz - component), Electric field (Ex - Component), Ion moments (Velocity - Vy). Figure 2(a) illustrates the Magnetic Field First Difference Filtered (FDF) Bz-component time series. In Fig. 2(b) the slopes of the correlation integral are presented, estimated for the (FDF) Bz time series and its surrogates using delay time r=1O, embedding dimension m=9, and Theiler's parameter w=lO, as a function of Ln(r). Fig. 2(c) shows the Lmax using delay time r= 10 and embedding dimension m=9, as a function of time (days). Fig. 2(d) represents the Electric field (Ex - Component). The slopes of the correlation integral and the maximum Lyapunov exponent of the Ex time series and its surrogates, are shown in Figs. 2(3-f). The parameters we used were delay time r=ll, w=17 and embedding dimension m=9. Figures 2(g-i) represent

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the time series of Ion Moment Velocity (Vy-component), the slopes of the Correlation Integral estimated for the Vy-component series and its Surrogate using m=8, r=13, w=13 and the Lmax of the Vy-Component series and its Surrogates using m=8, r=13. The results indicate the existence of chaotic dynamics corresponding to (FDF) Bz, Ex, Vy time series, as well as the existence of high dimensional dynamics corresponding to Bz time series, underlying the Magnetosphere. ~

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3.3. Earthquakes In this section results obtained by applying nonlinear analysis to three seismic data sets are presented. The data series correspond to 5400 seismic events that have occurred in the Hellenic region, using a threshold local magnitude of 4 Richter and were observed during the time period of 1968-2004, within the broad Hellenic area 34.00-42.00

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Nand 18.00-30.00 E. These data series, which correspond to the basic focal parameters (time, space, magnitude) of earthquakes, were constructed by the bulletins of the National Observatory of Athens. In Figure 3(a,d,g) the three time series of seismic events are illustrated, which correspond to the interevent series (time intervals between successive earthquakes (Fig. 3a», the Latitude (Fig. 3d) and the Magnitude (Fig. 3g). In these figures, it can be observed that the signals maintain a strong stationary and aperiodic profile. Figures 3(b-e) present the slopes of the correlation integral estimated for the Interevent and the Latitude time series and their corresponding surrogate data, using delay time r=l, and embedding dimension m=6 as a function of Ln(r), correspondingly. As it can be seen these slopes reveal efficient scaling and low saturation value, indicating few degrees of dynamical freedom whereas there is a significant discrimination between the two first Oliginal signals and their surrogates. However, in the case of Magnitude series, shown in Fig. 3(h), the slopes estimated using delay time

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r=1 and embedding dimensions m==3-8 as a function of Ln(r), do not reveal efficient scaling and saturation indicating strong stochastic dynamics. Figs. 3(c, f,i) illustrate the Lmax of the three time series using parameters r=1, m=6, 7, correspondingly. As it is shown, the Lmax is positive for all cases but therc exists a significant difference between the original time series and their surrogates, only for the Interevent and Latitude time series and not for the Magnitude series. The results indicate the coexistence of low chaotic and high dimensional dynamics underlying the Hellenic region.

3.3. HW1lan brain and cardiac rhythm In general, epileptic seizures may be associated with cardiac an~hythmias. The present study was aimed at evaluating Electrocncephalography (EEG)

Figure 4. a) EEG time series of patient with Ollt epileptic seizure. b) Stopes of the Correlation Integral estimated for the EEG series and its Surrogate (thick line). c) ECG time series of' patient without epileptic seizure. d) Slopes of the Correlation Integral estirrwtedfor the ECG series and its Surrogate (thick line).

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(b)

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and electrocardiographic (EeG) changes occurring before and during partial epileptic seizures. We describe the features of the brain and cardiac patterns that may have implications for understanding cardiac and neuroautonomic instability in epilepsy. Figure 4(a) presents an EEG (brain dynamics) time series of a patient without epileptic seizure. The duration of EEG segment is 20 sec (4000 points). Figure 4(b) represents the slopes of the correlation integral estimated for the EEG series using parameters m=6-1O, r=22, w=22 and its surrogate using parameters m=lO, r=22, w=22 (thick line). Figures (c-d) are similar to figs. 4(a-b) but correspond to an EeG (electrocardiographic) time series of a patient without epileptic seizure. The duration of the EeG segment is 20 sec (4000 points) and the slopes of the correlation integral were estimated using m=6-1O, r=6, w=6 for the EeG time series and m=lO, r=6, w=6 for its Surrogate (thick line). In particular, Figs. 5 (a-d) are similar to figs. 4(a-d) but represent an EEG and an EeG time series of a patient during epileptic seizure. The parameters used for the estimation of the slopes were m=6-1O, r=7 , w=7 for EEG time series and m=lO, r=7, w=7 for its

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surrogate (thick line), as well as m=6-1O, r=6, w=6 for the ECG time series and m=lO, r=6, w=6 for its surrogate (thick line). The results show in normal conditions, namely absence of epileptic seizure, the brain activity corresponds to high dimensional stochastic dynamics as indicated by the high values of the slopes of the correlation integral, shown in Fig. 4(b). On the contrary, the cardiac activity corresponds to low dimensional and periodic dynamics, as indicated from the low value saturation slopes profile. However, during the epileptic seizure crucial changes occur both in brain and cardiac dynamics, as shown in Figure Sea-d).

4. Summary of Data Analysis and Results The results presented previously showed: a) Coexistence of a SOC (stochastic) process and a hidden low dimensional chaotic process, revealed after the exclusion of high frequency component, underlying the solar activity. b) Evidence for low and high dimensional local plasma processes undelying the magnetosphere, that reveal the turbulent character of the plasma sheet, during Superstorm-substorm processes. c) Coexistence of low dimensional chaotic spatiotemporal processes and a strong stochastic component corresponding to magnitude, revealing the nonlinear and complex character of the Hellenic lithospheric dynamics. d) During healthy states, the brain activity is high dimensional corresponding to SOC dynamics, while during epileptic episodes corresponds to local and low dimensional chaotic dynamics. As for the cardiac dynamics our results indicate that during healthy states low dimensional and periodic dynamics prevail, while during epilepsy the dynamics become high dimensional and chaotic. Thus, our results showed two different and significant chaotic processes underlying all the distributed systems that were studied. The first corresponds to the development of local low dimensional chaos in a high dimensional SOC environment and was observed during supersubstorms in the magnetospheric system or epilepsy events in the human brain. The second one corresponds to the chaotic spreading or wandering of local avalanche spatiotemporal events and was observed in the solar flares and earthquake processes. The development of cardio chaos during epilepsy could be explained also by the spatial spreading of desynchronization local events or spreading of local defects, as we show in the next section.

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S. Conclusions and Theoretical Expanations Although solar flares, magnetospheric superstorms or space plasma turbulence, earthquakes, epilepsy and cardiac arrhythmias are processes in different physical systems, they reveal significant common features. They include local structures in distributed physical systems far from equilibrium. Moreover, these phenomena are related to far from equilibrium, first or second order, phase transition processes, with low chaotic or high dimensional chaotic-SOC spatiotemporal profile. The common theoretical presupposition underlying the above spatiotemporal phenomena is one of the main concepts supported in this study. This point of view was further supported by previous studies concerning the character of universality in solar flares and substorms or earthquake occurrence (Watkins, 2001[14], Archangelis et ai., 2006 [1]). This concept of universality is based on the experimental observation of power laws which were observed also in the brain dynamics. Generally, the universality of power law processes historically was used as the trace of SOC theory which was against chaos. In this study we presented strong experimental evidence for a wider theoretical framework which can compromise SOC and Chaos as two different, between others, manifestations of nonlinear dynamics of a distributed system. In the following we introduce and try to correlate the results of data analysis with some significant phenomena included in the far from equilibrium nonlinear dynamics.

5.1. Critical points and spinodal nucleation Although critical theory and spinodal decomposition are well known processes near equilibrium, their essential features can be also observed at far from equilibrium phenomena. In particular, nucleation from metastables states with deep quenches of their free energy reveal noncompact, dominant fluctuations, while the nucleating droplets are formed by the coalescence of clusters and are ramified with fractal dimension. Fluctations near a spinodal are generated by random walk of the number of fundamental clusters while there exist a raming onto random bond percolation. These concepts can be extended in the far from equilibrium processes including long-range interactions (mean or near-mean field). Spinodal decomposition processes and their emerging spatiotemporally correlated fluctuations can be used for the modeling of driven threshold systems where the avalanches can be explained as nucleation near the spinodal and percolation states. These theoretical

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concepts concerning the far from equilibrium spinodal nucleation, can be used to explain the low dimensional chaos or high dimensional chaos or SOC states, observed in the systems we studied, as critical dynamics and changes of critical states in the space of the relevant parameters of the nonlinear stochastic system (see also Seq. 2).

5.2. Local structures, turbulence, intermittency, anomalous diffusion and synchronization in extended systems Solar flares, super-substorms, earthquakes, epileptic episodes and the macroscopic cardiac arrhythmia are some examples of manifestation of local and dynamical structures of the underlying complex dynamical systems. On the other hand, complex dynamical systems with many degrees of freedom, composed of many parts interrelated in nontrivial way, exhibit a wealth of collective phenomena such as: ordered (coherent) and random (incoherent) behavior, hierarchy of structures over a wide range of lengths. The variety of complex phenomena in distributed systems with many (practically infinite) degrees of freedom has been revealed until now by studying different systems such as: deterministic or probabilistic cellular automata (CA), coupled map lattices (CML), PDE systems (Kuramoto - Shivashinsky (KS) equation), complex Ginzbourg-Landau equation, reaction-diffusion (RD) equation, NavierStokes equation, stochastic Kardar-Parisi-Zuang equation and other Langevin type equations). Synchronization has been one of the collective phenomena as interacting chaotic systems, in spite of their natural instability, can synchronize their trajectories. Synchronization or de synchronization in the system is produced by the competition between intrinsic disorder and the diffusive effect provoked by the coupling. At critical points toward synchronization we observe intermittent behavior where the synchronized state is interrupted by chaotic burst. The chaotic bursting accompanying the synchronization transition gives the behavior of on-off intermittency. This behavior is accompanied with dimension variability as it is characterized by the breakdown of the continuous splitting between stable and unstable manifold in the high-dimensional phase space of the coupled system. The synchronization or loss of synchronization process with on-off intermittency burst can be used as a theoretical explanation of the correlation dimension changes, observed in solar flares, magnetospheric substorms or space plasma turbulence, earthquakes,

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epilepsy and cardiac arrhythmias, according to our data analysis presented in Sec. 3. In general, our data analysis results reveal bursts of low dimensionality in a high dimensional environment with SOC or chaotic characteristics, in cases of solar flares, substorms, earthquakes, epilepsy, or oppositely high dimensional (SOC or chaos) bursts in a low dimensional environment in the case of cardiac arrhythmia. Moreover, chaotic bursts with positive Lyapunov exponents in a high dimensional environment, as we have observed in our study, indicate the existence of a strongly intermittent process known as sporadic chaos, as well as the existence of directed percolation process (Wang, 1995[13]). On the other hand, intermittent phenomena and sporadic chaos can be the manifestation of anomalous diffusion and non-Gaussian fluctuations with Levy motion profile (Wang, 1992[12]). Anomalous diffusion and Levy flight distributions indicate the nonextensive generalization of Boltzmann-Gibbs statistical mechanics for the nonequilibrium systems described here. In particular, the fluctuating dynamics at the onset (edge) of chaos and intermittent behavior is connected with Tsallis q-statistics and Mori's q-phase thermodynamics [Tsallis, 2005]. The physical systems that were analyzed in this study are driven and spatially extended systems. For such systems with many degrees of freedom and with chaotic fluctuations in space and time, as we have described previously, it is possible to observe continual creation, annihilation and interaction of topological defects. This behavior is known as weak turbulence, defect chaos and defect turbulence. Topological defects destroy the coherence of spatial structures as their appearance is linked to the onset of defect, mediated turbulence and onset of spatiotemporal chaos and spatial dislocations. The defects can move in space with chaotic wandering following diffusion, according to a stochastic Langevin process. This process of the chaotic defect dynamics can map the far from equilibrium spinodal nucleation process and the droplet dynamics, as well as the cluster dynamics in the directed percolation process (Miller & Huse, 1993[8]). Topological defects, as local structures in distributed systems, can be related also to anomalous diffusion and Levy flight processes according to the mapping of deterministic processes of the Kuramoto-Sivashinsky type to stochastic and diffusive processes of Kardar-Parisi-Zhang processes (Jayaprakash et at., 1993[6]).

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5.3. Chaotic itinerancy Chaotic itinerancy can be equivalent to the defect-turbulence or spinodal nucleation and directed percolation processes, as it constitutes a universal character of itinerant motion among varieties of low dimensional dynamical states interrupted by high dimensional dynamics (Kaneko & Tsuda, 2003[7]). The chaotic spatiotemporal spreading of local events as well as the development of local chaos events in a high dimensional environment as we have observed for the solar flares and earthquakes (first case) and for the magnetospheric super-substorms, the brain epilepsy and the heart arrhythmias (second case) can also be explained as a chaotic itinerancy event in the high dimensional phase space of the distributed dynamics. 5.4. Toward a theoretical unification The universe is a far from equilibrium process, creating fluctuations and correlations at any scale. The principle of scale invariance, expressed by the renormalization group theory, indicates the possibility for unification of physical processes at any level, while nonlinearity, chaos and dissipativity are present at any level (Castro, 2005[2]) creating spatiotemporal coherence and structures at microscopic, mesoscopic and the macroscopic level. Moreover, these characteristics indicate the unification of the deterministic nonlinear chaotic processes with stochastic diffusive processes (Jayaprakash et al., 1993[6], G't Hooft, 1999[5]). In the direction of such global unification of the far from equilibrium dynamics, the generalized Liouville equation as a master equation

d

-Ip dt r >=-Llpr >

(3)

where L is generally non-Hermitean, can be used as a prototype of unified non-equilibrium phenomena (Tsallis, 2005[ 11], Hinrichen, 2006[4]), including generalized statistics and far from equilibrium dissipative structures.

References [1] Archangelis et al. Universality in Solar Flare and Earthquake Occurrence. Phys. Rev. Lett. 96: 051102,2006. [2] Castro C. On non-extensive statistics, chaos and fractal strings. Physica A 347:184-204,2005.

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[3] Chang, T. Low-Dimensional Behavior and Symmetry Braking of Stochastic Systems near Criticality - Can these Effects be observed in Space and in the Laboratory. IEEE 20(6): 691-694,1992. [4] Hinrichen H. Non-equilibrium Phase Transitions. Physica A 369:1-28, 2006. [5] Hooft G. 't. The Holographic Principle. Class. Quant. Grav. 16:3283, 1999. [6] Jayaprakash C. et al. Universal Properties of the Two-Dimensional Kuramoto-Sivashinsky Equation. Phys. Rev. Lett. 71(1): 12-15, 1993. [7] Kaneko K. & Tsuda I. Chaotic Itinerancy. Chaos 13(3):926-936,2003. [8] Miller J. and Huse D. A., Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice. Phys. Rev. E 48(4): 25282535, 1993. [9] Pavlos, G.P. et al. Evidence for Chaotic Dynamics in the Jovian Magnetosphere. Planetary and Space Science 52:513-541, 2003. [10] Pavlos, G. P. et al. "Self Organized Criticality or / and Low Dimensional Chaos in Earthquake Processes. Theory and Practice in Hellenic Region," in Nonlinear Dynamics in Geosciences, eds. Tsonis A. & Elsner J. (Springer) pp. 235-259,2007. [11] Tsallis C. Nonextensive Statistical Mechanics, Anomalous Diffusion and Central Theorems. Milan}. math. 73:145-176, 2005. [12] Wang X.J. Dynamical sporadicity and anomalous diffusion in the Levy motion. Phys. Rev. A 45(12):8407-8417, 1992. [13] Wang X.J.Sporadic Chaos in space-time processes. Phys. Rev. E 55(2):1318-1324,1995. [14] Watkins W. et al. Testing the SOC Hypothesis for the magnetosphere. 1. Atm. Solar-Terrestrial Phys. 63:1435-1445,2001. [15] Wilson K.G. Problems in physics with many scales of length. Scientific American 241:158-179,1979.

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An Enhanced Tree-Shaped Adachi-Like Chaotic Neural Network Requiring Linear-Time Computations Ke Qinl,* and B. John Oommen 2 ,** 1

2

School of Computer Science and Engineering University of Electronic Science & Technology of China 610054, Chengdu, China (e-mail: [email protected]) School of Computer Science Carleton University K1S 5B6, Ottawa, ON, Canada (e-mail: [email protected])

Abstract: The Adachi Neural Network (AdNN) [1-5], is a fascinating Neural Network (NN) which has been shown to possess chaotic properties, and to also demonstrate Associative Memory (AM) and Pattern Recognition (PR) characteristics. Variants of the AdNN [6,7J have also been used to obtain other PR phenomena, and even blurring. A significant problem associated with the AdNN and its variants, is that all of them require a quadratic number of computations. This is essentially because all their NNs are completely connected graphs. In this paper we consider how the computations can be significantly reduced by merely using a linear number of computations. To do this, we extract from the original complete graph, one of its spanning trees. We then compute the weights for this spanning tree in such a manner that the modified tree-based NN has approximately the same input-output characteristics, and thus the new weights are themselves calculated using a gradient-based algorithm. By a detailed experimental analysis, we show that the new linear-time AdNN-like network possesses chaotic and PR properties for different settings. As far as we know, such a tree-based AdNN has not been reported, and the results given here are novel. Keywords: Chaotic Neural Networks, chaotic pattern recognition, Adachi Neural Network.

1

Introduction

Artificial Neural Network (ANN) is one of the four best approaches for PR. ANN attempts to use some organizational principles such as learning, gen* The work of this author was done while he was in Canada as a Visiting Scholar at Carleton University in 2008. The author gratefully acknowledges the support of the K. C. Wong Education Foundation of Hong Kong. H Chancellor's Professor; Fellow: IEEE and Fellow: IAPR. This author is also an Adjunct Professor with the University of Agder in Grimstad, Norway. The work of this author was partially supported by NSERC, the Natural Sciences and Engineering Research Council of Canada.

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eralization, adaptivity, fault tolerance, distributed representation, and computation in order to achieve the recognition. One of the limitations of most ANN models is the dependency on an external stimulation. Once an output pattern has been identified, the ANN remains in that state until the arrival of a new external input. This is in contrast to real biological NNs which exhibit sequential memory characteristics. To be more specific, once a pattern is recalled from a memory location, the brain is not "stuck" to it, it is also capable of recalling other associated memory patterns without being prompted by any additional external inputs. This ability to "jump" from one memory state to another in the absence oj a stirnulus is one of the hallmarks of the brain, and th'ls is one phenornenon that a chaot'ic P R systern has to ern·ulate. The goal of the field of Chaotic PR systems can be expressed as follows: We do not intend a chaotic PR system to report the identity of a testing pattern with such a "proclamation". Rather, what we want to achieve, on one hand, is to have the chaotic PR system give a strong peTiodic or rnoTe jTequent signal when a pattern is to be recognized. Further, between two consecutive recognized patterns, none of the trained patterns must be recalled. Finally, and most importantly, if an untrained pattern is presented, the system must give a chaotic signal. This paper deals with the Adachi Neural Network AdNN [1-5], which possesses a spectrum of very interesting chaotic, AM and PR properties, as described in [6,7]. The fundamental "problem" associated with the AdNN and its variants is its quadratic computational requirement. We shall show that by using spanning tree concepts and an effective gradient search strategy, this burden can be reduced to be linear, and yet be almost as effective with regard to the chaotic and PR characteristics.

2

Limitations of the Current Schemes

Although the works of Adachi et al and Calitoiu et al were ground-breaking, it turns out that, as stated in [9], the results claimed in the prior art l were not as precise as stated. Apart from this limitation, the computational burden is excessive, rendering it as an impractical machine. More precisely, we mention the following "drawbacks" of the state-of-the-art: 1. The AdNN's power as a PR system requires excessive computations of the order of O(N2). For the examples cited by Adachi et al and Calitoiu et al [1-7], which use 10 x 10 pixel arrays, this involves la, 000 computations peT tirne step. 2. The PR capabilities of the AdNN are limited, as will be explained presently. Our intention is to enhance these. 1

The state of the art of the AdNN and its variants can be found in [6,7,9J.

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3. It is also appropriate that the PR limitations of the work of Calitoiu et al [7] are mentioned. Although the experimental results cited in [7] displaya reduction in the transient phase (when compared to the AdNN) , unfortunately, the M-AdNN does not achieve PR for untrained patterns. It turns out that although unreported in [7], the M-AdNN always reproduces the input patterns periodically independent of whether the input is a trained or an untmined pattern. This has also been illustrated in [9].

3 3.1

Designing the L-AdNN The Topology of the L-AdNN

To minimize the computational burden of the AdNN, we shall first arrive at a topology with a linear number of edges. We arrive at the modified linear version in two steps. First of all, we reduce the number of edges to be linear by using spanning tree. The second step will involve the computation of the weights associated with this new structure, which we will address subsequently. Of course, the details of these steps cannot be presented here due to space limitations, but they can be found in [8]. Since the network topology has been changed and we want the L-AdNN to maximally approximate the original AdNN, we propose to use the Maximum Spanning Tree (MST) of the AdNN (with the edge weights given by their absolute values) to be the initial linear approximation to the completelyconnected AdNN. This is formalized by Algorithm Topology-.1-AdNN below. Unlike the AdNN and its variants, this initial approximation of the L-AdNN Algorithm 1 Topology -.1-AdNN Input: N, the number of neurons in the network, and a set of p patterns which the network has to "memorize". Output: The topology and initial weights of the L-AdNN. Method: 1: Create a completely connected graph 9 which is to represent the AdNN. 2: Compute the weights of the edges of g, {Wij}, by the following: Wij = ~ L~=l (2xi - 1) (2xj -1), where xi is the ith component of the sth stored pattern. 3: Create a non-negative modified set of edge-weights for the graph as: W;j = IWijl Vi,] 4: Compute the MST of 9 with the associated weights of {w;j} as the initial approximation for the L-AdNN. End Algorithm Topology _L-AdNN

is tree-shaped, since we have pruned the edges which we believe are the most "redundant" . In this regard, since that the original AdNN has N neurons and N 2 edges, we recommend the use of the Kruskal algorithm which is much more efficient than the Prim algorithm in such a setting.

Enhanced Tree-Shaped Adachi-Like Chaotic Neural Network

3.2

287

The Weights of the L-AdNN: Gradient Search

Since we have removed most of the "redundant" edges from the completelyconnected graph by using a MST, it is clear that the NN at hand will not adequately compare with the original AdNN. Thus, our next task is to determine a new set of weights so as to force the L-AdNN to retain some of its PR properties, namely those corresponding to the trained patterns. We briefly explain below (for more details the reader is requested to refer to [8]) the process for achieving this. The L-AdNN is defined by following equations:

xf(t + 1) = !(TJf(t + 1) + ~f(t + 1)),

(1)

wf x;(t),

(2)

TJf(t + 1)

=

kfTJf(t)

+

L eijET

(3)

where {wf}, xf, ~f and TJf are the weights, outputs, and state variables of the L-AdNN respectively, and have similar meanings to {Wij}, Xi, ~i and Tli of the AdNN. In order to find the optimal values of {wf}, we define the square error between the original output of the AdNN and new output at the nth step: N

1""

Ep = 2"

~(XiA ,p

-

XiL ,P(n)) 2 ,

(4)

k=l

where xt,p and .y;'P imply the outputs of the ith neuron when the pth pattern is presented to the AdNN network and the L-AdNN network respectively. The overall global error is defined by E = "L:=l Ep where P is the number of trained patterns. In order to adjust wfj to obtain the least global error E, we consider the gradient, Llwfj, and move wfj by an amount which equals Llwfj in the direction where the error is minimized. This can be formalized as below:

8x;'P(n)

owfj p

= (3 L(xt'P - x;,P(n)) . ~ . x;'P(n) . (1 - xf,p(n)) . xf,p(n), (5) p=l

where (3 is the learning rate of the gradient search. The formal algorithm which achieves the update is given in Algorithm 2. The results of a typical numerical experiment which proceeds along the above gradient search are shown in Figure 1. In these simulations, we have chosen the learning rate (3 to be 0.05. The reader will observe that the global error, E, converges very

K. Qin and B. 1. Dammen

288

rapidly. Indeed, after merely 60 iterations, the average value of the matrix of converges to a value arbitrarily close to 0 (from the perspective of the machine's accuracy), and becomes invariant, thereafter.

L1wt

Algorithm 2 Weights_L-AdNN Input: The number of neurons, N, a set of P patterns, and the initial weights {wt} of the L-AdNN. These initial weights are {wj} for the edges in the MST, and are set to zero otherwise. The parameters and the setting which we have used are the learning rate (3 = 0.05, c = 0.015, Do = 10, k f = 0.2 and kr = 1.02. Output: The weights {wt'} of the L-AdNN.

Method: 1: Compute the outputs of the L-AdNN corresponding to the P trained inputs. 2: For all edges of the L-AdNN, compute L1w{j as per Equation (5). Otherwise, set L1wt = 0 .

3: wt

<-

wt

+ L1wt·

4: Go to Step 1 until E is less than a given value or L1wt ;:::;: O.

End Algorithm Weights_L-AdNN

<10 '

f2

" ~

~~

W 107 0

C

v

rul06

""

1-'D5

80

100

Time

120

140

160

ISO

2'00

SO

100

120

140

16.0

Time

Fig. 1. The figure on the left shows the variation of the average of L1wt (averaged over all values of i and j) over the first 200 iterations of the gradient search scheme. The average converges to a value arbitrarily close to zero after 60 time steps. The figure on the right shows the variation of the global error over the same time frame. Observe that this quantity does not converge to zero.

4

Lyapunov Exponents Analysis of the L-AdNN

We shall now briefly submit a Lyapunov analysis of the L-AdNN. Due to space considerations, many of the details are omitted here, but are in [8]. For a dynamical system, its sensitivity to the initial conditions is quantified by its so-called Lyapunov exponents. Although the calculation of the

Enhanced Tree-Shaped Adachi-Like Chaotic Neural Network

289

Lyapunov exponents by directly using a Jacobian analysis is not feasible. However , the specific topology and state update equations of the L-AdNN makes the task feasible, as we shall presently see. Considering the Jacobian matrix J of the L-AdNN whose structure, topology and weights have been determined in Section 3. J can be seen to have the form:

J=( [J~] [J~ ] ) [Jij ] [Jij ]

where each [Ji~] is an N x N sub-matrix of J, for 1 ~ k ~ 4, 1 ~ i ~ Nand 1 ~ j ~ N. Each [Ji~] is a result of taking the partial derivatives of 1]f(t + 1) and

~f( t + 1) with regard to 1]1 (t)

and

~1 (t)

respectively as:

Ji~ (t)= °b~b\~)l) , .1

j2 () _ 01J f(t+l) J3 ()_O';f(t+l) J 4 () o.;f(t+l) £. . ij t o';f(t) , ij t - 01Jf(t)' ij t o';f(t)' or all 1 ~ t ~ Nand 1 ~ j ~ N; By taking the partial derivatives after considering the explicit forms of the state space equations given by (1)- (3), we see that the following closed

form expressions for Ji~(t) result: (a): JMt) when i

=

j; (b): Ji~(t)

L*

=

W~j

L*

+ ~ . xf(t)(l - xf(t)) Tf(t)(l - xf(t)) when i # j; (c): Ji~(t) = =

kf

• •

L*

~xf(t)(l - xf(t)); (d): Ji~(t) = -%xf(t)(l - xf(t)) when i = j; (e): = 0 when i # j; (f): Jt(t) = kr - %xf(t)(l - xf(t)) when i = j; (g): Ji~(t) = 0 when i # j. To get a better insight of the dynamics, we observe, first of all, the term xf(t)(l - xf(t)) has an asymptotic value of zero for all i. Secondly, since the topology of the L-AdNN is tree-shaped, wI;' should be 0 if eij is not in T. Further, since the spanning tree topology does not contain any circles, this implies wf;' = 0 for all j = i. Thus, the Jacobian matrix has the form: J~(t)

kj 0 ... 0 0

0

o

0

o ...... o

kf 0 0 kr 0

o o

o

0

o

0 0 . . .. .. kr

Consequently, J(xo) = J(xd = ... = J(Xk)' The Lyapunov exponents can be calculated by the eigenvalues of matrix L(xo) ,

L(xo) = lim [J(k) (J(k))TP /2k = lim [Jk(JT)k]1/ 2k = 1. k---+ oo

k--HXJ

(6)

290

K. Qin and B. 1. Oommen

Observe that this matrix has 2N eigenvalues. thus the Lyapunov exponents are: Al = A2 = ... = AN = log kj, AN+l = AN+2 = ... = A2N = log k r . By choosing the values of the coefficient as k f = 0.2, kr = 1.02, we can force the LLE to be AN+l = AN+2 = ... = A2N = log 1.02> 0, rendering the L-AdNN to be chaotic.

5

Chaotic and PR Properties of the AdNN s

We now briefly report the comparative PR properties of the L-AdNN and AdNN, and refer the reader to [8] for more detailed results. These properties have been gleaned as a result of examining the Hamming distance between the input pattern and the patterns that appear in the output.

5.1

Comparative Chaotic Properties

In the ideal setting we would have preferred the L-AdNN to be chaotic when exposed to untrained patterns, and the output to appear periodically or more frequently when exposed to trained patterns. Besides yielding this phenomenon, the L-AdNN also goes through a chaotic phase and a PR phase as some of its parameters change. We summarize the results for the L-AdNN

l........ l~ l~ lr-l l'"

l~~l~~l~~l~iL.i l~~ 2 4 6 810

2 4 6 810

2 4 6 810

2 4 6 810

2 4 6 810

Fig. 2. The set of patterns used in Adachi et ai's and our experiments. The first four patterns constitute the set used by Adachi et al. Pattern 5 is an untrained pattern, which represents the digit 5.

by stating that by using different settings of a and ai, the latter demonstrates the following amazing result in Figure 3. From this Figure we see that if the L-AdNN is presented with a trained pattern, the output is the same trained pattern occurring Frequently. But the output is Chaotic for untrained patterns. While this phenomenon is not observed when ai = 2 (when the L-AdNN is always Chaotic), the Frequent behaviour is noticeable when ai = 2 + 6Xi. Since a finer measure of how closely the output mimics the input is their Hamming distance (which is exactly zero if the output is an precise replica of the input), we have also tabulated the frequency of the Hamming distance within the first 1! 000 time intervals in Table 1 for the various input patterns. Table 1 demonstrates that this distance is close to zero (i.e., between 0 and 5) quite infrequently, implying that under these settings, the L-AdNN is chaotic. It is worth mentioning that most of the Hamming distances are in

Enhanced Tree-Shaped Adachi-Like Chaotic Neural Network

(a)

291

(b)

Fig. 3. The Hamming distance between the output and the trained patterns or untrained patterns while using different settings. The input is Pattern 4 in both (a) and (b). The external stimulation weights are ai = 2 and ai = 2 + 6Xi in (a) and (b) respectively. The output graph are displayed for pattern 1, pattern 4 and pattern 5. Table 1. The frequency distribution of the Hamming distance between the input and the output as the measure occurs within 10 intervals, when the external input is ai = 2, and the refractory factor ct = 10. HamIlling distance Input Pattern [0,5J 116,19J [20,29J [30,39J [40,49J [50,59J [60,69J [70,79J 1180,89J [90,100J Pattern 1 1 25 129 279 3 397 143 22 1 0 Pattern 2 3 5 33 121 291 374 146 24 3 0 Pattern 3 1 2 15 0 117 342 358 146 18 1 Pattern 4 1 12 6 157 338 339 121 24 2 0

the interval [40,59] - i.e., almost 50% erroneous pattern, which is what we expect from a truly noisy pattern. If the same settings are used in the AdNN, the system behaves as an AM, which has been well illustrated by Adachi and his co-authors in their paper [1].

5.2

Comparative PR Properties

If the parameters of the AdNN are set as prescribed in [1], and the AdNN weights the external inputs with ai = 2 + 6Xi, the system becomes closer to mimicing a PR phenomenon. Adachi et al claim that such a NN yields the stored pattern at the output periodically with a small periodicity. However, it turns out that if the input is an untrained pattern, the AdNN can still retrieve it and the other four trained patterns with relatively lower frequency. This property is more or less one which resembles an AM system instead of a PR system. This can be seen from Table 2. As opposed to this, for the L-AdNN, the most interesting scenario occurs when C\' = 10 and the weight for the external input is ai = 2 + 6Xi' In this case, the system's output is noticeably chaotic. But the input pattern can be retrieved very frequently as long as the input is one of the trained patterns.

292

K. Qin and B. 1. Oommen

Table 2. The AdNN: In this table, the input is the untrained Pattern 5 in Figure 2. The numbers in this table are the frequencies of each pattern being retrieved within the first 1, 000 time intervals.

We clarify this by explaining Figure 3 (b) in greater detail. The input is Pattern 4, the output displays the input (Pattern 4) or a reasonably similar pattern with a very high frequency - as demonstrated by the low Hamming distance. The Hamming distance between the output and all the other patterns - the trained and the untrained pattern - is noticeably very large (i.e., with almost 50% noise). The corresponding table of Hamming distances is given in Table 3. Consider the first column of Table 3. The reader will observe that the trained input patterns can be approximately retrieved almost 20% of the time. On the contrary, when encountering an untrained input pattern (Pattern 5 in Figure 2), it is observed in the output with a much lower frequency - less than 5%. Adachi and his co-authors did observe this for the AdNN, namely that the system with the external stimulations yielded a stored pattern with a relative higher frequency, and an untrained pattern with relative lower frequency - which phenomenon can be utilized to achieve PR. The amazing point here is that the L-AdNN displays the same phenomenon even though it merely has a linear- number of connections, and the fact that the weights used are not directly obtained from the patterns themselves, but by migrating from the latter by using a gradient search algorithm. As far as we know, analogous properties have not been reported earlier for any type of NN (chaotic or otherwise). The unabridged paper [8] also presents a comparative analysis of the quasi-energy function properties of the L-AdNN and the AdNN, omitted here due to space considerations. Table 3. The frequency distribution of the Hamming distance between the input and the output as the measure occurs within 10 intervals, when the external input is 0i = 2 + 6x; and the refractory factor ex = 10. Input Pattern Pattern 1 Pattern 2 Pattern 3 Pattern 4 Pattern 5

6

Hamming distance [0,5] [6,19] [20,29] [30,39] [40,49] [50,59] [60,69] [70,79] [80,89] [90,100] 217 342 226 112 63 28 10 2 0 0 197 369 222 131 21 1 58 J 0 0 198 251 325 130 57 32 3 4 0 0 196 357 235 126 54 25 2 5 0 0 46 495 334 104 18 2 0 1 0 0

Conclusions

In this paper we have concentrated on the field of Chaotic Pattern Recognition (PR), which is a relatively new sub-field of PR. Such systems, which have only recently been investigated, demonstrate chaotic behavior under

Enhanced Tree-Shaped Adachi-Like Chaotic Neural Network

293

normal conditions, and resonate when it is presented with a pattern that it is trained with. The network which we have investigated is the Adachi Neural Network (AdNN) [1-5], which has been shown to possess chaotic properties, and to also demonstrate Associative Memory (AM) and Pattern Recognition (PR) characteristics. Because its structure involves a completely connected graph, the computational complexity of the AdNN is quadratic in the number of neurons. In this paper we have considered how the computations can be significantly reduced by merely using a linear number of computations. To achieve this, we have extracted from the original completely connected graph, one of its spanning trees and then computed the best weights for this spanning tree by using a gradient-based algorithm. Apart from a Lyapunov analysis and an analysis of the quasi-energy function found in [8], by a detailed experimental suite, we show that the new linear-time AdNN-like network possesses chaotic and PR properties for different settings.

References 1.M. Adachi and K. Aihara. Associative dynamics in a chaotic neural network. Neural Networks, 10(1):83-98, 1997. 2.M. Adachi and K. Aihara. Characteristics of associative chaotic neural networks with weighted pattern storage-a pattern is stored stronger than others. In The 6th International Conference on Neural Information Processing, Perth, Australia, volume 3, pages 1028-1032, 1999. 3.M. Adachi, K. Aihara, and M. Kotani. Pattern dynamics of chaotic neural networks with nearest-neighbor couplings. In Circuits and Systems, 1991, IEEE International Sympoisum on, Westin Stanford and Westin Plaza. Singapore, volume 2, pages 1180-1183, June 1991. 4.M. Adachi, K. Aihara, and M. Kotani. An analysis of associative dynamics in a chaotic neural network with external stimulation. In Proceedings of 1993 International Joint Conference on Neural Networks, Nagoya, Japan, volume 1, pages 409-412, 1993. 5.K. Aihara, T. Takabe, and M. Toyoda. Chaotic neural networks. Physics Letters A, 144:333-340, 1990. 6.D. Calitoiu, B.J. Gommen, and D. Nussbaum. Desynchronzing of chaotic pattern recognition neural network to model inaccurate parception. IEEE Transaction on Systems, Man, and Cybernetics-part B:Cybernetics, 37(3):692-704, 2007. 7.D. Calitoiu, B.J. Gommen, and D. Nussbaum. Periodicity and stability issues of a chaotic pattern recognition Neural Network. Pattern Anal Applic, pages 175-188, 2007. 8.Qin Ke and B. J. Gommen. Adachi-like chaotic neural networks requiring lineartime computations by enforcing a tree-shaped topology. In Unabridged Version of This paper, 2008. 9.Qin Ke and B. J. Gommen. Chaotic pattern recognition: The spectrum of properties of the adachi neural network. In The 12th International Workshop on Structural and Synaptic Pattern Recognition and 7th International Workshop on Statistical Pattern Recognition, December 2008.

294

Clustering of Inertial Particles In 3D Steady Flows Themistoklis Sapsis 1 and George Haller 2 1

2

Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139, USA (e-mail: sapsis@mit. edu) Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139, USA (e-mail: [email protected])

Abstract: We study the motion of inertial particles in three-dimensional steady fluid flows that contain a family of two-dimensional invariant manifolds. Using results from Ergodic Theory we derive a condition that predicts if the considered invariant manifold for the flow will persist as an invariant manifold for inertial particles. We illustrate our results for the three-dimensional ABC flow with paremeters corresponding to a non-integrable case. Keywords: inertial particles, Ergodic Theory, Melnikov Theory.

1

Introduction - Setup

The motion of small particles in fluid flows is a common phenomenon encountered both in nature (e.g. contaminant dispersion in the atmosphere) as well as in many technological applications (e.g. chemical systems involving particulate reactant mixing). In dilute flows with particles having different density from the fluid, a particularly interesting behavior that has been observed both numerically and experimentally is particle clustering. This refers to the tendency of particles to concentrate in narrow bands and is due to the fact that the equations of motion for inertial particles are considerably different from those for fluid parcels, as inertial and viscous effects playa significant role. Several works have analyzed inertial particle dynamics in either analytically defined or numerically generated fluid flows (cf. Haller and Sapsis [6]' Burns et al. [3], Yannacopoulos et al. [11], Rubin et al. [9], Maxey and Riley [8]). In this work we study clustering properties of small spherical particles in three-dimensional steady flows that contain a single or a family of closed invariant manifolds. In particular, we derive a necessary condition for the persistence of an invariant manifold that has no stagnation points as an attractor for inertial particles. Let u (x) represents the velocity field a three-dimensional steady fluid flow of density Pf. The fluid fills the compact spatial region D with boundary aD;

Clustering of Inertial Particles in 3D Steady Flows

295

we assume that V is a uniformly bounded smooth manifold for all times. We also assume that u (x) is a smooth function for all of its arguments. We denote the material derivative of the velocity field by

Du Dt

= (\7u) u,

(1)

where \7 denotes the gradient operator with respect to the spatial variable

x. We also assume that u (x) is volume preserving, i.e. \7. u

=

o.

For a particle p of density Pp immersed in the fluid we denote with x (t) its path. Let the particle be spherical with radius a « L, with L denoting a characteristic length scale in the flow, Re the Reynolds number, and v (t) = x (t) the Lagrangian velocity of the spherical particle. In this case the particle satisfies the Maxey-Riley equation of motion (cf., e.g. Maxey and Riley [8], Benczik et al. [2] or Babiano et at. [1]) (2a)

x= v

. 3R Du (x, t) EV= U (x, t) - V+E ,

2

(2b)

Dt

where 1

E

= - « 1, fJ

R

St

fJ = St'

2 (a)2 =9 L Re.

Note that the larger the inertia parameter fJ, the less significant the effect of inertia; in the fJ --> 00 limit, (2b) describes the motion of a passive ideal tracer particle. The density ratio R distinguishes neutrally buoyant particles (R = 2/3) from aerosols (0 < R < 2/3) and bubbles (2/3 < R < 2). In what follows we will consider only non-neutrally buoyant particles. Using geometric singular perturbation theory Haller and Sapsis [6] proved that for E > 0 small enough, equation (2b) admits a globally attracting invariant slow manifold that has the form

3R ME = { (x, Vs) : Vs = u (x, t) + E ( 2

-

) Du(x , t)

1

Dt

2 }

+ O(E)

.

(3)

By reducing the dynamics on the slow manifold ME it is shown that particle motion is governed by the inertial eq'uation

x=u(X,t)+E (

23R -

1

) Du(x,t) 2 Dt +O(E).

(4)

T. Sapsis and C. Haller

296

2

Attracting invariant manifolds for inertial particles

For two-dimensional, steady, flows the motion of inertial particles is governed by the autonomous, two-dimensional ODE (4) from which powerful conclusions about the asymptotic nature of finite-size particles can be concluded (cf. Rubin et al. [9], Haller and Sapsis [6]). In what follows we provide a general result for three-dimensional, volume preserving flows having a family of invariant manifolds concerning the persistence of these slow manifolds as attractors for inertial particles. Theorem 1. Let u (x)

be a three-dimensional, autonomous, volumepreserving, vector field and So a compact, two-dimensional, invariant manifold, with unitary normal vector fi. (x) on which the motion is non-resonant. Mor-eover, we assume that Ilu (x)11 > 0 for all x ESo. Then a necessary condition for the persistence of So as an invariant manifold for inertial particles is I (So)

== lim

eR-1) iT Ilu 2

T--+CXJ

T

0

Du (ip (t, xo))ll fi. (ip (t, xo)) (ip (t, xo)) dt Dt

=

0

(5) where xoESo. In this case the flow on So is ergodic with probability density function for the position of an inertial par-ticle on So given by

f(x) = Iso Ilu(Y)II-1d!r(y)'

(6)

x ESo.

Proof. Let SE be a compact, two-dimensional, invariant manifold with a well defined interior for the steady inertial equation (4). We assume that SE varies smoothly for f :::0: O. The phase space volume bounded by SE does not change in time, therefore

r

\7. [u (x)

+f

(3R _ 1) Du (x)

f (3R _

=

2

Dividing by

f

1) r

Dt

2

Jlnt(S,)

\7.

1

f

Int(So)

---+

dV (x)

[DU (x) + O(f)] dV (x) Dt

Jlnt(S,)

and taking the

+ 0(f 2 )]

0 limit, we obtain that

Du (x) \7·--dV (x) = 0 Dt

must hold for the compact streamsurface So of u (x). By applying the divergence theorem on Int(So) we obtain

1

Du (x) \7·--dV (x) = Int(So) Dt

1 So

Du (x) ~ - - n (x)da (x) = 0 Dt

Clustering of Inertial Particles in 3D Steady Flows

297

where dO" (x) is the infinitesimal area element (Riemannian volume) of So at x. Let us now consider the flow of the dynamical system on the two dimensional torus So. As we have assumed the motion on So is non-resonant. In this case, as it is proven by Kolmogorov [7] (see also Sinai [10D, there exists an infinitely differentiable change of coordinates where the system reduces to dii

diJ

(ii, iJ) E S2

-=a

dt = 1,

dt

(7)

where a is such that the pair (1, a) are Diophantine frequencies or equivalently a is normally approximated by rational numbers (cf. Sinai [10]). Hence, any trajectory of (7) will fill out densely the torus S2 preserving the Lebesgue measure on it. Thus, the flow on So will be ergodic and by Liouville Theorem (see e.g. Sinai [10]) the invariant measure will be given by J.L (x)

= Ilu (x)II- 1 dO" (x)

x E So.

Hence, we can apply the Birkhoff-Khinchin Ergodic Theorem to obtain

r

r DUD(X) ii (x)dO' (x) = lim -T iso t T~oo io 1

lIu (

for every Xo E So. Moreover, by considering the Radon-Nikodym derivative of the invariant measure J.L (x) we obtain the probability density function for a finite-si ze particle on So, i.e. the concentration of inertial particles on So

f

(x) =

J~o Ilu (y) 11 - 1 dO' (y) ,

x ESo .

This completes the proof of the theorem. To estimate the quantity I (So) from a finite time interval we exploit the Central Limit Theorem for dynamical systems (cf. Burton and Denker [4]). More specifically by setting

IT (So,xo)

=

eR-1)j,T T 2

Du a Ilu('P(t, xo))IIfi.(
there exists a positive number

ju

~

such that for every (3 E lR.

f(x)dO" (x) ~ T

{l

1

~

v 27r

1(3

exp

(X2) -2 dx

-<Xl

Ul

where = {x ESo : ~T [IT (So, x) - I(So)] < (3}. Thus, the rate of convergence of IT (So, x) to I (So) is of order 0 (T - 1) . The stability of a streamsurface So for which I (So) = 0 can be verified by checking whether the phase space volume grown or shrinks on either side of So. For a stable SE we must have

(9)

298

T Sapsis and G. Haller

"

I

'\.

\

I

\

I \

\ I '\

-

....... "",'"

I /

Fig. 1. Numerical calculation of the normal vector fi (x) .

for all closed streamsurfaces 50~ , 5 0+ close enough to 50 with the property Int(50) C Int(50+}. At the same time , for an unstable 5 e , we must have Int(50~) C

(10) For the special case oftwo-dimcusional , steady, velocity field u (x) , x E ]R2 we have flex) = U..L (x) lI u(x) l l~l where U ..L (x) = (- v ,u). Hence, the nonlinear functional (5) reduces to the following Melnikov-type form

I (So) = }~~

=

e T- 1) .far u(cp(t,xo)) R

T

2

A

Du

Dt (cp(t,xo))dt

(¥T~ 1) laTaU(cp(t , XO)) A ~~ (cp (t,XO)) dt

where Ta is the period of a fluid particle on the closed streamline and A is the wedge product defined in ]R2 as x A y = :rl1/2 - :t)lX2.

3

Application to the ABC flow

In general, the streamlines of non-integrable steady flows may have a complicated Lagrangian structure which will not allows us to express analytically the streamsurfaces and integrate over them. A typical example is the ArnoldBeltrami- Childress (ABC) flow where as it has been observed (see Dobre at al. [5]) there is a set of closed helical streamlines each of which is surrounded by a finite region of KAM invariant surfaces. For the ABC flow the velocity field is given by ASin Z + CCOS Y) u (x) = ( Bsinx + Acos z C sin y + B cos x

(11)

Clustering of Inertial Particles in 3D Steady Flows

299

a)

>-

x

Fig. 2. a) Poincare Map of the ABC flow; with color we interpret the computed value of IT for every closed streamsurface. b) The streamsurface that attracts inertial particles obtained by direct numerical simulation of the inertial equation for t( 1) = -0.01.

3: -

For the computation of the normal vector we exploit the ergodic property of the flow on the invariant streamsurface and we express it as

nT(ip(i,xo)) =O:U(ip(t,xo)) x [rp (t,xo) - ip(s ,xo)] where 0: > a is a normalization constant such that is defined as 8=0;-1 [ inf

I.,.-tl>f)

lin (x)11 = 1 and 8

Ilip(t,XO)-ip(T, Xo)ll]

(12 ) E [0, T]

300

T. Sapsis and U. Haller

:I

x 10"

'1.5 ~~

f;~~11A

I:: :~1'1

1

'1 '0.9

0,8

x -4

-3

-2

,.-,

0,(

Fig. 3 . Invariant streamsllrface for inertial particles. The color shO\vs the proba-, bility density function of the position of an inertial particle.

with Ot (s) == 11'P(t,xo) - 'P(s,xo)11 and () the characteristic time for a particle to run over a typical diameter dso of 8 0 (Figure 1). In this ease the ergodic property of the flow and the Central Limit Theorem gua.rantee that IlnT'(x) - 11 (x) II~'O as T-., 00 with rate of convergence of order 0 (r-l) . To illustrate our results we consider the ease A 2 1, 8 2 = ~) C 2 = by Dobre at aL [5]. For this choice of parameters the phase space is devided into 'ordered' and 'chaotic' regions and the system is not integrable. vVe first compute the Poincare section defined by the plane z = 1\ and sub"equently calculate the integral IT' (1' = 50 ) over the ordered regions where closed streamsurfaces exist. ~ studied

In Figure 2 we present the F)oincare map for the flow, superimposed by a colornlap that represents the value of h· for every closed streamsurface. As we are able to observe there is a pair of dosed streamsurfaces in S
Clustering of Inertial Particles in 3D Steady Flows

301

according to the probability density function calculated through equation (6).

4

Conclusions

We have studied clustering properties of inertial particles in three-dimensional, steady flow fields. Using results of Ergodic theory we have derived a necessary condition for the persistence of any closed invariant manifold as a two-dimensional invariant manifold for inertial particles. This condition has the form of a nonlinear functional that acts on the unitary normal bundle of the streamsurface and any non-resonant trajectory that lies on it. We have illustrated our theoretical findings through a non-integrable case, the ABC flow, and we have determined clustering regions for non-neutrally buoyant particles. Acknowledgements This research was supported by NSF Grant DMS-04-04845, AFOSR Grant AFOSR FA 9550-06-0092 , and an MIT Presidential Fellowship.

References 1.A. Babiano, J. H. E. Cartwright, O. Piro, and A. Provenzale. Dynamics of a small neutrally buoyant sphere in a fluid and targeting in hamiltonian systems. Phys. Rev. Lett., 84:5764, 2000. 2.1. J. Benczik, Z. Toroczkai, and T. Tel. Selective sensitivity of open chaotic flows on inertial tracer advection: Catching particles with a stick. Phys. Rev. Lett., 89:164501, 2002. 3.T. Burns, R. Davis, and E. Moore. A perturbation study of particle dynamics in a plane wake flow. J. Fluid Mech., 384:1, 1999. 4.R. Burton and M. Denker. On the central limit theorem for dynamical systems. Trans. Amer. Math. Soc., 302:715, 1987. 5.T. Dombre, U. Frisch, J. M. Greene, M. Hinon, A. Mehr, and A. M. Soward. Chaotic streamlines in the abc flows. J. Fluid Mech., 167:353, 1986. 6.G. Haller and T. Sapsis. Where do inertial particles go in fluid flows? Physica D, 237:573, 2008. 7.A. N. Kolmogorov. On dynamical systems with an integral invariant on the torus. Dokl. Akad. Nauk, 93:763, 1953. 8.M. Maxey and J. Riley. Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids, 26:883, 1983. 9.J. Rubin, C. K. R. T. Jones, and M. Maxey. Settling and asymptotic motion of aerosol particles in a cellular flow field. J. Nonlinear Sci., 5:337, 1995. W.YA. G. Sinai. Introduction to ergodic theory. Princeton University Press, 1976. 11.A. N. Yannacopoulos, G. Rowlands, and G. P. King. Influence of particle inertia and basset force on tracer dynamics: Analytic results in the small-inertia limit. Phys. Rev. E, 55:4148, 1997.

302

A Two Population Model for the Stock Market Problem Christos H. Skiadas Technical University of Crete, Chania, Crete, Greece (e-mail: [email protected]) Abstract: The development of the last year disaster in the Stock Markets all over the world gave rise to reconsidering the previous models used. It is clear that, even in an organized international or national context, large fluctuations and sudden losses may occur. This paper explores a two populations' model. The populations are conflicting into the same environment (a Stock Market) by following the main rules present, that is mutual interaction between adopters, potential adopters, word-of-mouth communication and of course by taking into consideration the innovation diffusion process. The proposed model has special futures expressed by third order terms providing characteristic stationary points. Keywords: chaotic modeling, the stock market problem, stock market, innovation diffusion modeling, Lotka-Volterra, simulation, chaotic simulation.

1. Introduction Several attempts to model the stock-market problem can be found in the literature. Between these models the Lotka-Volterra modeling approach is of considerable interest. The model can be used if we assume two interacting populations and can be generalized to more than two. The case of two interacting populations is explored by using two coupled differential equations and the results found give rise to a limit cycle and the corresponding oscillating graphs over time, Skiadas, 2009 [7]. However, the Lotka-Volterra system of differential equations, a non-linear system of the second order, fails to explain the sudden growth and decline resulting into the stock-market environment and even more the stability to high or low values (capital gains or losses). In view of the last year losses in the stock markets globally it would be very important to reconsider the classical Lotka-Volterra theory by introducing into the corresponding equations terms having to do with the diffusion aspect of the communication process. The proposed model has special futures expressed by third order terms providing characteristic stationary points. It should be noted that a seminal paper was published by Harold Hotteling [1] during March 1929 few months before the big crash in the New York stock exchange. Hotelling, in this paper explored the "stability in competition" problem. The analysis proposed was based in a simplification of the problem. Only two markets where competing; however, the results where extremely important and clarified the situation. In this paper we accept the same methodology supposing that there is a stock market with two interacting populations and we explore the arising case. That is new is the introduction of the theories of the last decades on the

A Two Populatiol1 Model for the Stock Market Problem

303

adoption-diffusion of innovations in developing the equations of the proposed model.

2. The Model and Simulations To model the specific situation we take into account that two populations x and yare present into the stock market and interact each other. Even more to simplify the case we suppose that the variables XI and y, stand for the number of players or for the number of transactions or for the values of the stocks belonging to each part of the players at a specific time period t. The aim is to explore the behavior of the two populations during time and especially in the limits. A model including the main characteristics of two interrelating or even conflicting popUlations into a stock market can be expressed by the following set of differential equations:

i

= a Y + a 2 xy + a 3 x(l- y) + a 4 yx(k j

x)

y =b x+b2 xy+b3 y(k -x)+b4 xy(l- y), j

where aj, a2, a3, a4 and bj, bz, b3 , b4 are parameters expressing the mutual interaction of the populations X and y. The parameters k and I express the upper limit of the popUlations x and y respectively. The first term to the right stands for the flows from the one part to the other whereas the second term expresses the mutual interaction between the active parts of the populations x and y. If we retain the first two terms to the right the proposed model is of the Lotka-Volterra type. The non active parts of the populations are (k - x) for x and (l - y) for the population y. The interaction of these parts with the active popUlations y and x is expressed with the third terms to the right of the above differential equations (see [2, 3,4, 5, 6]). The fourth terms to the right include the terms x(k - x) and y(l - y) multiplied by y and x respectively. These terms account to the rates of adoption-diffusion that is extremely important in order to express the word-of-mouth communication between adopters (the active part of the players) and potential adopters (the nonactive part of the players). Looking back to the set of the two differential equations above we see that we have two third degree equations for x and y. In the following we check the influence of the third order term to the stock-market model. To this end the above model is simplified to the following form: •

= -ay + cyx 2 Y. = bx-cxy 2 , X

304

C. H. Skiadas

Where a, band c are parameters of interaction. Especially the parameter c is selected to be the same in both equations expressing the coupling of both populations x and y. The set of the last equations takes the form:

i = -ay + cyx 2 = -cyra;;(.Ja/ c - x)- c.xy(Ja/ c - x)

y = bx -

cxy2

=

cx~(.Jb/ c - y)+ c.xy(~ - y),

The last equations include in the right part side two basic parts of the interaction

ra;;

and the diffusion process where x = and y = .Jb / c are the upper limit or the maximum for x and y respectively. There is a fixed point at (0, 0) and four other characteristic points at

(x

= ra;; , y = .Jb / c),

(x

= ra;; ,y = -~ )

(x=-ra;; ,y =~), (x=-ra;; ,y =-~) The last differential equations give the following differential equation for x and y:

dy

bx-cxy2

dx

-ax+cyx 2

The solution is

(l-b/c)(X2 -a/c)=h Where h is the integration constant. The next Figure illustrates characteristic graphs of the last equation. The four characteristic points define a rectangle. Into this space are drawn few of the trajectories of the process. The trajectories outside of the rectangle diverge to infinity.

A Two Population Model for the Stock Market Problem

305

) --

---~.

-~

---Figure 1. Characteristic Trajectories for various h. As it is presented in Figure 1 there appear to be positive and negative values for the stock-market process. To be realistic we will move the main part, that is the rectangle space, inside the first quarter (the positive space) of the Cartesian Coordinates. This is achieved by introducing the transformation x*=x-el and y*=y-e2·

The resulting set of differential equations is:

x = -a(y -

e2 )

+ c(y - e2 )(x - ej )2

y = b(x -

) -

c(x - ej )(Y - e2 )2 ,

ej

Where x stands for x* and y for y*. We can also find the corresponding difference equation form for this system by observing that

. X

dx

Lix

x t+1 - x t

dt

M

(t + 1) - t

= - "" - =

i1y . dy Y = dt "" ---;;

= Xt+1

-

xr

Yr+l - Yr

= (t + 1) - t = Yt+j

- Yr

The resulting difference equations set is: X t+1

=xr -a(Yr -e 2 )+C(Yr -e 2 )(x r _e j )2

Yr+l

= Yr +b(xr -el)-c(xr -ej)(Yr -e 2)2,

An illustration of this difference equations' model appear in Figure 2. There is an unstable fixed point located at (e" e2) and the four fixed points at the corners

306

C. H. Skiadas

of the rectangle. The process can stabilized at any of the four points depending on the starting values or on values of the parameters. That is extremely important is that we may have four specific cases when aJter some time the process is stabilized in which: I. (x = e 1

-+;;;t;. y:::: e2 -+ -Jb/ c); both gains

2. (x:::: e J

+ -Ja/ c,)'::::; e2 - -Jb/ c

eJ -- J;;iz, . y =

);x gains,y losses

+,Jb / c ); x losses, y gains

3. ( X

:::

4. (x

= e -,J a / c ,y : -. : e2 -.Jb / c ); both losses

e2

j

In Figure 2 the third case appears by means that y gains and x losses. The upper part of the figure illustrates the charactelistic rectangle whereas the time varying development of x and y is presented. The original sinusoidal oscillations give rise to step like oscillations and finally to a stable process where x retain losses and y keep their gains. The process has stopped at the upper left hand side of the charactclistic rectangle indicated by a small circle (see fjgure 2). The starting point was inside the characteristic rectangle and the movement was counterclockwise.

-~----V U

(-.J ' - - - - - - - - - - - - -

Figure 2. The Difference Equations model (Case 3) For a simpler presentation of the findings we use the previous difference equation without the linear movement expressed by ej and ez by setting el=O and ez=O. The resulting form is

x r+] Yt+l The Jacobian of this map is

J

:::::

XI -

aY t -+ eytx t -

2

==)'/

+bXt -ext)'t '

A

l(x,),) =

7\1'0

Population Modelfor the Stock Marker Problem

307

] +2cxy 1

b-cy~

The value of the Jacobian at the characteristic point (0, 0) is: J (0,0) =: 1+ ab . Assuming positive parameters a and h the point (0,0) is unstable. The Jacobian at the other 4 characteristic points

(±,ja/ c ,±,jb/ c)

is

J (0,0) =: 1- 4ab, and for appropriate positive values of the parameters the 4 points are stable (attracting). The resulting graphs are presented in the following four figures (Fig.3-Fig.6) where the four different cases of gains and losses appear for various statiing values of Xo and Yo. The parameters selected are a=0.06, h=O.03 and c=0.025. A small circle in the appropriate comer of the charactetistic rectangle indicates the stopping case. In Figure 3 this circle is at the upper light hand comer, in Figure 4 is at the lower right part of the rectangle whereas, in Figure 5 the circle is at the upper left part and in Figure 6 the circle is located at the lower left hand side. In the latter case both x and y is stabilized at the lower values of the process with losses.

lih frt, (T--===

tt~fJt\-m:\-t== 1].1 l::t' L':t------

¥@Tl1tj-------

Fig. 3 (xo=O.5, Yo=O)

Fig. 4 (xo=O.55, Yo=O)

5 (xo=O.6, Yo=O)

Fig. 6 (xo==O.65, Yo=O) Model Simulations

308

C. H. Skiadas

3. Conclusions A model expressing two conflicting populations in the stock-market is developed. A general model is formulated and a simpler one is explored and simulated. The results support the well known process of fluctuations, oscillations and further sudden growth and decrease of gains-losses of the two conflicting populations. It was derived that it could arise that both would be stabilized to gains or losses or to stabilize the one in gains and the other to losses.

References [I] H. Hotelling. Stability in Competition, Economic Journal, vol. 39, no. 153,41-57, 1929. [2] C. H. Sldadas. Two generalized rational models for forecasting innovation diffusion. Technol Forecast Soc Change 27:39-61, 1985. [3] C. H. Skiadas. Innovation diffusion models expressing asymmetry and/or positively or negatively influencing forces. Technol Forecast Soc Change 30:313-330, 1986. [4] C. H. Sldadas. Two simple models for the early and middle stage prediction of innovation diffusion . IEEE Trans Eng Manage 34:79-84,1987. [5]

c.

[6]

c. H. Skiadas, A.

H. Skiadas and A. N. Giovanis. A stochastic bass innovation diffusion model for studying the growth of electricity consumption in Greece. Appl Stoch Models Data Anal 13:85-101, 1997 N. Giovanis and J. Dimoticalis. A sigmoid stochastic growth model derived from the revised exponential. In: Janssen J, Sldadas CH (eds) Applied Stochastic Models and Data Analysis. World Scientific, Singapore, pp 864- 870, 1993.

[7] C. H. Skiadas and C. Sldadas. Chaotic Modeling and Simulation: Analysis of Chaotic Models, Altractors and Forms, Taylor and Francis/CRC, London, 2009.

309

Electron Quantum Transport through a Mesoscopic Device: Dephasing and Absorption Induced by Interaction with a Complicated Background Valentin V. Sokolov Budker Institute of Nuclear Physics 630090 Novosibirsk, Russia (e-mail: [email protected])

Abstract: Effect of a complicated many-body environment is analyzed on the chaotic motion of a quantum particle in a mesoscopic ballistic structure. The absorption and dephasing phenomena are treated on the same footing in the framework of a schematic microscopic model. The single-particle doorway resonance states excited in the structure via an external channel are damped not only because of the escape onto such channels but also due to ulterior population of the long-lived background states. The transmission through the structure is presented as an incoherent sum of the flow formed by the interfering damped doorway resonances and the retarded flow of the particles re-emitted by the environment. Keywords: quantum transport, chaos, dephasing, absorption.

1

Introduction

Extensive study of the electron transport through ballistic micro-structures [1] has drawn much attention to peculiarities of chaotic wave interference in open mesoscopic set-ups. It is well recognized by now that the statistical approach [2-4] based on the random matrix theory (RMT) provides a reliable basis for describing the universal fluctuations which, in particular, manifests itself in the single-particle resonance chaotic scattering and the transport phenomena. However experiments with the ballistic quantum dots reveal appreciable and persisting up to very low temperatures deviation from the predictions of the standard RMT, which indicates some loss of the quantummechanical coherence. This effect is called dephasing. Two different methods of accounting for the dephasing have been suggested, which give different results. In the phenomenological Biittiker's voltage-probe model [5] an subsidiary randomizing scatterer has been introduced with M¢ channels each with a sticking coefficient T¢. Even assuming all such channels to be statistically equivalent we are still left with two independent parameters. This results in an ambiguity since quantities of physical interest (e.g. the conductance distribution) depend, generally, on M¢ and T¢ separately whereas the de phasing phenomenon is controlled by the unique

310

v. V. Sokolov

parameter: the dephasing time T¢ which is fixed by their product. On the other hand, only one parameter: the strength of a uniform imaginary potential governs the Efetov's model [6] which points at the absorption as the cause of the loss of quantum coherence. The difference and deficiencies of the two models [7,8] have been analyzed in [8]. A prescription was suggested how to get rid of uncertainties, and simultaneously, to accord the models by considering the limit M¢ ---; 00, T¢ ---; 0 at fixed product Mq;I'¢ == r¢ in the first model and by compulsory restoration of the broken because of the absorption unitarity in the second. The construction proposed infers a complicated internal structure of the Biittiker's probe which should, in particular, possess a dense energy spectrum. Otherwise, the assumed limit could hardly be physically justified. If so, the typical time Tp spent by the scattering particle inside the probe, being proportional to its mean spectral density, forms a new time scale different from the dephasing time. Below we propose a microscopic schematic model of absorption and dephasing phenomena induced by interaction with a very complicated environment with very high density of the energy levels.

2 2.1

Chaotic resonance scattering against a complicated background Fragmentation of the resonance states because of the influence of the background

Let D be the mean level spacing between the single-particle resonance states in an open cavity with perfectly reflecting walls and no environment. The cavity is supposed to be attached to two similar leads which support M channel states. An open system of such a kind is described by an effective nonHermitian Hamiltonian H(s) = H(s) - ~AAt where the rectangular matrix A consists of the M(s) column vectors of coupling amplitudes between internal and channel states. Further, let the Hermitian matrix H(e) represent the Hamiltonian of a a many-body environment with a very small mean level spacing 5. The latter fixes the smallest energy scale which will never appear explicitly but ensures unitarity of the scattering matrix S(E) = 1- iT(E). The total system: the cavity interacting with the environment, is described by the extended non-Hermitian effective Hamiltonian H =

(

vt )

H(S) V H(e)

.

The corresponding transition matrix equals

T(E)

=

AtQD(E)A

where QD (E) stands for the upper left block

QD(E)

=

E

_1-{(8~ -

E(E)

Electron Quantum Transport through a Mesoscopic Device

311

of the resolvent Q(E) = E~H of the extended non-Hermitian Hamiltonian H. The subscript D means" doorway" and marks the states in the ideal cavity, which are directly connected to the scattering channels. In zero approximation V == 0, only these states have complex eigenenergies, [n = en - ~rn. The environment states get excess to the channels only due to the mixing with the doorway resonances exclusively through which they can be excited or decay. The matrix E(E) = vt E~H V accounts for transitions cavity +--7 environment and remains Hermitian (and, correspondingly, the scattering matrix remains unitary) as long as the energy spectrum of the environment is discrete, i.e. the mean level spacing 6 f O. In the single-particle mean field approximation, (quasi)electrons move in the background in a field which is random because of impurities. The dimension N(e) of the Hilbert space of the quasi-particle in the background is much larger than that N(s) of the particle in the mesoscopic cavity, N(e) » N(s). Correspondingly, the single-particle mean level spacing d is much smaller than the doorway mean spacing, d « D, although it is very much larger, d > > > 6, than the many-body level spacing 6. Supposing, therefore, that the coupling matrix elements are random,

(V/Lm> = 0, (V1;m Vvn >=

~r)}-6/Lv6mn' 2 7r

the matrix E reduces after such an averaging to the function

d 1 g(E) = ;Tr E _ H(e)

.

Here the subscripts m, nand /-L, v mark the doorway and the background single-particle states respectively and = 27r (1~12) is the spreading width which characterizes the scale of the fine-structure fragmentation of the doorway states because of the coupling to the background. The loop function g(E) is real so that the V-averaging does not destroy the unitarity. The transition amplitudes reduce to the sums of the doorway resonant contributions

rs

(1)

For the certainty sake, we restrict ourselves for some time to the case of systems with time reversal symmetry. Then the decay amplitudes A~ are real and the matrix of non-Hermitian effective Hamiltonian HCs) is symmetric. The appearing in Eq. (1) amplitudes A~ are the matrix elements of the coupling matrix A = IJrT A with IJr being the orthogonal (IJrTIJr = 1) matrix of the eigenstates of the effective Hamiltonian H(s). Unlike the real matrix elements of the matrix A, those of the matrix A are complex quantities [9]. The exact resonance spectrum {[o:} is found from the equation

312

V. V. Sokolov

Each doorway state is fragmented to ~ rs / d narrow fine-structure resonances. Finally, transition amplitudes can be represented as coherent sums of interfering contributions of the exact resonances fa,

Literally, no dephasing takes place yet. Phases are tuned in such a way that the unitarity condition is fulfilled. However, as distinct from the case of ideal cavity the interference pattern depends now on the two additional parameters: the fine-structure level spacing d and the spreading width rs·

3

A veraging over the fine-structure scale

Let us suppose that the energy resolution l1E is not perfect and do not allow us to resolve the fine structure of the doorway resonances, d « l1E though l1E « D. Then only averaged cross sections

are measured. To carry out the energy averaging explicitly, we neglect the level fluctuations on the fine structure scale and assume the uniform spectrum, E" = JLd (vicket fence approximation). This yields immediately g(E) = cot ("'f) . 3.1

Isolated doorway resonance

In the case of an isolated doorway resonance with the width r = Le r e « D though r » l1E, whieh is situated at the energy E res = 0 the transition cross section equals (Jab (E)

=

IT

ab (E)1 2

= ____ r_ar_b_-,,----_ _ [E - ~ rs cot (7r:) ]2 + ~ r 2

and the energy averaging yields

The phase coherence is destroyed by the averaging and the result is given by a sum of two incoherent contributions. The first one corresponds to a single resonance widened because absorption by the environment. This contribution is obtained with the aid of shifting by the distance ~rs in the upper part

Electron Quantum Transport through a Mesoscopic Device

313

of the complex energy plane. The second term accounts for the particles reinjected from the background. There is no net loss of particles. All of them are, finally, back. The environment looks from outside as a black box which swallows particles and spits them back after some time. The transport through the cavity is characterized by the quantity [3,4]

G(E) =

L aEI,dE2

aab(E) = rITz A(E)

+

rITz

~ = TI2 + TIsTs2

r l + r 2 A(E)

TIs

+ Ts2

(2)

i

where A(E) = E2 + (r + rs)2. The second term in the final expression is expressed in the terms of the subsidiary transitions probabilities

k = 1, 2,

rl + r2 = r

(3)

entailed by an additional fictitious channel with the partial width rs. The result (2) is identical to that of the Biittiker's voltage probe model [5] of dephasing phenomenon with the dephasing rate given by 'Ys = ~ rs = rsTD where TD is the mean delay time in the cavity. One can simulate such a situation by introducing an additional fictitious (M(s) + l)th channel with the transition amplitude A(s) = ,;r; which connects the resonance state to the environment. It is easy to check that the fictitious scattering matrix S(E) = 1- iT(E) built in such a way is unitary. Nevertheless, one should remember that the similarity is not perfect. Indeed, in the case of transition from an individual initial channel b onto a subgroup of the final channels the model cross section equals

'"' (jcb(E) L.cEsub

= 1

+ rsl LCEsnb r 1 + rs l r

c '"' acb(E) > '"' aCb(E). L.- L.cEsub cEsub

The equality takes place only if the summation over the final channels is extended to all of them. The single-particle approximation used above is well justified only when the scattering energy E is close to the Fermi surface in the environment. For higher scattering energies, many-body effects should be taken into account, which result , in particular, in finite lifetime of the quasi-particle. The simplest way to do this is to attribute some imaginary part to quasi-particle's energy, E" = f.Ld - ~re. Strictly speaking, this suggestion destroys the unitarity of the S-matrix in contradiction with what has been suggested above. However, an initially excited single-particle state with the energy E" evolves because of many-body effects quite similar to a quasi-stationary state till the time 27r I O. A particle delayed for such a long time can be considered as irreversibly absorbed. The resonant denominator in the Eq. (1) equals in this case [10]

Dres(E)

1 2 7) i ( 1 + T)2 ) = E - E res - "2rs(1- ~ ) 1 + ~2T)2 +"2 r + rs~ 1 + eT)2

314

V. V. Sokolov

where the following notations have been used: ~

= tanh

( 7rr 2de) '

The total averaged transport cross section G(E) still retains its form (2) but the subsidiary transition probabilities (3) read now as

where

1 1 A(E) 1 + /'£ A(E) rrs

1

The parameter

/'£

(4)

is defined as

with Td = 27r / d being the mean delay time of the particle in the background. The absorption is small if the particle's lifetime Te = 1/ r e in the environment is noticeably larger than the delay time Td . In the opposite case the quasiparticle has enough time to decay and irreversible absorption takes place. The transition probabilities (4) vanish in such a case and our consideration reproduces in this limit the result of Efetov's imaginary potential model [6] with the strength a = fJ rs = ~/'s. Notice that the crossover from the first to the second regime is very sharp because of the exponential dependence on the dimensionless parameter /,e == reTd. 3.2

Overlapping doorway resonances

In the general case of overlapping doorway resonances, the average cross section reads accordingly to the Eq. (1)

ab(E) - "'"'A a *Ab *AaAb (J"

-

~

n'

n'

1 n n V~,(E)Vn(E) .

(5)

nn

A bit tedious calculation yields D~,

------,---'1~ D ;', (E)D n (E)

(E)D n (E )

[1 + -irs

r; _

_

(E~, - En )+t< D;" (E)D n (E)

]

Electron Quantum Transport through a Mesoscopic Device

Let us suppose first that two identities

K,

=

315

O. Then taking into account the following

we can represent the averaged cross section (5) as a sum, (J"ab(E) (J"~b(E), of incoherent flows the first of which,

= (J"d:b(E) +

(6) describes the contribution of the doorway states damped because of the capture by the environment when the second,

accounts for the particles which spend some time tr in the background, repopulate the doorway levels, and finally escape from the cavity via the channel a. With the help of the Bell-Steinberger relation

contribution Gr(E) = La El,dE2 (J"~b(E) of the re-inj ected particles to the 1 -+ 2 transmission can be transformed to

Here U = IJrtlJr is the matrix of non-orthogonality of the overlapping doorway states and the subsidiary amplitudes

implicate the fictitious channel. The quantities r a = U~n IA~12 satisfy the condition La r a = r and can be interpreted as the partial widths. In the case of moderately overlapping doorway resonances the matrix Un'n ;:::; 0n'n and only probabilities 1
316

4

V. V. Sokolov

Ensemble averaging

If the particle motion in the cavity is classically chaotic, which is the case of the main interest, the ensemble averaging should be carried out. It can be shown then that such an averaging fully eliminates the spreading width from the mean cross section as long as the many-body effects in the background are fully neglected. Indeed, the ensemble averaged cross section (6) is expressed in the terms of the S-matrix two-point correlation function Cf;b(c:) =C8 b(c:irs) as

where the subscript V indicates the coupling to the background and the function K8 b(t) is the Fourier transform of the correlation function C8 b(c:). Using the identity

e-iEntr

1

-_-- = - erstr/2 'Dn(E)

1

00

a

27r

dt eiEt

1

00

-00

dE'

e-iE'(tHr) - - 0_: ; - - - -

'Dn(E')

one can convince oneself that

and, finally,

(aab(E)) =

Jooo dte - rst K8 b(t) + rs Jooo dtr Jooo dte- rst K8 b(t + t r ) = Jooo dt K8 b(t) = C8 b(O) == aob(E) .

(7)

Separating in Eq. (5) the part which does not depend of the spreading width after the ensemble averaging we arrive at the conclusion that (aab(E)) = aob(E) + Lla~b(E) where

(8)

In spite of the presence of absorption the cross section (8) can still be expressed in terms of the Fourier transform K8 b (t) of the two-point correlation function C8 b (c: ). To do this we first formally expand the expression in the second line into power series with respect to the parameter K, and then make use of the relations: 1 = (E* -E /k + l)

n'

"

_.i..

roo dt erst (-it)k e-iV~,(E)t eiVn(E)t.

k! Jo

jj~(E)eiVn(E)t =

eiVn(E)t

=

(-ift)k eii),,(E)t; _eiEt ...1..., Joo dE' ,,-iE't 27ft -00 Dn(E') .

,

Electron Quantum Transport through a Mesoscopic Device

317

The following up summation brings us to the result

L1(J"~b(E) = - (2~)2 J~oo dtr Jooo dters(tr+t) x

[e-

2'! at~~t2

froo dEl froo dE2 e iE ,(t r +t2)-E2(tr+it) Co (El o 0

e iE (t I -t2)

E2 - ir )] s

it =t2=1

which after a change of the variables:

and integration over the variables E and

€

yields

(Notice that the form-factors K8 b (t) monotonously decrease with the time t.) Being presented in such a form this expression is equally valid for both the orthogonal (COE) as well as the unitary (CUE) symmetry groups. To simplify subsequent calculation we will consider the case of an appreciably large number M » 1 of statistically equivalent scattering channels all with the maximal transmission coefficient T = 1. Then the channel indices a, b can be dropped and the characteristic decay time of the function K(t) tw = 1/ rw = tH / M is much shorter then the Heisenberg time tH = 27r / D (here rw = E,M is the so called Weisskopf width). It is convenient to represent the function K 0 (t), which is real, positive definite and satisfies the conditions Ko(t < 0) = 0, Ko(O) = 1 in the form of the mean-weighted decay exponent [11]

Ko(t) =

1

00

dr e- rt w(r) .

Rigorously, the weight functions w(r) are different in the intervals t < tH and t > tH' However contribution of the latter domain vanishes as e- M [11] when the number of channel grows. Neglecting such a contribution we obtain

If the parameter and reduces to

K

»

(A~R:)2 the found expression ceases to depend on it

318

V. V. Sokolov

Independently of the symmetry class considered, this contribution perfectly compensates the second term in the r.h.s. of the Eq. (7). The resulting mean cross section is identical to that of the Efetov's imaginary-potential model

[6]. On the contrary, in the limit of weak absorbtion '" which reads as

«

(~~;::J2 the result (10)

strongly depends on presence or absence of the time-reversal symmetry. In the first case the well known asymptotic expansion [2] of the two-point correlation function yields in the considered case of perfect coupling to the continuum [11] W(GOE)(r)

= 5(r - rw) - ~5'(r - rw) + ~ 5//(r - rw) + ... tH

2tH

(11)

whereas in the second case W(GUE)(r)

= 5(r -

rw)

+ ...

(12)

where contributions of the omitted terms in (8) are O(1 /(M- 7 / 2 )) . Within this accuracy we obtain for the transport functions (2): C(GOE) (E) =

M!%[2

[ (1-~)

C(GUE)(E) =

M!%I2

-

(1 _

~

(1-

~~)J,

~).

The difference i1C(E)

= cC GU E) (E) _

C (GOE) (E) = JvhM2 M2

(1 _~! 8

"'''is) 4M

(13)

is referred to as the weak localization whereas suppression of this term implies dephasing. We conclude, therefore, that the decay of quasi-particles in the environment not only induces reduction of the total outgoing flow but accounts also for the dephasing effect. The regime of weak a bsorption considered above is restricted to the range o < '" ~ (r~~;::)2 which is very narrow if one out of the two widths r s , rw appreciably exceeds another. Indeed , if for example F., » rw and therefore '" « 4rw / rs « 1 the probability to penetrate into the environment prevails and t he particle can finally escape from the cavity through an open channel only if the decay width of quasi-particle in the environment is small enough, "ie « 4"is/M . Otherwise absorption dominates. On the contrary, if rw » rs and therefore", « 4rs/ rw = 4"is/ M the penetration probability is small and particles mostly leave the cavity before penetrating. In the both cases the parameter ~ « and the influence of the environment in Eq. (13)

k

Electron Quantum Transport through a Mesoscopic Device

319

:s

is negligible. The discussed range becomes maximal, 0 < "" 1, when In reality, the spreading width rs which describes relatively weak influence of the environment is expected to be smaller than the characteristic escape width if the number of open channels M » 1. But the adduced arguments show that the role of the considered mechanism of suppression of the weak localization increases in the case of small number of open channels, which is the most interesting one from the practical point of view. It should be stressed that, rigorously, the asymptotic expansions (11, 12) for the weight functions w(r) are not justified [11] in the case of few number of channels. However, at least qualitatively, our arguments remain valid.

rs : : :; rw·

rw

References 1.A.G. Huibers, S.R. Patel, C.M. Marcus, P.W. Brouwer, S.L Durui:iz, and J.S. Harris (Jr.). Phys. Rev. Lett., 81 1917 (1998). 2.J.J.M. Verbaarschot, H.A. Weidenmiiller, and M.R. Zirnbauer, Phys. Rep. 129 367 (1985). 3.C.W.J. Beenakker, Rev. Mod. Phys. 69 731 (1997). 4.Y. Alhassid, Rev. Mod. Phys. 72895 (2000). 5.M. Biittiker, Phys. Rev. B 33 3020 (1986). 6.K.B. Efetov, Phys. Rev. Lett. 742299 (1995). 7.E. McCann and LC. Lerner, J. Phys. Condo Matter 8 6719 (1996). 8.P.W. Brouwer and C.W.J. Beenakker, Phys. Rev. B 55 4695 (1997). 9.V.V. Sokolov and V.G. Zelevinsky, Nuc!. Phys. A 504 562 (1989). 1O.Valentin V. Sokolov and Vladimir Zelevinsky, Phys. Rev. C 56 311 (1997). 11.Valentin V. Sokolov, AlP Conf. Proc. 995 85 (2008).

320

Composing Chaotic Music from a Varying Second Order Recurrence Equation Anastasios D. Sotiropoulos and Vaggelis D. Sotiropoulos School of Music, Composition-Theory Division University of Illinois at Urbana-Champaign, Urbana Illinois 61801, USA (e-mail: sotirop [email protected], [email protected]) Abstract: Music is composed algorithmically from a varying second order recurrence equation which defines sound frequency, x, at each step n knowing sound frequency and amplitude, y, of the previous step, n-l, as: Xn = C - aXn_l 2 + bYn-l where the amplitude is given as Yn-l= f (x n_], Xn-2) with f an arbitrary function, and a, b, and c are also arbitrary constants to be chosen by the composer. This compositional model includes as special cases well-known recurrence equations as the logistic map, the Henon map, and the delayed Julia set. Another feature of our model is that the sound amplitude does not necessarily have the same dependence on previous frequencies in each step. In fact, by the use of discrete form boxcar functions it can change either in consecutive steps or after a finite number of steps. Compositional patterns are also incorporated by use of specific equations for the change of frequency. The effect of initial frequencies chosen is examined as well. Numerical examples are presented to show the chaotic behavior of our model. A final score is presented in musical notation .

1. Introduction Music composition is a set of notes each of which is described by its loudness (amplitude) and frequency. Any mathematical formula that relates the amplitude and frequency of a note to the amplitude and frequency of the previous note(s) and applied repeatedly constitutes an algorithm and results in a complete music composition. When the mathematical formula that relates the new sound to the previous sound is non-linear, the resulting composition can exhibit the phenomena of bifurcation and chaos. Previous works in the literature have dealt with musical algorithms. In the 50's, Iannis Xenakis was the pioneer in introducing mathematics and patterns in music composition which is described in his book, Xenakis [1]. Other recent books such as the ones by Taube [2] and Karlheinz [3] exemplify the wide use of algorithmic music composition of which mathematical based models include, among many others, the magic squares by Browning [4] and manifolds by Tipei [5]. The aesthetics of algorithmic compositions was addressed by Garnett [6].

Composing Chaotic Music from Varying Second Order Recurrence Equation

321

The phenomena of bifurcation and chaos in music composition have been examined in the literature mostly for well known recurrence equations as the logistic map or the Henon map. A comprehensive description of the use of the logistic equ~tion in composing music is posted in the world wide web by Elaine Walker [7]. In a very recent paper of ours [8], another recurrence relation that leads to chaos was proposed based on a frequency amplitude triangle, where the frequency of a note is given as the square root of the side of a triangle whose other two sides are given respectively by the frequency and sound of the previous note. In the present paper, we propose yet a new compositional model based on a varying second order recurrence relation that leads to chaos.

2. The model The compositional model we propose is based on a varying second order recurrence equation which defines sound frequency, x, at each step n knowing sound frequency and amplitude, y, of the previous step, n-l,as:

where f is an arbitrary function and a, b, and c are also arbitrary constants to be chosen by the composer. We note that the second order effect comes from the dependence of the sound amplitude on the previous sound frequency. For given a, band c, the variations in the equation come from choosing, in general, different functions f at different steps. The composer needs to also choose the initial two frequencies. It is convenient from the computational point of view to use small numbers for x that are used in the mathematical literature for recurrence equations and do not correspond to actual sound frequencies. The mathematical frequencies Xn are then converted to standard MIDI (Musical Instrument Digital Interface) note numbers ranging from 0 to 127 by using the following equation midikey = floor (127(2+x)/4 +0.5)

(2)

where midikey is the MIDI note number and floor(z) is the largest integer not greater than z. We note that equation (3) is valid for x between -2 to +2. If the largest value of x that comes out of the recurrence equation (1) is greater than 2 then, in equation (3), x is adjusted accordingly to yield note numbers between 0 and 127. The

322

A. D. Sotiropoulos and V. D. Sotiropoulos

actual frequency, f, of notes in an A440 equal temperament is given in terms of MIDI note numbers by the fonnuia

f I (440 Hz) = imidikey-69)!l2

(3)

3. Analysis, numerical results, and discussion The reCUlTence equation proposed given by equation 0), includes as special cases well known classical recurrence relations that lead to chaos. If c=l and Yo-I= XII-I. then equation (l) reduces to the well known Henon equation. If a = 0, b =1 and Yn-[= XII}, then equation (1) becomes the delayed Julia set, whose chaotic features have been discussed in a recent book by Skiadas and Skiadas [9]. We will now consider in detail a special case of the recurrence equation (1), defined by Xn+l

=

I I Xn-I

A-

l'

(4)

where A and r are arbitrary constants. We note that when A =2, equation (4) results in a delayed real domain Julia set which for 1'=2, is well known that it yields chaos. The choice of the initial two frequencies has a pronounced effect on the composition. If X2=Xj, then the frequencies of the composition repeat once. If X2:f:Xj, then there is no frequency repetition. When X2 = XI f, the resulting frequencies repeat once as welL For A =2, the resulting frequencies for fOlty steps when the two initial frequencies are equal were computed and are now presented graphically:

I I" -

2.5 2 1.5 1

0.5 frequency 0 (xn) -0.5 -1

-1.5 -2

-2.5

Composing Chaotic Musicfrom Valying Second Order Recurrence Equation

323

The frequency repetition is apparent. When 1'=1 a period doubling is observed. When r=2, as expected, there is chaos. For comparison, the same cases are obtained and are shown here when the initial two frequencies are not equal:

2.5 2 1.5 1

0.5 !frequency 0 (x n) -0.5 -1 -1.5 -2

-2.5

Thc frequency repetition has disappearcd with the other features remaining the same as before. Next, the effect of the degree of non-linernity is shown graphically with the two initial frequencies not being equal. First, the case of Ie =5/4 is presented:

0.5

o -0.5 ifrequency {Xn}

-1

-1.5 -2 step (n)

It is noted that there is a =3/2:

4th

period doubling. Following is the case of Ie

324

A. D. Sotimpoulos and ED. Sotiropoulos

1

0.5

0 (:xn)

-0.5 ·1

-1.5 -2

A chaotic behavior may be observed. Compositional pattems or, otherwise, called fillers may be obtained by choosing the appropliate recurrence equation. When Xn+l == Xu 2 with Xn < 2, it can be easily shown that for Xl <0, the resulting frequencies are negative altemating frequencies, i.e .. Xn+2 == Xl! and Xn+3 == Xn+l. For Xj>O, the third frequency is mirrored reflected of the first ( X3 = -xd, whereas from the second frequency onwards the frequencies are negative alternating frequencies (Xn+3 == Xn+l , X2n+3 == X2n+l)' It is remarked here that mirror rel1ection is equivalent to midikey note inversion. That is, x' s as -3, -2, -1 that are mirror reflected about 0 to 3, 2, 1 respectively, as midekey notes these x's are reflected about 0 from 125 , 126, 127 to 1, 2,3 respectively. Another example of a composition fiIler is given by the recurrence relation Xn+l == 1 - (2xnr 1 which yields Xn+4 == Xn. i.e., four notes repeating. As a final example, a varying reCUlTence equation model is used to demonstrate the effect of chaos. For five consecutive steps, we use the relation Xn+! ;;:;; I X n-l 1 312 - 2. Then, for the next two consecutive steps we use )(n+1 ;;:;; Xn-l 2. This model results in the seventh note being equal to the fifth note when the fifth note is negati ve, and in the seventh note being the mirror ret1ection of the fifth note when the fifth note is positive. Here, this result is shown graphically where, for comparison, is also shown the delayed Julia set recurrence discussed above. The two resulting sets of frequencies are distinctly different, however with both being chaotic. The varying model exhibits a chaotic behavior with an imbedded pattern created by the filler, as discussed above.

I 1-

I I

I I-

Composing Chaotic Music/rom Varying Second Order Recurrence Equation

325

frequency (xn) 0

-1

-2

The mathematical frequencies obtained in the previous example for the varying recurrence model, were converted to actual notes frequencies using equations (2) and (3). Then, these frequencies were entered in the FINALE computer software program that produced the actual sounds for the musical score, which was played on the 4th of June 2009 for the audience of the Chaos 2009 international conference in Chania, Greece, when our paper was presented. In music notation, the composition reads as: AJJouo •• -- _••••

-r; ;r I:: : I::~ I ~; : I: ..8 4. . . -- -- --

.

- -- -- - - - - - -

-- -- --- -- -- - --- -_..

. . J ...

.

.

@

.

~

326

A. D. Sotiropoulos and V. D. Sotiropoulos m __________ ------------------------~~- -,

.!..

. ------- ---- ----- ------ -- -- -------- ---"

4:ii¥!J'"

• /'; -

.

1-,.

____

--~

~

...

rit: ----- --~; --- --- ----- ~ --- - ------ ------,

References [1] 1. Xenakis. Formalized music: thought and mathematics in composition (Harmonologia Series No.6). Hillsdale, NY: Pendragon Press. 2001. f2] H. K Taube. Notes from the metalevel: an introduction to algorithmic music composition. Studies in New Music Research, Routledge, N. York, 2004. [3] Karlheinz Essl: Algorithmic composlllon. in: Cambridge Companion to Electronic Music, cd. by N. Collins and J. d' Escrivan, Cambridge University Press 2007. [4] Z. D. Browning. B01~jaxed. Capstone Records, N. York, 1999. [5) S. Tepei. Manifold compositions: formal control, intuition, and the case for comprehensive software. Proceedings 2005 International Computer Music Cmfference, Copenhagen, Denmark, 2007. [6] G. E. Garnett. The aesthetics of interactive computer music. Computer Music Journal, 25:21-33, 200l. [7] E. Walker. Chaos melody theory. www.ziaspace.com/eiaine!ChaosMeiodyTheory.pdj [8] V. D. Sotiropoulos and A. D. Sotiropoulos: Music composltlon from the cosine law of a frequency-amplitude triangle. in: Chaotic Systems: Theory and Applications, ed. by C. H. Skiadas and 1. Dimotikalis, World Scientific, 2009. 191 c. H. Skiadas and C. Skiadas. Chaotic Modelling and Simulation. CRC Press, N. York, 2009.

327

Music Composition from the Cosine Law of a Frequency-Amplitude Triangle Vaggelis D. Sotiropoulos and Anastasios D. Sotiropoulos School of Music, Composition-Theory Division University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA (e-mail: [email protected], [email protected]) Abstract: We propose a frequency-amplitude triangle whose cosine law is used as a recurrence equation to algorithmically compose music. The triangle is defined by two sides and the angle between them; one of the sides is the sound frequency, x, at a step, the other side is the sound amplitude, y, at the same step, while the angle between them is a free parameter to be chosen by the composer. The resulting third side of the triangle is defined as the square root of the sound frequency, x, of the next step. The sound amplitude, y, at each step depends, in general, on the amplitude and frequency of the same or previous step(s) in any way the composer chooses. To test our proposition for bifurcation and chaos against known solutions, we consider in detail the case where the amplitude depends only on the frequency of the same or some previous step as Yn = b(x" ..,)' , with b being a free parameter, m either zero or a positive integer, and 'A a real number greater than or equal to zero. If 'A is taken as zero, the resulting equation after a change of variables becomes the one parameter logistic recurrence equation. From the known range of the parameter for bifurcation and chaos, we obtain the range of values for sound amplitude and angle 8 for periodic and chaotic music. For specific values, the frequencies corresponding to musical notes are obtained from the proposed triangular recurrence equation and musical chaotic scores are composed and presented. Other 'A's are also considered and their effect on the periodicity of the frequencies is examined.

1. Introduction In the present paper we are concerned with algorithmic music composition. Each note in a music composition is described by its loudness (amplitude) and frequency. Any mathematical formula that relates the amplitude and frequency of a note to the amplitude and frequency of the previous note(s) and applied repeatedly constitutes an algorithm and results in a complete music composition. When the mathematical formula that relates the new sound to the previous sound is non-linear, the resulting composition can exhibit the phenomena of bifurcation and chaos. The principles of algorithmic composition have been used since medieval times. An example of a musical algorithm is the round, of which the earliest traced is Sumer Is Icumen In (1260). In the round, singers are instructed to sing the same melody line, but starting at set delays with respect to each other. In the 20 th century with the appearance of computers, algorithmic music composition blossomed. In the 50's, Iannis Xenakis was the pioneer in introducing mathematics and patterns in music composition which is described in his book, Xenakis [1]. Other recent books such as the ones by Taube [2] and Karlheinz [3] exemplify the wide use of algorithmic music composition of which mathematical based models include, among many others, the magic squares by Browning [4] and manifolds by Tipei [5]. Moreover, the aesthetics of algorithmic compositions was addressed by Garnett [6].

328

V. D. Sotiropoulos and A. D. Sotiropoulos

Particular attention is focused in the present paper in the phenomena of bifurcation and chaos in music compositions. Many musical works, when dealing with chaotic music, have been using the classical mathematical recurrence models of the logistic equation or the Henon equation. A comprehensive description of the use of logistic equation in composing music is posted in the world wide web by Elaine Walker [7]. Our work departs from this approach in that it proposes a new mathematical model to algorithmically compose music. This model is based on geometry and, specifically, on a triangle which is formed by its two sides and the angle between them; the two sides representing the sound amplitude and frequency separated by an arbitrary angle between them. The frequency of the next sound is represented by the square root of the third side of the triangle. Applied repeatedly, this model defines a recurrence equation whose solutions result in a complete music composition.

2. The model We propose a frequency-amplitude triangle whose cosine law is used as a recurrence equation to algorithmically compose music. The triangle is defined by two sides and the angle between them; one of the sides is the sound frequency, x, at a step (note), the other side is the sound amplitude, y, at the same step, while the angle between them is a free parameter to be chosen by the composer. The resulting third side of the triangle is defined as the square root of the sound frequency, x, of the next step (note). This frequency, using the cosine law, is given in terms of the frequency and amplitude of the preceding step (note) as: xn+ 1 = Xn 2 + Yn2 - 2xny ncos8

(I)

The sound amplitude, y, at each step depends, in general, on the amplitude and frequency of the same or previous step(s) in any way the composer chooses, that is, in any functional manner: Yn = f (xn, Xn-l , ... , xn-m) (2) where m=0,1,2,3, ... , n=m+l,m+2, ... Once the composer chooses the initial frequency or frequencies, the recun-ence equation (I) yields all the sound frequencies Xn and, thus, the complete music composition since the sound amplitudes Yn are given by equation (2). It is convenient from the computational point of view to use small numbers for x that are used in the mathematical literature for recurrence equations and do not correspond to actual sound frequencies. The mathematical frequencies Xn are then converted to standard MIDI (Musical Instrument Digital Interface) note numbers ranging from 0 to 127 by using the following equation midikey = floor (127(1 +x)/2 +0.5)

(3)

where midi key is the MIDI note number and floor(z) is the largest integer not greater than z. We note that equation (3) is valid for x between -1 to +1. If the largest value of x that comes out of the recurrence equation (1) is greater than 1 then, in equation (3), x is adjusted accordingly to yield note numbers between 0

Music Composition from Cosine Law of Frequency-Amplitude Triangle

329

and 127. For example, for 0:Sx:S4, l+x in equation (3) is substituted by xl2. The actual frequency, f, of notes in an A440 equal temperament is given in terms of MIDI note numbers by the formula f I (440 Hz) = imidikey -

(4)

69)1l2

3. Analysis First, we refer to two special cases of our frequency - amplitude triangle composition model given by the recurrence equation (1): i)

(Sa, b)

in which the amplitude of a note being linearly proportional to the frequency results in frequencies that fall on a single curve. The resulting frequencies, only after a few steps, either go to zero or infinity depending on the values of b, e, and the initial frequency. For cose 0:: bl2 which is satisfied if b :S 2, the resulting frequencies, only after a few steps, go to zero if the initial frequency is smaller than one. ii)

(6a, b, c)

Yn = 1 , cose = 0.5 ,

which, for initial frequency equal to 2, is Sylvester's sequence with resulting frequencies that are equal to 2, 3, 7, 43, 1807, 3263443, ... To test our proposed composition model for bifurcation and chaos, we consider in detail the case:

Yn

= b,

(7a, b)

that is, the case where note loudness remains constant throughout the composition. With the following change of parameters and variables r = I ±,j g(b, cose), g(b, cose) = 4b 2(cos 2e -1) + 4bcose +1

a = -I Ir,

c =0.5+bcose/r

(8a, b) (9a, b) (10)

z=ax+c equation (7b) becomes

(11 )

which is the well known logistic recurrence equation whose resulting solutions are of meaning for r 0:: 1. This, together with equations (8a, b), yields g(b, cose) = 4b 2( cos 2 e -1) + 4bcose + 1 0:: 0

(12)

which is satisfied for all e's such that cose 0:: 1- 1/(2b) ,

bo:: 0.5

(13a, b)

330

V. D. Sotiropoulos and A. D. Sotiropoulos

where (l3b) is the requirement for acute angles. Therefore, our proposed triangular recurrence model defined by equation (7b) yields a composition for any angle 8 when the loudness b is smaller than 0.5 and for angles satisfying (13a) for all other b's. Having knowledge from the classical logistic equation (11) of the values of r which yield no bifurcation or multiple bifurcations means that we can know a priori the composition's features as the loudness b and the angle 8 are related to r by the equation (14) If we choose an angle 8 having a value satisfying (13a) for a chosen b equal to

or greater than 0.5 and arbitrary for all other b' s, equation (14) will tell us which r we are picking in the classical logistic map and, thus, what will be the features of our composition in terms of bifurcation and chaos. However, there are infinite pairs of values (b, cos8) which if substituted in equation (14) yield the same r. Therefore, it is important to examine equation (14) from the point of view of determining the characteristics of the values of band cos8 for a given rand, thus, a given roo Solving equation (14) for cos8 yields (15) which is to be taken together with the conditions b 2:

.y(l- r02)/2,

ro:S 1

(16)

and (17) in order to have acute and real angles respectively. Equation (15) readily yields the angle 8n for which the nth period doubling (bifurcation) occurs as where (I8b) are known from the classical logistic map. It is understood that the loudness b satisfies condition (17). Another feature to point out is the existence of a minimum cos8 for a given ro> 1 which is obtained from the zero slope of cos8 of equation (15) as (19)

which occurs at (20) When the amplitude b is smaller than this value, cos8 approaches one and, in fact, becomes one at the value of b given by the equation in (17). For b larger than the value defined in (20), cos8 approaches again one in a fashion described by (l3a).

Music Composition from Cosine Law of Frequency-Amplitude Triangle

331

Therefore, to compose music using the specific triangular recurrence equation (7b) with a priori knowledge of its bifurcation and chaos features, we need to choose the value of ro, then select the sound amplitude b to satisfy (17) for most cases of interest, next calculate the triangular model's angle 8 using (15), and last compute all the frequencies Xn of the composition from iterating equation (7b). The proposed general triangular composition model as expressed by the recurrence equation (1) will result in frequencies Xn whose features in terms of its periodicity will depend on the chosen functional dependence of the amplitude on frequency or frequencies expressed by equation (2). For example, for a dependence on the frequency of the same step (note) to a power deviating slightly from zero, there is a dramatic change on the periodicity of the resulting frequencies. Let's take, as an example, Yn=2x n1/8 to represent equation (2). Then, on changing to a new mathematical frequency z=x7/8 12, equation (1) for the case when the angle of the triangular model is zero becomes simply (21) From the form of this recurrence equation it is expected, on comparison with the classical logistic recurrence equation as was done for other recurrence equations by Skiadas and Skiadas[8], that the resulting frequencies will be chaotic, as indeed they are. This will be demonstrated numerically in the following section. However, when the angle of the triangular model is non-zero, the frequencies do need be chaotic as is the case when cos8=O.95 for which the frequencies exhibit a one-period doubling. This will be demonstrated numerically in the section that follows. We examined other fractional power models for the functional dependence of the amplitude on frequency as well. We found that the frequency tends to a single value for a range of powers smaller or equal to one independent of the triangle's angle, in agreement with the result obtained analytically in case i above where the power is equal to one.

4. Numerical Results and Chaotic Compositions The parameter range affecting the presence of single or multiple periodicity of the composition's frequencies for the constant amplitude triangular model was analytically examined in detail in the previous section. Here, this parameter range is depicted graphically. In Figures 1, 2 the cosine of angle 8, the angle between the amplitude and frequency for each note defining the triangular compositional model is plotted versus the amplitude's magnitude b for ro defined by (14) smaller or equal to one, and greater than one respectively. Figures 1,2 represent graphically equation (15).

332

V. D. Sotiropou/os and A. D. Sotiropoulos

0.8

I leose 0.4 0.2

o

Fig.I. The dependence of the triangle's angie 0 on the sound amplitude for the eOllstunt amplitude model.

For fo smaller or equal to one, Figure 1 shows that cosO, which is a parameter chosen by the composer, may vary between zero and one but is enveloped above and helow by the curves obtained by setting 1'0 equal to one and zero respectively, if the triangular compositional model is to produce a composition. The lower curve of the envelope is given by the equation in (13a) whereas the lowest possible amplitude b for each 1'0 is given by the equation in (16). Since 1'0= r-1 ,"vith r the classical logistic reClU'fenCe equation parameter, Figure 1 shows the envelope of the range of values of cose and of the amplitude b for which the resulting composition will approach towards its end one single note with no periodicity at aU.

2.5

sound amplitude (b)

Fig.2. The dependence of the triangle' s angle (l on the sound amplitude for the constant amplitl!de model.

In Figure 2, in contrast to Figure 1, fo is greater than one. For each such value of 1'", there is a minimum value of cose given by (19), and a minimum value of the amplitude b given by (17). Independent of 1'0 and for large b's, cose obeys the equation in (l3a), which is drawn as the lowest curve in the Figure, as it approaches one, The values of r" chosen in the Figure, define from lower to higher the first, second, and infinite period (chaos) doubling according to

Music Compositionjinm Cosine Law of Frequency-Amplitude Triangle

333

equation (18a). If, for example, the composer chooses the values of cosO and of the amplitude b to lie in the region between the curves of the two lower values of r", then the composition will approach two notes which will repeat themselves until the end of the composition. The effect of changing the angle 6 on the frequencies that result from iterating the recurrence equation (7b), is demonstrated in Figure 3 where the frequencies x are plotted for the first 100 steps (notes). The amplitude chosen is equal to 2 whereas the cosS is equal to 1.0 and 0.95 respectively for the two curves (compositions). For both cases the initial frequency was taken as 0.1.

4

3 ifrequency (Xn) 2 !

1

o o

10

20

30

40

50

60

70

step (n) Fig.3. Variation of souud frequency for the first 100 steps for a constant amplitude model.

It is seen that tllis slight change of cosS has a dramatic effect on the resulting composition. However. both compositions are chaotic since the chosen values of cose and b for both cases result. from equation (14), in values of r equal to 9 and 7.04 respectively, that define chaotic regimes in the classical logistic bifurcation diagram. After the notes' actual frequencies are computed using consecutively equations (3) and (4) with l+x substituted by x/2 since the mathematical frequencies vary from () to 4, the notes themselves are entered in the FINALE computer software program that produces the actual sounds for the composition. The notes of the two compositions which we made follow respectively the logistic equation with r=4 and our tliangular model with b::::2 and cose=O.95. In both compositions the loudness (ampHtude) was taken to be forte for all notes while the rhythm was our choice. The two compositions respectively are:

334

V. D. Sotiropoutos and A. D. Sotiropoulos

ScOTe

lr. : .~ I· .... '\~t~"~~~~~~, (~~~~-; =:[~=:~~~~

Music Composition.timn Cosine La,!' of Frequency-Amplitude Triangle

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Tt is remarked that our notation is non-conventional and is not intended for anyone else other than the computer to play it. Both compositions were played on June 4, 2009 for the audience of the Chaos 2009 international conference in Chania, Greece during the presentation of our paper, A different amplitude model and, thus, a different recurrence equation is chosen next to demonstrate graphically its effect on the resulting frequencies of the composition. The amplitude of each note (step) is taken to depend on the frequency of the same note raised 10 the power of 118, with its magnitude being equal to 2 when the mathematical frequency is equal to 1. Then, for a zero triangular angle the recurrence equation is given by equation (21), the iterations of which yield chaotic hequencics as may be seen in Figure 4 where the frequencies x axe plotted for the first 100 notes (steps) having picked an initial frequency value equal to O. J _

336

V. D. Sotiropoulos and A. D. Sotiropoulos

o

10

20

30

40

50 60 step (n)

70

80

90

Fig.4. Variation of sound frequency for the first 100 steps for a varying amplitude model.

It is observed that the changing of the angle to an angle whose cosine is equal to 0.95 dramatically affects the frequency characteristics and, in fact, it introduces a one-period doubling as is clearly seen in the second curve of Figure 4, in contrast to the other chaotic curve of zero triangle angle. 5. Conclusions We proposed a compositional algorithmic model for music based on a triangle fonned by two sides and the angle between them, one of the sides being the frequency of a note (step) and the other side the amplitude of the same note. The composer needs to choose the angle and the amplitude dependence on frequency of the same note or a combination of frequencies of the same and previous notes in a functional form. The resulting third side is defined as the square root of the frequency of the next note. It was demonstrated analytically and numerically that the model, depending on the composer' s choices, can lead to notes with no periodicity, mUltiple periodicity or chaotic behavior. Detailed analytical results were obtained and discussed for the case of constant note amplitude which was transformed to the classical logistic map. Numerical results and music compositions were presented for different choices of the amplitude dependence on frequency. Future research is encouraged on our proposed gcneral triangular composition model based on the results of the present paper. References [1] 1. Xenakis. Formalized music: thought and mathematics in composltwll (Harmonologia Series No.6). Hillsdale, NY: Pendragon Press .. 2001. [2] H. K. Taube. Notes from the metalevel: an introduction to algorithmic music composition. Studies in New Music Research, Routledge, N. York, 2004.

Music Compositionjrom Cosine Law oj Frequency-Amplitude Triangle

337

[3] Karlheinz Essl: Algorithmic composition. in: Cambridge Companion to Electronic Music, ed. by N. Collins and J. d'Escrivan, Cambridge University Press 2007. [4] Z. D. Browning. Banjaxed. Capstone Records, N. York, 1999. [5] S. Tepei. Manifold compositions: formal control, intuition, and the case for comprehensive software. Proceedings 2005 International Computer Music Conference, Copenhagen, Denmark, 2007. [6] G. E. Garnett. The aesthetics of interactive computer music. Computer Music Journal, 25:21-33,2001. [7] E. Walker. Chaos melody theory. www.ziaspace.com/eiaine!ChaosMelodyTheory.pdf [8] C. H. Skiadas and C. Skiadas. Chaotic Modelling and Simulation. CRC Press, N. York, 2009.

338

Chaos in a Fractional-Order Jerk Model Using Tanh Nonlinearity Banlue Srisuchinwong Sirindhorn International Institute of Technology, Thammasat University Pathum-Thani, 12000, Thailand (e-mail: [email protected]) Abstract: Chaos in a fractional-order jerk model using hyperbolic tangent nonlinearity is presented. A fractional integrator in the model is approximated by linear transfer function approximation in the frequency domain. Resulting chaotic attractors are demonstrated with the system orders as low as 2.1. Keywords: chaos, fractional order, jerk model, hyperbolic tangent nonlinearity.

1. Introduction Since the discovery of the famous Lorenz chaotic attractor in 1963, chaos has been extensively studied and found ubiquitous applications in scientific, engineering and mathematical communities, for example, in astronomy [1], galaxies [2], quantum chaos [3], circuits and systems [4], chaos-based communications [5-9], medical science [10-12] and chaotic oscillators [13IS]. An integer-order chaotic system, according to the Poincare-Bendixon theorm [19], must have a minimum order of 3 for chaos to appear. For example, the well-known Chua's circuit is based on three first-order ordinary differential equations (ODEs), whilst Sprott [14] has alternatively proposed chaotic oscillators based on a single third-order ODE called a "jerk" (time derivative of acceleration) function of the form [14-17] : d 3 x +A d 2 x + dx =G(x) dt 3 dt 2 dt

(1)

or

(2)

where 'x', 'v' and 'a' may represent position, velocity and acceleration of an object, respectively, and G(x) is a nonlinear component such as GJ

j

X)=IXI-2

G(x)= G2 (x)=1.2s-4.5sgn(x) G3 (x)=-1.2x+2sgn(x) G4 (x)=-x+2tanh(x)

;A=0.6

[14])

;A=0.6 ;A=0.6

[14] [14]

(3)

;A=0.19 [20]

Recently, fractional-order chaotic systems of orders less than 3 have been demonstrated based on fractional calculus [21,22]. For example, fractional-

Chaos in a Fractional-Order Jerk Model Using Tanh Nonlinearity

339

order Chua's circuit of order as low as 2.7 can produce a chaotic attractor [23], whilst chaos in a fractional-order jerk model [24] of orders as low as 2.1 has been demonstrated. The fractional-order jerk model is of the form: dv d"'x -G( ) -da + A -+--x dt

dt

(4)

dt'"

d"'x

where

-

-=v

dt'"

(5)

and m is a fractional number (0 < m ~ I). Equation (5) can be integrated using a fractional integrator in the frequency domain. Although chaos in the fractional-order jerk model (4) has been illustrated using the similar nonlinear components GJ(x), G2(x) and G3(x) of (3), chaos in the fractionalorder jerk model (4) has never been demonstrated using the nonlinear component Gix) = -x + 2tanh(x) and A = 0.19 of(3). In this paper, chaos in a fractional-order jerk model using the hyperbolic tangent nonlinearity G(x) = -x+2tanh(x) is presented. The fractional integrator in the model is approximated by linear transfer function approximation in the frequency domain. Resulting chaotic attractors are demonstrated with the system orders as low as 2.1.

2. A Fractional Integrator In the frequency domain (s = jro), a fractional integrator of order 'm' can be represented by a transfer function [22]: I

(6)

F(s) = sl1l

where the magnitude in the Bode plots is characterized by a slope of -20m dB/decade. The transfer function (6) may be approximated by pole-zero pairs (Pi, Zi) of the form [22-24],

I

nN-I( I+~z, ) n ,~O

I

s)'"

-;;; = ( 1+-

PT

N

i=O

(

N-I

n(s+z,) G '-0 o N

1+ - s. ) PI

(7)

n( s+ p,) i=O

where PT is the corner frequency, Go is a constant and N is given by I N = 1+ Integer [ og

(wmaX)J

Po

log(ab)

(8)

340

B. Srisuchinwong

(9) (10)

b=

10(yIlOm)

(11)

where y is the discrepancy (dB) between the actual and the approximated lines, and IDmax is the frequency bandwidth of the system. Such pole-zero pairs are found as -

Zo - Po Zi

10[yIlO(l-m)]

= apo(ab)i

Pi = Po (ab/

(12) (13) (14)

Table I summarizes a list of integer-order approximation of fractional integrators of order 'm' using y = 2 dB, ~nax = 100, PT = 0.01. Table!. List of integer order approximation of fractional integers of order 'm' using y = 2 dB, IDmax = 100, PT = 0.01, Y = 2 dB. _'" 1584.8932(s+0.1668)(s+27.83) SOl (s + O.l)(s + 16.68)(s + 2783) _

79.4328(s + 0.05623)(s + I)(s + 17.78)

?2 = (s +0.03162)(s +0.5623)(s + lO)(s + 177.8) 1 _ 39.8107(s +0.0416)(s +0.3728)(s + 3.34)(s + 29.94) = (s + 0.02154)(05 + 0.1931)(s + 1.73)(s + 15.51)(s + 138.9)

S03

1 _

35.4813(s + 0.03831)(s + 0.261)(s + 1.778)(05 + 12.12)(s + 82.54)

~ = (05 +0.01778)(s +0.1212)(s + 0.8254)(s + 5.623)(s + 38.31)(s + 261) _

15.8489(s + 0.03981)(05 + 0.2512)(s + 1.585)(s + 63.1)

?s = (s + 0.01585)(s + 0.1)(05 + 0.631)(s + 3.981)(s + 25.12)(s + 158.5) I _ 1O.7978(s +0.04642)(s +0.3162)(s + 2.154)(s + 14.68)(s + 100) = (s + 0.0\468)(05 + O.I)(s + 0.6813)(s + 4.642)(s + 31.62)(s + 215.4)

S06

I _

?

9.3633(s +0.06449)(s +0.578)(s + 5.179)(s + 46.42)(s + 416)

= (s + 0.01389)(s +0.1245)(s + 1.116)(s + lO)(s + 89.62)(s + 803.1)

1 _

5.3088(s+0.1334)(s+2.371)(s+42.17)(s+749.9)

?s = (s +0.01334)(05 + 0.2371)(s + 4.217)(s + 74.99)(s + 1334) I _

2.2675(s + 1.292)(s + 215.4)

~ = (s + 0.01292)(s + 2.154)(s + 359.4)

Chaos in a Fractional-Order Jerk Model Using Tanh Nonlinearity

341

3. Fractional-Order Jerk Model using Tanh Nonlinearity Figure 1 shows a structural implementation of the fractional-order jerk model described in (4) and (S) using G(x) = -x + 2tanh(x) and K = A = 0.19. For simplicity, let X = position 'x', Y = velocity 'v' and Z = acceleration 'a' of the jerk model. In Fig. 1, the pole-zero block diagram represents the fractional integrator described in (7) and may be approximated by the polezero pairs shown in Table 1 where y = 2 dB. ~ ________~L __________~Z Y

~----~----~y

x

X

Scope

x x

Gain 3

Trigonometric Function

Figure 1. A fractional-order jerk model using Tanh nonlinearity.

4. Numerical Results The fractional-order jerk model shown in Fig. 1 is simulated and the order m is reduced from 1 to 0.1. This result in a total system order reduced from 3 to 2.1. For m = 1, Figs 2 and 3 show the resulting chaotic attractor in XY and XZ axes, respectively. For m = 0.9, Figs 4 and S show the trajectories in XY and XZ axes using Go = 3 and y = 2 dB. Note that Go is simply a scaling factor and therefore may not be the same as Go shown in Table 1. It can be observed that the double-scroll-like attractor is preserved and is not completely destroyed by order reduction. For m = O.S, Figs 6 and 7 display the trajectories in XY and XZ axes using Go = 30 and y = 2 dB. By using Table 1 where the discrepancy y = 2 dB, the chaotic attractor can be observed for m = 0.9 to 0.2. For m = 0.1, the chaotic attractors can be observed using A = 0.02 and the discrepancy y = 3 dB, i.e. S01.14(s +0.6811)

}T= (s+O.3162)(s+681.1)

(15)

In (IS), Go = SO 1.14. Figures 8 and 9 show the trajectories in XY and XZ axes using m = 0.1, Go = IS00. Figures 10 and 11 show the trajectories in XY and XZ axes using m = 0.1, Go = 2000. Figures 12 and 13 show the trajectories in XY and XZ axes using m = 0.1, Go = 3000.

342

B. Srisuchinwong

U>

.~

0

N -1 -2

-~4

-2

0 X-axis

2

4

Figure 3. Trajectories in XZ axes for ill = 1.

Figure 2. Trajectorie in XY axes for 111 = 1.

3

:r f/)

1f i

'P

I

2

1

I

I

'iii'"

'x of

N

>:1

·1

J

-~~ ----~ 2---0~--~ 2-~

0

-2 -2

2

0 X-axis

4

4

X-axis

Figure 4. Trajectorie in XY axes for 111 = 0.9, Go = 3, Y= 2 dB.

Figure 5. Trajectories in XZ axes for ill = 0.9, Go = 3, Y= 2 dB .

I -1

Figure 6. Trajectorie in XY axes for ill = 0.5, Go = 30, Y = 2 dB.

I

0 X-axis

3

Figure 7. Trajectories in XZ axes for ill = 0.5, Go = 30, Y = 2 dB . 0.8

06

1

0.61'

0.6

OAf

0.4

0.21"

0.2

.~

~

>-

0

-0.2

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-0.4

·0.6

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I

-0 8 L-----~~~--~

-3

-2

-1

0

X-axis

Figure 8. Trajectorie in XY axes, for ill = 0.1, Go = 1500, Y = 3 dB, A = 0.02.

-2

-1

0

__, __ ~_ _i 1

2

X-axis

Figure 9. Trajectories in XZ axes, for ill = 0.1, Go = 1500, Y == 3 dB, A=0.02.

3

Chaos ill a Fractional-Order Jerk Model Using Tanh Nonlinearity

-O:~. 5

'2 -1 ~. S ~ - I -~S-I~~1.5-~ 2

2.5

X-axis

·2

Figure 10. Trajectorie in XY axes, for M = 0.1 , Go = 2000, Y = 3 dB, A=0.02.

-():~.5

-=2'''---::ts--:-1-:O.-s

-o~- -t5-~I--t5--X-axis

343

2

Figure 12. Trajectorie in XY axes, for M = 0.1 , Go = 3000, Y = 3 dB, A= 0.02.

-1.5

-1

-0.5

0 X-axis

0.5

1

1.5

2

2.5

Figure II. Tr~jectorie in XY axes, for m = 0.1, Go = 2000, Y = 3 dB , A=O.02.

2.5

Figure 13. Trajectorie in XY axes, for M = 0.1, Go =3000, y = 3 dB , A = 0.02.

5. Conclusions In this paper, chaos in a fractional-order jerk model using the hyperbolic tangent nonlinearity has been presented. A fractional integrator in the model has been approximated by linear transfer function approximation in the frequency domain. Resulting chaotic attractors have been observed with the system orders as low as 2.1.

Acknowledgements This work is supported by Telecommunications Research and Industrial Development Institute (TRJDI), National Telecommunications Conunission (NTC), Thailand.

References [1] G. Contopoulos, Order and. Chaos in Dynamical Astronomy, New York: Springer, 2004. [2] G. Contopoulos, and N. Voglis, Eds_ Galaxies and Chaos, Lecture Notes in Physics, Germany: Springer, Germany, 2003. l3] H, -J. Stockmann, Quantum Chaos : an introduction, UK: Cambridge University Press, 2000.

344

B. Srisuchinwong

[4] G. Chen and T. Ueta, Chaos in Circuits and Systems, Singapore: World Scientific, 2002. [5] F. C. M. Lau, and C. K. Tse, Chaos-Based Digital Communication Systems, Germany: Springer, 2003 . [6] S. MandaI, and S. Banerjee, Analysis and CMOS Implementation of a ChaosBased Communication System, IEEE Trans. Circuits and Systems,S 1(9): 17081722,2004. [7] G. Kolumban, M. P. Kennedy and L. O. Chua, The Role of Synchronization in Digital Communications Using Chaos - Part I: Fundamentals of Digital Communications, IEEE Trans. Circuits and Systems, 44(10):927-936, 1997. [8] G. Kolumban, M. P. Kennedy and L. O. Chua, The Role of Synchronization in Digital Communications Using Chaos - Part II: Chaotic Modulation and Chaotic Synchronization, IEEE Trans. Circuits and Systems, 45(11): 1129-1140, 1998. [9] P. Stavroulakis, Chaos Applications in Telecommunications, New York: Taylor & Francis, 2006. [10] H. Adeli, S. Ghosh-Dastidar, N. Dadmehr, A Wavelet-Chaos Methodology for Analysis of EEGs and EEG Subbands to Detect Seizure and Epilepsy, IEEE Trans Biomedical Engineering, 54(2):205-21 I, 2007. [11] S. Ghosh-Dastidar, H. Adeli, N. Dadmehr, Mixed-Band Wavelet-Chaos-Neural Network Methodology for Epilepsy and Epileptic Seizure Detection, IEEE Trans. Biomedical Engineering, 54(9):1545-1551 , 2007. [12] A. Dalgleish, The relevance of non-linear mathematics (chaos theory) to the treatment of cancer, the role of the immune response and the potential for vaccines, Q. 1. Med., 92:347-359,1999. [13] M. P. Kennedy, Robust Op Amp Realization of Chua's Circuit, Frequenz, 46(34):66-80, 1992. [14] J. C. Sprott, A New Class of Chaotic Circuit, Physics Letters A, 266:19-23, 2000. [IS] B. Srisuchinwong, and C. -H. Liou, High-Frequency Implementation of Sprott' s Chaotic Oscillators Using Current-Feedback Op Amps, Proceedings of International Symposium on Signals Circuits and Systems - ISSCS 2007, 1:9799,2007. [16] B. Srisuchinwong, C. -H. Liou, and T. Klongkumnuankan , Prediction of Dominant Frequencies of CFOA-Based Sprott' s Sinusoidal and Chaotic Oscillators, Proceedings of CHAOS 2008, Chaotic Modeling and Simulation International Conference, 3-6 June, Crete, Greece, 1-8, 2008. [17] B. Srisuchinwong, C. -H. Liou, and T. Klongkumnuankan, Prediction of Dominant Frequencies of CFOA-Based Sprott's Sinusoidal and Chaotic Oscillators, in Topics on Chaotic Systems: Selected Papers from CHAOS 2008 International Conference, c. H. Skiadas, Ed. Greece: World Scientific, (in press). [18] B. Srisuchinwong, and W. San-Urn, A Chua's Chaotic Oscillator Based on a Coarsely Cubic-Like CMOS Resistor, Proceedings of Asia-Pacific Conference on Communications -APCC 2007,2007,47-49,2007. [19] M. W. Hirsch, and S. Smale, Differential Equations: Dynamical Systems and Linear Algebra, New York: Academic Press, 1974, Chapter 1 I , 238-54. [20] J. C. Sprott, Simple chaotic systems and circuits, Am. 1. Phys. 68(8):758-763, 2000. [21] K. B. Oldham and J. Spanier, The Fractional Calculus, New York: Dover Publications, 1974.

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[22] A. Charef, H. H. Sun, Y. Y. Tsao, and B. Onaral, Fractal System as Represented by Singularity Function, IEEE Trans. Automatic Control, 37(9): 1465-1470, 1992. [23] T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, Chaos in a Fractional Order Chua's System IEEE Trans. Circuits and Systems, 42(8):485-490,1995. [24] W. M. Ahmad, and 1. C. Sprott, Chaos in Fractional-Order Autonomous Nonlinear Systems, Chaos, Solitons & Fractals, 16:339-351 , 2003.

346

Nonlinear Systems in Brunovsky Canonical Form: A Novel Neuro-Fuzzy Algorithm for Direct Adaptive Regulation with Robustness Analysis Dimitris C. Theodoridisl, Yiannis Boutalis 1 ,2, and Manolis A. Christodoulou3 1

2

3

Department of Electrical and Computer Engineering Democritus University of Thrace 67100 Xanthi, Greece (e-mail: [email protected] ) Department of Electrical, Electronic and Communication Engineering Chair of Automatic Control University of Erlangen-Nuremberg 91058 Erlangen, Germany (e-mail: [email protected]) Department of Electronic and Computer Engineering Technical University of Crete 73100 Chania, Crete, Greece (e-mail: [email protected] )

Abstract: The direct adaptive regulation of unknown nonlinear dynamical systems in Brunovsky form with modeling error effects, is considered in this chapter. The method is based on a new Neuro-Fuzzy Dynamical System definition, which uses the concept of Fuzzy Adaptive Systems (FAS) operating in conjunction with High Order Neural Network Functions (HONNF's). Since the plant is considered unknown, we propose its approximation by a special form of a Brunovsky type fuzzy dynamical system (FDS) assuming also the existence of disturbance expressed as modeling error terms depending on both input and system states. The development is combined with a sensitivity analysis of the closed loop in the presence of modeling imperfections and provides a comprehensive and rigorous analysis of the stability properties of the closed loop system. Simulations illustrate the potency of the method and its applicability is tested on well known benchmarks, where it is shown that our approach is superior to the case of simple Recurrent High Order Neural Networks (RHONN's). Keywords: direct control, Neuro-Fuzzy Algorithm, Brunovsky form, robustness analysis.

1

Introduction

Adaptive control theory has been an active area of research over the past years [2,5-7,9,11,17,18,20,25,27]. The identification procedure is an essential part in any control procedure. In the Neuro or N euro - Fuzzy adaptive

Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm

347

control two main approaches are followed. In the indirect adaptive control schemes, first the dynamics of the system are identified and then a control input is generated according to the certainty equivalence principle. In the direct adaptive control schemes [5,9,18] the controller is directly estimated and the control input is generated to guarantee stability without knowledge of the system dynamics. Also, many researchers focus on robust adaptive control that guarantees signal boundness in the presence of modeling errors and bounded disturbances [2,5-7,9,11,18,25,27]. It has been established that neural networks and fuzzy inference systems are universal approximators [8,19,26]'i.e., they can approximate any nonlinear function to any prescribed accuracy provided that sufficient hidden neurons and training data or fuzzy rules are available. Recently, the combination of these two different technologies has given rise to fuzzy neural or NeuroFuzzy approaches, that are intended to capture the advantages of both fuzzy logic and neural networks. Numerous works have shown the viability of this approach for system modeling [3,10,13,14,16].

The neural and fuzzy approaches are most of the time equivalent, differing between each other mainly in the structure of the approximator chosen. Indeed, in order to bridge the gap between the neural and fuzzy approaches several researchers introduce adaptive schemes using a class of parameterized functions that include both neural networks and fuzzy systems [3,10,14,16]. Recently [4,12], HONNF's have been proposed for the identification of nonlinear dynamical systems of the form i; = f(x, u), approximated by a Fuzzy Dynamical System. This approximation depends on the fact that fuzzy rules could be identified with the help of HONNF's. The same rationale has been employed in [1,23,24]' where a Neuro-Fuzzy approach for the indirect and direct control of square unknown systems has been introduced assuming only parameter uncertainty. In this chapter HONNF's are also used for the Neuro-Fuzzy direct control of nonlinear dynamical systems in Brunovsky canonical form with modeling errors. In the proposed approach the underlying fuzzy model is of Mamdani type [15]. The structure identification of the underlying fuzzy system is made off-line based either on human expertise or on gathered data. However, the required a-priori information obtained by linguistic information or data is very limited. The only required information is an estimate of the centers of the output fuzzy membership functions. Information on the input variable membership functions and on the underlying fuzzy rules is not necessary because this is automatically estimated by the HONNF's. This way the proposed method is less vulnerable to initial design assumptions. The parameter identification is then easily addressed by HONNF's, based on the linguistic information regarding the structural identification of the output part and from the numerical data obtained from the actual system to be modeled. We consider that the nonlinear system can be expressed in Brunovsky canonical form. We also consider that its unknown nonlinearities could be

348

D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou

approximated with the help of two independent fuzzy subsystems. We also assume the existence of disturbance expressed as modeling error terms depending on both input and system states. Every fuzzy subsystem is approximated from a family of HONNF's, each one being related with a group of fuzzy rules. Weight updating laws are given and we prove that when the structural identification is appropriate and the modeling error terms are within a certain region depending on the input and state values, then the error reaches zero very fast. Also, an appropriate state feedback is constructed to achieve asymptotic regulation of the output, while keeping bounded all signals in the closed loop. The chapter is organized as follows. Section 2 presents the Fuzzy systems description using indicator functions and HONNF's. The direct adaptive Neuro-Fuzzy regulation of systems being in Brunovsky canonical form is given in section 3, where the problem formulation and the weight updating laws are derived. Section 4 presents simulation results and comparisons while section 5 concludes the work.

2

FUzzy system description using rule firing indicator functions and HONNF

Consider a nonlinear function f(x) E R, x E X c R n approximately described by a Mamdani-type Fuzzy System (FS). Let .fl;"h, ... ,j" be defined as the subset of x E X belonging to the (jl,j2, ... ,jn)th input fuzzy patch and pointing - through the function fe) - to the subset which belong to the [th output fuzzy patch. In other words, .fl;,,12, ... ,j" contains input values x that are associated through a fuzzy rule with output values f(x). Furthermore, the FS receiving as input the n-tuple of x = (Xl, X2, ... ,xn ) gives as output an approximate of f (x) using fuzzy rules and well known fuzzy inference procedure. According to the above notation the Rule Firing Indicator Function(RFIF) l . or simply Indicator Function (IF) connected to.flJl . is defined as fol,J2,··· ,In lows: l . I . [.. . ( x ()) t = { a(x(t)) if (x(t)) E.fl)1,)2,···,)" (1) )1,)2,.·.,)n 0 otherwise where a(x(t)) denotes the firing strength of the rule. According to standard fuzzy system description, this strength depends on the membership value of each Xi in the corresponding input membership functions /-LFji and more specifically [26], a(x(t)) = min[/-LFj1 (Xl (k)), ... /-LFj" (xn(k))]. Then, assuming a standard defuzzification procedure (e.g. weighted average), the functional representation of the fuzzy system can be written as

f(x(t)) =

L !J" ...

,jn X

(I1)~" ... ,jn (x(t))

(2)

Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm

349

where the sumation is carried over all the available fuzzy rules. l)" ... ,j" is any constant vector consisting of the centers of fuzzy partitions of f determined by l and (I1)~" ... ,j" (x(t)) is the IF defined in (1) divided by the sum of all IF participating in the summation of 2. Based on the fact that functions of high order neurons are capable of approximating discontinuous functions [4] and [12] use high order neural network functions HONNF's in order to approximate an IF. A HONNF is defined as: L

N(x(t); w, L)

L

=

II

Whot

hot=l

p~j(hot)

(3)

jEhot

where hot = {h, h , ... , h} is a collection of L not-ordered subsets of {1,2, .. .,n}, dj(hot) are non-negative integers. Pj are the elements of the following vector,

(4) where

S

denotes the sigmoid function defined as: 1

s(x) -- a 1 + e-{3x -

(5)

rv I

with a, /3, 'I being positive real numbers and W := [Wl" weights. Eq. (3) can also be written as

'WL]T

are the HONNF

L

N(x(t); w, L)

=

L

WhotShot(X(t))

(6)

hot=l where Shot(X(t)) are high order terms of sigmoid functions of x. Following the above notation (II)~, ,.. ,j" in (2) can be approximated by

1 . (x) = N(x(t);wJJ ,···,jn;I,LJI,···,jn;l ) N J1,···,]n So, Eq. (2) can be rewritten as ~-l

f(x(t)) = L

h ,... ,jn

I

x N j"

.. ,jJx(t))

(7)

From the above definitions and Eq. (7) it is obvious that the accuracy of the approximation of f(x) depends on the approximation abilities of HONNFs and on an initial estimate of the centers of the output membership functions. These centers can be obtained by experts or by off-line techniques based on gathered data. Any other information related to the input membership functions is not necessary because it is replaced by the HONNFs.

3 3.1

Direct adaptive neuro-fuzzy regulation Problem formulation and neuro-fuzzy representation

Problem formulation We consider nonlinear dynamical systems of the Brunovski canonical form

350

D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou ::i;

= Aex + bc[j(x) + g(x) . u]

(8)

f

where the state x E Rn is assumed to be completely measured, the control input u E R, j and 9 are scalar nonlinear

functionS[O~~f..the tate

volved in the dynamic equation of x n . Also , Ae =

o o

ein:

onl~oo~'lll]-

0

0 0

and be = [0 The state regulation problem is known as our attempt to force the state to zero from an arbitrary initial value by applying appropriate feedback control to the plant input. However, the problem as it is stated above for the system (8), is very difficult or even impossible to be solved since the j, 9 are assumed to be completely unknown. To overcome this problem we assume that the unknown plant can be described by the following model arriving from a neurofuzzy representation described below, plus a modeling error term w(x, u)

(9) where x j , Xg are vectors containing the centers of the partitions of every fuzzy output variable of j(x) and g(x) respectively, determined by the designer and Wj, W; are unknown weight matrices. 8 j, 8 g are sigmoidal vectors containing high order sigmoidal terms terms with parameters which also determined by the designer. Therefore, the state regulation problem is analyzed for the system (9) instead of (8). Since, Wj and W; are unknown, our solution consists of designing a control law u(Wj, W g , x) and appropriate update laws for Wj and Wg to guarantee convergence of the state to zero and in some cases, which will be analyzed in the following sections, boundedness of x and of all signals in the closed loop. The following mild assumptions are also imposed on (8), to guarantee the existence and uniqueness of solution for any finite initial condition and u E Uc . Assumption 1. Given a class Ue C R of admissible inputs, then for any

u E Ue and any finite initial condition, the state trajectories are uniformly bounded for any finite T > O. Meaning that we do not allow systems processing trajectories which escape at infinite, in finite time T, T being arbitrarily small. Hence, Ix(T) I < 00. Assumption 2. The j, 9 are continuous with respect to their arguments and satisfy a local Lipschitz condition so that the solution x( t) of (8) is unique for any finite initial condition and u E Uc .

Nonlinear Systems ill Brullovsky Canonical Form: A Neuro-Fuzzy Algorithm

351

Neuro-fuzzy representation We are using a fuzzy approximation of the system in (8), which uses two fuzzy subsystem blocks for the description of f( x) and g( x) as follows

f(x) = ~ Xfl. ~

. xNfl.

}1,}2"",)n

. (x)

J1,)2,· ·,)n

(10)

(11) where the summation is carried out over the number of all available fuzzy l . . rules, N f , N g are appropriate HONNF's and the meaning of indices. )1 ,J2,··· ,)n has already been described in Section 2. In order to simplify the model structure, since some rules result to the same output partition, we could replace the NNs associated to the rules having the same output with one NN and therefore the summations in (10),(11) are carried out over the number of the corresponding output partitions. Therefore, the system of (8) is replaced by the following equivalent Brunovsky form Fuzzy - High Order Neural Network (F-HONN), which depends on the centers of the fuzzy output partitions xfi and xgi

(12) where Npf and Npg are the number of fuzzy partitions of f and g respectively. Or in a more compact form

(13) where, similar to (9), xf, Xg are vectors containing the centers of the partitions of every fuzzy output variable of f (x) and g( x) respectively, sf (x), Sg (x) are vectors containing high order combinations of sigmoid functions of the state x and Wf, Wg are matrices containing respective neural weights according to (6) and (12). The dimensions and the contents of all the above matrices are chosen so that both xfWfsf(x) and XgWgSg(x) are scalar. For notational simplicity we also assume that all output fuzzy variables are partitioned to the same number, m, of partitions. Under these specifications x f is a 1 x m row vector of the form

where xfp denotes the center or the fuzzy p-th partition of f. These centers can be determined manually or automatically with the help of a fuzzy c-means clustering algorithm as a part of the off-line fuzzy system structural identification procedure mentioned in the introduction. Also, sf(x) = [sj,(x) s!k(x)f, where each Sfi(X) with i = {1,2, ... ,k}, is a high

352

D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou

order combination of sigmoid functions of the state variables and Wf is a m x k matrix with neural weights. Wf can be also written as a collection of column vectors Wi, that is Wf = [W} Wi Wi]' where l = 1,2, ... , k. Similarly, Xg is a 1 x m row vector of the form

where xgk denotes the center or the k-th partition of g. Wg, Sg(x) have the same dimensions as Wf, sf (x) respectively. 3.2

Adaptive regulation with modeling error effects

In this subsection we present a solution to the adaptive regulation problem and investigate the modeling error effects when the dynamical equations have the Brunovsky canonical form. Assuming the presence of modeling error the unknown system can be written as (9). The regulation of the system can be achieved by selecting the control input to be (14) with

v = kx

(15)

where k is a vector of the form k = [k n ... k2 klJ E R n be such that all roots ofthe polynomial h(s) = sn + k1s n - 1 + ... + k n are in the open left half-plane. Define now, the regulation error as ~ =-x

(16)

After substituting Eq. (14) to the n - th state equation of Eq. straightforward manipulations we have that

(9) and

where ilc = Ac - bck is a matrix with its eigenvalues on the left half plane. Wf = Wf - Wj and Wg = Wg - W;. Wf and Wg are estimates of Wj and respectively and are obtained by update laws which are to be designed in the sequel. After substituting Eq. (16), (17) becomes

W;

~ = ilc~ + bc[x f Wfs f(x)

+ Xg WgSg(x)u -

w(x, u)J

(18)

v = ~e p~ + 2~1 tr { Wf T Wf } + 2~2 tr { WJWg}

(19)

To continue, consider the Lyapunov candidate function

Where P > 0 is chosen to satisfy the Lyapunov equation

Pile

+ il~ P = - I

Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm

353

If we take the derivative of Eq. (19) with respect to time we obtain

. 1 2 V = -211~11

+~

T

-

PbcXfWfSf(X) +

T T ~ PbcXgWgSg(X)u-~ Pbcw(x,u)+

~1 tr { WfTWf } + ~2 tr { Wg TWg } Hence, if we choose (20) (21 )

V becomes

v ~ -~ 11~112 + 11~llllPllllw(x, u)11

(22)

It can be easily verified that Eqs. (20) and (21) after making the appropriate operations, result in the following weight updating laws (23)

(24) where ~ is the vector defined in (16), u is a scalar and 11, 12 are positive constants expressing the learning rates. The above equations can be also element wise written as: a) for the elements of Wf

(25) or equivalently wj = -11~nPn (xf)T sf, (x) for alll = I, 2, ... , k, j = 1, 2, ... , m, is the n - th element of the vector ~ and Pn is the last n - th row of P. b) for the elements of Wg

~n

(26)

WJ

or equivalently = -12~nPn (Xg)T Sg, (x)u, for all l = 1,2, ... , k and j = 1,2, ... ,m. Furthermore, we can make the following Assumption. Assumption 3. The modeling error term satisfies

I w(x, u)

I~ e~

Ixi + e~ lui

where e~ and e~ are known positive constants. Also, we can find an a priori known constant

eu

> 0, such that

354

D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou

and Assumption 3 becomes equivalent to (27) where

£1 = £~

+ £~£u

(28)

is a positive constant. Employing Assumption 3, Eq. (22) becomes

v : :; -~ 11~112 + £1 11~llllPllllxll since

Ilxll

= II~II

(29)

then Eq. (29) becomes

v : :; -~ 11~112 + £1 IIPIIII~112 :::; -

(~ -

£1

IIPII)

11~112

(30)

Hence, if we chose IIPII < 2t then the Lapyunov candidate function becomes negative. We are now ready to prove the following theorem Theorem 1. The control law (14) and (15) together with the update laws (23) and (24) guarantee the following properties

• limt-+oo ~(t)

provided that

= 0,

limt-+oo x(t)

IIPII < 2}!

=0

and Assumption 3 is satisfied.

Proof. From Eq. (30) we have that V E Loo which implies ~,Wj, Wg E Loo. Furthermore Wj = Wj + Wi E Loo and Wg = Wg + Wg * E Loo. Since ~ E Loo this also implies x E Loo. Moreover, since V is a monotone decreasing function of time and bounded from below, the limt-+ oo V(t) = Voo exists so by integrating V from 0 to 00 we have

which implies that

~

=

I~I E

L 2 . We also have that

Ac~ + bc[xj WjSj(x)

+ Xg WgSg(x)u -

w(x, u)]

Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm

355

Hence and since U E L oo , the sigmoidals and center matrices are bounded by definition, Wf , Wg E Loo and Assumption 3 hold, so since ~ E L2 n Loo and ~ E L oo , applying Barbalat's Lemma we conclude that limt->oo ~(t) = O. Now, using the bounded ness of u, sf(x), Sg(x), x and the convergence of ~(t) to zero, we have that Wf, Wg also converge to zero. Hence we have that limt---+oo x(t)

= -limt->oo ~(t) = 0

Thus, limt---+oo x(t) = O.

Remark 1. We can't conclude anything about the convergence of the synaptic weights Wf and Wg to their optimum values Wi and W; respectively from the above analysis. The existence of signal u is associated with conditions which guarantee that XgWgSg(x) # O. It can be shown that using appropriate projection method [9], the weight updating laws can be modified so that the existence of the control signal can be assured. However, this development is not presented in this chapter. The relevant results along with the ones given here are to be presented shortly in a forthcoming work.

4

Simulation results

To demonstrate the potency of the proposed scheme we present two simulation results. One of them is the well known benchmark "Inverted Pendulum" and the other one is the "Van der pol" oscillator. Both of them present comparisons of the proposed method with the well established approach on the use of RHONN's [21]. The comparison shows off the regulation superiority of our method under the presence of modeling errors. 4.1

Inverted pendulum

Let the well known problem of the control of an inverted pendulum. Its dynamical equations can assume the following Brunovsky canonical form [22]

rnlx~ cos Xl sin Xl

.

X2 =

'Tnc+ rn

gSlnxl

1(1- =C08 3

with

e

-

2

Xl)

rHC+rn

d = 5X2

~

+

=C+ m

I

U

+d

=c082 " " ) 4 ( '3rrtC+ m

+ 10sin(lOx2) + sin(2u)

(31)

where Xl = and X2 = iJ are the angle from the vertical position and the angular velocity respectively. Also, 9 = 9.8 m/ S2 is the acceleration due to

356

D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou

gravity, m e is the mass of the cart, m is the mass of the pole, and I is the half-length of the pole. We choose me = 1 kg, m = 0.1 kg, and I = 0.5 min the following simulation. It is our intention to compare the direct control abilities of the proposed Neuro-Fuzzy approach with RHONN's [21] (page 62-71). We also, make the appropriate changes to RHONN's in order to be equivalent with F-HONNF's (brunovsky form) for comparison purposes. For the proposed F-HONNF approach we use the adaptive laws which are described by Eqs. (23) and (24) and the control law described by Eqs. (14) and (15). Numerical training data were obtained by using Eqs. (31) with initial conditions [Xl (0) X2(0)] = [ - ;2 0] and sampling time 10- 3 sec. We are using the proposed approach with Eq. (13) to approximate Inverted Pendulum dynamics and send data to sigmoidal terms. Our NeuroFuzzy model was chosen to use 5 output partitions of f and 5 output partitions of g. The number of high order sigmoidal terms (HOST) used in HONNF's (for F-HONNF and RHONN) were chosen to be 5: (s(xd, S(X2), s(xd .

S(X2)' s2(xI), S2(X2)). In order the proposed model to be equivalent with RHONN's regarding to other parameters except the initial weights (Wf(O) = 0 and Wg(O) = 1 for F-HONNF and Wf(O) = 0 and Wg(O) = 0.016 for RHONN) we have chosen the updating learning rates 1'1 = 5 and 1'2 = 1 and the parameters of the sigmoidal terms such as al = 0.1 , a2 = 3, bl = b2 = 1 and Cl = C2 = O. Also, after careful selection, kl = 2 and k2 = 80 . Fig. (1) shows the regulation of states Xl and X2 (with blue line for F-HONNF and red line for RHONN's), while fig. (2) gives the evolution of control input u, the errors el, e2 and the modeling error d(X2' u) respectively. The mean squared error (MSE) for RHONN and F-HONNF approaches were measured and are shown in Table 1, demonstrating a significant (order of magnitude) increase in the regulation performance of F-HONNF against RHONN's. Example RHONN F-HONNF Inverted Pendulum MSExj 0.0010 6.0794 . 10 - 4 MSEx2 0.9252 0.3368

Table 1. Comparison of RHONN's and F-HONNF approaches for the Inverted Pendulum.

4.2

Van der pol

Van der Pol oscillator is usually used as a simple benchmark problem for testing control schemes. It's dynamical equations are given by

Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm

357

0.2 ,\

* x

II

jll\A

0.1

~~~~~~~~-----------1

0

II

(J5

~

-0.1 -0.2

o

2

3

4

5

1\ . il

4 1

*

';i

2 0

II,.

mly:0,\P\iXX'7VV'''XI''NV~v'''A-~~-

,Ii V

(J5 -2

-4

\/ V

_6L---__L -_ _ _ _ _ _

o

~

_ _ _ _ _ _ ~_ _ _ _ _ _~ _ _ _ _ _ _ ~

2

3

4

5

Time (sec)

Fig. 1. Evolution of angle state variable Xl and angular velocity for RHONN's (red line) and F-HONNF approach (blue line)

X2 = X2 . with

d

(a - xi) . b -

Xl

X2

respectively,

+ U+ d

= 2X2 + 4sin(5x2) + 5sin(10u)

(32)

The procedure of the approximation was the same as that of Inverted Pendulum. The proposed Neuro-Fuzzy model was chosen to have initial conditions, [Xl (0) X2(0)] = [0.4 0.5] and the number of high order sigmoidal terms (HOST) used in HONNF's (for F-HONNF and RHONN) were chosen to be 2 (s(xd, S(X2)' In order our model to be equivalent with RHONN's regarding to other parameters except the initial weights (Wf(O) = 0 and Wg(O) = 0.1 for FHONNF and Wf(O) = 0 and Wg(O) = 0.05 for RHONN) we have chosen the updating learning rates {I = 0.1 and {2 = 10 and the parameters of the sigmoidal terms such as al = 0.1, a2 = 4, bI = b2 = 1 and CI = C2 = O. Also, kI = 2 and k2 = 80 after careful selection. Fig. (3) shows the regulation of states Xl and X2, while fig. (4) presents the evolution of control input u, the errors eI, e2 and the modeling error d. The mean squared error (MSE) for RHONN and F-HONNF approaches were measured and are shown in Table 2 demonstrating as before (Inverted

358

D. C. Theodoridis. Y. BOlltalis, and M. A. Christodoulou

5

~ _: Hhilu-v II

~

j'-\D<~~;~~'=~.~~~~~>~.~_~

o

2

40 S

N 2"5'0

__

<_ _• •

<_ _. _ ._ _ _ _ • _ _ • _ _ • • _____

3

-~----<-----,----------,--

4

I

5

-------,---

20

0 -20 _40~--------L--------L-------~--------J-------~

o

2

3

4

5

Time (sec)

Fig. 2 . Evolution of control input n, errors c], e2 and disturbance d(:r;2' 'U) for RHONN's (red line) and F-HONNF a.pproach (blue line)

Pendulum case), a significant (order of magnitude) increase in the control performance. Conclusively, the comparison between llHONN and F-HONNF'fl leads to a superiority of F-HONNF'fl regarding the control abilities. Especially, if we choose smaller learning rate and reduced number of high order terms, then the difference between the two methods becomes huge. Examples RHONN FHONNF Van del' polA[S'Ex] 0.0024 9.5226 . 104~ M 8£:';2 0.8417 0.5930

Table 2 . CompaTison of RHONN's am! F-HONNF a.pproacbes for the Van del' pol oscillator.

5

Conclusions

A direct ada.ptive control scheme was considered in this chapter, '-liming at the regulation of nonlinear unknown plants of Brunovsky canonical form with the presence of modeling errors. The approach is based on a new Neuro-Fuzzy

Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm

359

0.6r---~--~---~--~---'-----,

-0.2 -0.4 L--_ _-'--_ _---'_ _ _ o 0.5 1.5

~

_ ___'__ _ _~_ _.....J

2

2.5

3

2

2.5

3

5

)(

*

(jj

ti(\

0

N

-5

-10

r,

\j .\li V 0

0.5

1.5 Time (sec)

Fig. 3. Evolution of state variable approach (blue line)

Xl

and

X2

for RHONN's (red line) and F-HONNF

Dynamical Systems definition, which uses the concept of Fuzzy Adaptive Systems (FAS) operating in conjunction with High Order Neural Network Functions (F-HONNF's). Since the plant is considered unknown, we first propose its approximation by a special form of a Brunovsky type fuzzy dynamical system (FDS) where the fuzzy rules are approximated by appropriate HONNF's. This practically transforms the original unknown system into a neuro-fuzzy model which is of known structure, but contains a number of unknown constant value parameters known as synaptic weights. The proposed scheme does not require a-priori experts' information on the number and type of input variable membership functions making it less vulnerable to initial design assumptions. vVcight updating laws for the involved HONNF's are provided, which guarantee that the system states reach zero exponentially fast, while keeping all signals in the closed loop bounded. Simulations illustrate the potency of the method while the applicability is tested on well known benchmarks where it is shown that by following the proposed procedure one can obtain asymptotic regulation quite well. Compared to simple RHONN's direct control, proves to be superior.

References l.y. S. Boutalis, D. C. Theodoridis, and M. A. Christodoulou. A new neuro fds definition for indirect adaptive control of unknown nonlinear systems using a method of parameter hopping. IEEE Transactions on Neural Networks, 20(4):609-625, 2009.

360

D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou

i'"lJ

,

!::: 2000

e §

()

1000 0

1

~

o

0.5

2

1.5

2.5

3

~ ~l~rrL-.~ · L-.'--------'----~,l 1JrQj~~__n_n~ ~ o

0.5

1

1.5

2

2.5

3

2.5

3

_ _ _ _

o

0.5

1

1.5

2

. o

0.5

1.5 Time (sec)

., 2

2.5

1 3

Fig. 4. Evolution of control input u and disturbance d for RHONN's (red line) and F-HONNF approach (blue line)

2.M. Chemachema and K. Belarbi. Robust direct adaptive controller for a class of nonlinear systems based on neural networks and fuzzy logic system. International Journal on Artificial Intelligence Tools, 16:553-560, 2007. 3.K. B. Cho and B. H. Wang. Radial basis function based adaptive fuzzy systems and their applications to system identification and prediction. Fuzzy Sets and Systems, 83:325-339, 1996. 4.M. A. Christodoulou, D. C. Theodoridis, and Y. S. Boutalis. Building optimal fuzzy dynamical systems description based on recurrent neural network approximation. In Conference of Networked Distributed Systems for Intelligent Sensing and Control, pages 82-93, Kalamata, Greece, June 2007. 5.Y. Diao and K. M. Passino. Adaptive neural/fuzzy control for interpolated nonlinear systems. IEEE Transactions on Fuzzy Systems, 10:583-595, 2002. 6.S. S. Ge and W. Jing. Robust adaptive neural control for a class of perturbed strict feedback nonlinear systems. IEEE Transactions on Neural Networks, 13(6):1409 - 1419, 2002. 7.W. M. Haddada, T. Hayakawaa, and V. Chellaboina. Robust adaptive control for nonlinear uncertain systems. Automatica, 39:551 556, 2003. 8.K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2:359-366, 1989. 9.P. Ioannou and B. Fidan. Adaptive contml tutorial. SIAM, 2006. 1O.C. F. Juang and C. T. Lin. An on-line self-constructing neural fuzzy inference network and its applications. IEEE Transactions on Fuzzy Systems, 6:12-32, 1998.

Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm

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11.Y. T. Kim and Z. Z. Bien. Robust adaptive fuzzy control in the presence of external disturbance and approximation error. Fuzzy Sets and Systems, 148:377393, 2004. 12.E. B. Kosmatopoulos and M. A. Christodoulou. Recurrent neural networks for approximation of fuzzy dynamical systems. Int. Journal of Intelligent Control and Systems, 1:223-233, 1996. 13.C.T. Lin. A neural fuzzy control system with structure and parameter learning. Fuzzy Sets and Systems, 70:183-212, 1995. 14.Y. H. Lin and G. A. Cunningham. A new approach to fuzzy-neural system modelling. IEEE Transactions on Fuzzy Systems, 3:190-197, 1995. 15.E. Mamdani. Advances in the linguistic synthesis of fuzzy controllers. Int. Journal of Man-Machine Studies, 8(6):669678, 1976. 16.S. Mitra and Y. Hayashi. Neuro-fuzzy rule generation: survey in soft computing framework. IEEE Transactions on Neural Networks, 11:748-768, 2000. 17.H. N. Nounou and K. M. Passino. Stable auto-tuning of hybrid adaptive fuzzy /neural controllers for nonlinear systems. Engineering Applications of Artificial Intelligence, 18:317-334, September 2005. 18.R. Ordonez and K. M. Passino. Adaptive control for a class of nonlinear systems with time-varying structure. IEEE Transactins on Automatic Control, 46(1):152-155,200l. 19.K. Passino and S. Yurkovich. Fuzzy Control. Addison, 1998. 20.G. A. Rovithakis and M. A. Christodoulou. Adaptive control of unknown plants using dynamical neural networks. IEEE Transactions on Systems, Man and Cybernetics, 24:400-412, 1994. 2l.G.A. Rovithakis and M.A.Christodoulou. Adaptive Control with Recurrent High Order Neural Networks (Theory and Industrial Applications), in Advances in Industrial Control. Springer Verlag London Limited, 2000. 22.J. J. E. Slontine and W. Li. Applied Nonlinear Control. Prentice-Hall, New Jersey, 1991. 23.D. C. Theodoridis, Y. S. Boutalis, and M. A. Christodoulou. Direct adaptive control of unknown nonlinear systems using a new neuro-fuzzy method together with a novel approach of parameter hopping. Kybernetica, 45(3) :349-386, 2009. 24.D. C. Theodoridis, M. A. Christodoulou, and Y. S. Boutalis. Indirect adaptive neuro - fuzzy control based on high order neural network function approximators. In In Proccedings 16th Mediterranean Conference on Control and Automation - MED08, pages 386-393, Ajaccio, Corsica, France, 2008. 25.S. Tong and T. Chai. Direct adaptive control and robust analysis for unknown multivariable nonlinear systems with fuzzy logic systems. Fuzzy Sets and Systems, 106:309-319, 1999. 26.L. Wang. Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Prentice Hall, NJ, 1994. 27.Y. Yang. Direct robust adaptive fuzzy control (drafc) for uncertain nonlinear systems using small gain theorem. Fuzzy Sets and Systems, 151:79-97, May 2004.

362

Routh-Hurwitz Conditions and Lyapunov Second Method for a Nonlinear System Omur Umut Abant Izzet Baysal University 14280 Bolu, TUrkey (e-mail: [email protected] ) Abstract: In this paper , a Lyapunov function is generated to determine the domain of asymptotic stability of a system of three first order nonlinear ordinary differential equations describing the behaviour of a nuclear spin generator (NSG). The generated Lyapunov function, is a simple quadratic form, whose coefficients are chosen so that the Routh-Hurwitz criteria are satisfied for the corresponding linear differential equations. Keywords: Routh-Hurwitz conditions, Lyapunov method, Lyapunov function , nuclear spin generator, domain of stability, Cartwright's method.

1

Introduction

The well-known Routh-Hurwitz criteria can be easily applied to determine the conditions for the stability of a linear autonomous system. The simplicity of these conditions is a consequence of the fact that stability, for a linear system, is a global property. Since the stability of nonlinear systems is not necessarily global, no simple criteria exist for its determination, and it becomes necessary to seek domains of the state space within which the system behaves in a stable fashion. The direct or second method of Lyapunov [1] is a general approach to the question of stability. The essence of the method consists in determining sufficient conditions of stability not by explicit knowledge of the solutions of the differential equations describing the system, but by using potential-like functions, called Lyapunov functions. Since the solutions of the differential equations are not necessary, the method is particularly well suited for stability analysis of nonlinear systems. The generation of Lyapunov functions appropriate for a particular problem is the critical part of the study. A great deal of activity has been devoted to the problem of generating function since 1950's [2]-[5]. In this paper, Cartwright method is used for the determination of the domain of asymptotic stability of NSG.

2

Fundamental concepts

The general nth. order autonomous system can be represented by

Routh-Hurwitz Conditions and Lyapul10v Second Method

x=

f(x),

X E

Rn

363

(1)

The singular points x = 0 represent states of equilibrium of the physical system, any of which can always be situated at the origin of the state space by an appropriate coordinate transformation so that f(O) = O. It is desired to study the stability of a position of equilibrium; that is, to say, to investigate the behaviour of the system trajectories in the vicinity of equilibrium point x = O. These trajectories display only four distinct types of behaviour as time increases: (a) They may approach the origin; (b) the trajectories may tend to remain within a neighborhood of the origin, without ever converging to it; (c) they may approach a different equilibrium point; or (d) the trajectories may tend to infinity. The first case represents the type of behaviour a designer would desire a physical system to possess, and such a system would be called asymptotically stable. It is desirable to devise a method that will be capable of discriminating this case from the others. Since the systems to be analyzed are nonlinear, and their stability local, it becomes necessary that the method used be also capable of yielding a domain of asymptotic stability, a simply connected region of the state space containing only the equilibrium point x = 0, and such that any trajectory initiating within this domain will tend to this equilibrium point as time increases. The second or direct method of Lyapunov is particularly appropriate for these purposes. The method is based on the investigation of sign definiteness of certain functions which determine the stability of the equilibrium point , as well as a domain of asymptotic stability. It therefore becomes unnecessary to integrate the equations of motion, a remote possibility, but it is necessary to determine some appropriate functions, Lyapunov functions, for the differential system being investigated. The theorems of interest here are the followings [1]-[5]. Theorem 1. Let x = 0 be an equilibrium point for (1) and D c R n be a domain containing x = O. Let V : D --7 R be a continuously differentiable function such that

V(O) = 0 and V( x ) > 0 in D - {O} V(x)

= grad(V(x)) .f(x) ::; 0 in D

Then, the origin is stable. Moreover, if V(x)

= grad(V(x)).f(x) < 0

in D - {O}

then the origin is asymptotically stable.

The requirement that V be nonpositive assures that the system trajectories in D either remain on surfaces of constant V or cross them toward the equilibrium position.

364

O.

Umut

Theorem 2. (Barbashin and Krasovski) Let x = 0 be an equilibrium point for (1). Let also V : D -* R be a continuously differentiable positive definite function on a domain D containing the origin x = 0, such that 11 :::; 0 in D. Let S = {x ED: 11 = O} and suppose that no solution can stay identically in S other than trivial solution. Then the origin is asymptotically stable.

Theorem 2 implies that the function V(x) cannot have a relative maximum inside the domain D since grad( v( x)) is a vector with components VX i , i = 1,2, ... , n all of which cannot be zero simultaneously in D except at the equilibrium position.

3

NSG system

NSG is a high frequency oscillator which generates and controls the oscillations of the motion of a nuclear magnetization vector in a magnetic field. It was first carried out by Sherman, [6] in 1963. This system has the form

X = -;3X + Y y = -X - ;3Y(1- kZ) Z = ;3[a(1 - Z) - ky2]

(2)

where X, Y, Z are the components of the nuclear magnetization vector in the X, Y , Z -directions, respectively. a , ;3 and k are parameters where ;3a :::: 0, and ;3 :::: 0 are linear damping terms, while the nonlinearity parameter ;3k is proportional to the amplifier gain in the voltage feedback. Physical considerations limit the parameter a to the range 0 < a :::; 1. The equilibrium points of the system (2) are found by equating the right side of (2) to zero. The point A(O , 0,1) is an equilibrium point of (2) for all parametric values. Depending on the value of the parameters the point A(O, 0,1) may be a spiral or a node, stable or unstable, for example it is a stable focus when a = 0.15, ;3 = 0.25 and k = 4 as shown in Fig.l

Fig. 1. The point A(O, 0,1) is a stable focus when a = 0.15, /3 = 0.25 and k = 4.

However if,

(3)

"

Routh-Hurwitz Conditions and Lyapunov Second Method

365

the system (2) possesses two additional equilibrium points, namely

1 + ,82

B±(±al,±,8al, ~)

(4)

where a 1 --

(a[,82(k - 1) - I])! k,82

Notice that B+ and B_ are images of each other under the symmetry (x,y,z) --7 (-x, - y ,z) of the basic system (2). All these equilibrium points are unstable when condition in (3) holds. Sachdev and Sarathy studied in great detail the NSC system in [7]. The equilibrium point A(O, 0, 1) is transferred to the origin through the coordinate transformations x = X , Y = Y, z = Z - 1 and system (2) becomes :i; = - ,8x

iJ

+y

-x - ,8(1 - k)y + ,8ky z i = -a,8z - ,8ky2 =

(5)

Now, a Lyapunov function will be generated for system (5) using Cartwright's method.

4

Cartwright's method

This method, [5] is essentially one of the generalising from the linear to the nonlinear case, so that the Routh-Hurwitz stability conditions are satisfied for the linear case. Linearization of (5) is :i; = -,8x

+y

iJ

= -x -

i

= -a,8z

,8(1 - 2k)y

(6)

Let the following quadratic form with constant undetermined coefficients ki' i = 1, 2, ... ,6 be selected as a Lyapunov function for the system (6)

2V = k 1 x 2 + k2y2 + k3 Z2 + k 4xy + k5XZ + k 6yz

(7)

According to eqn (7) . 1 2 1 )]2 V = - 2(2k 1 ,8 + k4)X + 2 [2k2,8(2k - 1 + k4 y -

k3a,8z2 - [k2 - kl 1

+ k4,8(1 -

2[k5,8( 1 + a) + k6]XZ +

1

k)] x y -

2[k5 + k6 ,8(2k - 1 - a)]yz

(8)

366

O. Umut

Let

11 be constrained to

be a n.d (or n.s.d) function of the state variables

x and z . This results in the following set of equations

2k 2(3(2k - 1) + k4 = 0 kl-k2-(3(I - k)k4 0 k5(3(1 + 0:) - k6 0 k5 - (3(1 + 0: - 2k)k6 0 The last two equations imply k5 = k6 = O. 11 can be constrained as a n.s.d function of the state variables x and z by setting k2 = 1 and k3 = 1, arbitrarily. So the solution of the above equations yields kl = 1 + 2(32(1 k)(1 - 2k) and k4 = 2(3(1 - 2k) . Substitution of these values in (7) and (8) yields

and (10) Inspection of eqns (9) and (10) shows that the stability requirements of the 1 + (32 Lyapunov theorem are satisfied for 0: > 0, (3 > 0 and 0 < k < ~. This is not difficult to understand, since stability is a global property for a linear system. For linear systems Parks, [9] has derived the Routh-Hurwitz stability criterion using the second method of Lyapunov. For the nonlinear system (5), a Lyapunov function and its time derivative are 1 1 1

V = 2(1 +,82[2-3k(l+z)+k2(1+z)2])x2+2y2+2z2+,8[1 - k(1+z)] xy (11)

and

11 =

-[(3(1

+ ,82(1 -

,83k3(1

k)(l - 2k))

+ Z)y2]X 2 -

[o:,8z

+ 2,8(1 -

k(l

+ z))

-

~0:,83kz - ~,83k2y2 + 2

+ ,8ky2 + 0:,82k2(1 + z)x 2 -

2 0:,82kxy]z

V is n.d in a certain neighborhood of the origin and V is p.d in the same neighborhood. If Theorem 2 is applied to the Lyapunov function (11), a boundary for the stable domain is determined. The gradient of the function is

(1

\7V

=

[

+ ,82[2 -

+ z) + k 2(1 + Z)2)]X + ,8[1- k(l + Z)]y] y + ,8[1 - k(1 + z)]x z + ~[,82( -3k + 2k2(1 + z)]x 2 - ,8kxy 3k(1

which vanishes at the origin (0,0,0) and at the points

( ± k~2 )2(1 + (1 - k),82], ± k~3 )2[1 + (1 - k)(32], 1 :,8~2 - 1)

Routh-Hunvitl. Conditions and Lyapunov Second Method

367

Thus, I> is identically equal to zero at these three points. vVhen 0: = 0.15, f3 = 0.25 and /;; = 4 nontrivial critical points are 13 (±J26, ±4v26. 4 ) and Lyapunov fUIlction is

It is found that the largest value of c represents a dosed surface is 5.28125. Hence, a domain of asymptotic stability for the system (5)

+0.5(1+z)2]x 2 +O.5 y 2+0.5z 2 - (.z+O.75):z:y = 5.28125 ( 1~~) A three dimensional portrail of this surface is given in Fig.2 [0.5625-0.6667(1+

Fig. 2. Stabilit.y domain

Since I> is negative everywhere, all trajectories inside the bounded domain tend toward the origin and the solutions are asymptotically stable, Fig.a. Outside of the domain, the solutions mayor Hl.ay not tend toward equilibrium, Fig.4, As with the original criterion, Theorem 2 gives only sufficient conditions for stability, Another Lyapunov function may give another domain of stability.

1:~

1.0'

_::.:[.,;;; :;M~o:-"g~Cl;~)-J~O ~-'---'-~~'~'~'~. -..-.I- t-I.O~l

(

2040

fiO

t

/,.-"

~O.5 ,r: 20 40 60 80 100 120 ~ 0.[,5 '.1 j/ ~.O 8

80 100 120 -- 1.0,

~25'v

Fig. 3. The time variations of X', II and ,:;; for the initial conditions inside the stability region, Xo = 1.3, yo = 1.5, Zo = 1.0

368

O. Umut y

x

-0.5 -1.0 -1.5 -2.0 -2.5 -3.0

20 40 60 80

100120tj:6l' ~~ 2.5 2.0 1.5 1.0 0.5

t

-I -2 -3 -4

20 40

80 100 120

20 40 60 80 100120 t_5

Fig. 4. The time variations of x , y and z for the initial conditions outside the stability region, Xo = -3.3, yo = 3.5, Zo = -2.75

5

Conclusions

Using Cartwright method , a Lyapunov function is generated in a systematic way for nuclear spin generator system under suitable assumptions and a three dimensional region, in which stability is guaranteed by the Lyapunov theorem, is portrayed graphically and the results are verified by computer solutions.

References 1.J .P.LaSalle and S.Lefschetz, Stability of Lyapunov's direct method, Academic Press, New York N.Y., 1961. 2.R.D.Ingwerson, A modified Lyapunov method for nonlinear stability analysis, IRE Trans. voI.AC-6, 1961 , pp.199-21O. 3.G.P.Szego, A contribution to Lyapunov second method; nonlinear autonomous systems, ASME paper No.61-WA-192, 1961. 4.Z.V.Rekasius and J.E.Gibson , Stability analysis of nonlinear control systems by the second method of Lyapunov, IRE Trans. voI.AC-7, 1962, pp.3-15 . 5.M.L.Cartwright, On the stability ofsolutions of certain differential equations of fourth order, Quar. J. Mech. Appl. Math., 9(2), 1956, pp.185-194. 6.S. Sherman, A third-order nonlinear system arising from a nuclear spin generator, Contrib.Diff. Eqn., 2, 1963, pp.197-227. 7.P.L.Sachdev and R. Sarathy, Periodic and chaotic solutions for a nonlinear system arising from a nuclear spin generator, Chaos, Solitons and Fractals, voI.4(1l), 1994, pp.2015-2041. 8.W.G. Cunningham, Introduction to nonlinear analysis, McGraw-Hill, 1958. 9.P.C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Lyapunov, Proc. Cambridge Phil. Soc., 58 , 1962, pp.694-702.

369

Atomic de Broglie-Wave Chaos V. O. Vitkovsky, L. E. Konkov, and S. V. Prants Laboratory of Nonlinear Dynamical Systems Pacific Oceanological Institute of the Russian Academy of Sciences 690041 Vladivostok, Russia (e-mail: [email protected])

Abstract: Coherent motion of cold atomic wave packets is studied in a onedimensional optical potential created by a standing laser wave. Their evolution is interpreted in the dressed-state basis where it is the motion in a bipotential. Manifestations of de Broglie-wave chaos, caused by nonadiabatic transitions between components of this bipotential, are found. The probability of those transitions is large in that range of the atom-field detuning and recoil frequency where the classical center-of-mass motion is shown to be chaotic in the sense of exponential sensitivity to small variations in initial conditions or parameters. Keywords: cold atom, matter wave, quantum chaos .

1

Introduction

Cold atoms in a laser standing wave are ideal objects for studying fundamental principles of quantum physics, quantum-classical correspondence, and quantum chaos. The proposal [2] to study atomic dynamics in a far-detuned modulated standing wave made atomic optics [4] a testing ground for quantum chaos. A number of impressive experiments have been carried out in accordance with this proposal [3]. New possibilities are opened if one works near the atom-field resonance where the interaction between the internal and external atomic degrees of freedom is intense. A number of Hamiltonian nonlinear dynamical effects have been found in Refs. [6,1 ] with point-like a~oms in a near-resonant rigid optical lattice: chaotic Rabi oscillations, chaotic walking, dynamical fractals, Levy flights, and anomalous diffusion. Dynamical chaos in classical mechanics is a special kind of random-like motion without any noise and/or random parameters. It is characterized by exponential sensitivity of trajectories in the phase space to small variations in initial conditions and/or control parameters. Such sensitivity does not exist in isolated quantum systems because their evolution is unitary, and there is no well-defined notion of a quantum traj ectory. Thus, there is a fundamental problem of emergence of classical dynamical chaos from more profound quantum mechanics which is known as quantum chaos problem and the related problem of quantum-classical correspondence. In a more general context it is a problem of wave chaos. It is clear now that quantum

370

V. O. Vitkovsky, L. E. KOllkov, and S. V. Prants

chaos, microwave, optical, and acoustic chaos [5] have much in common. The common practice is to construct an analogue for a given wave object in a semiclassical (ray) approximation and study its chaotic properties (if any) by well-known methods of dynamical system theory. Then, it is necessary to solve the corresponding linear wave equation in order to find manifestations of classical chaos in the wave-field evolution in the same range of the control parameters. If one succeeds in that the quantum-classical or the wave-ray correspondence are announced to be established. In this paper we perform this program with cold two-level atoms in a onedimensional standing-wave laser field and show that evolution of the atomic matter waves in a standing wave is really complicated in that range of the control parameters where the corresponding classical point-like atomic motion can be strictly characterized as a chaotic one. The effect is explained by non adiabatic transitions in the optical bipotential whose probability increases greatly in that range.

2

Atomic wave-packet motion in a laser standing wave

The Hamiltonian of a two-level atom, moving in a one-dimensional classical standing-wave laser field, can be written in the frame rotating with the laser frequency W f as follows:

where G-±,z are the Pauli operators for the internal atomic degrees offreedom, X and P are the atomic position and momentum operators, Wa and [l are the atomic transition and Rabi frequencies, respectively. We will work in the momentum representation and expand the state vector as follows:

Itlf(t)) =

J

[a(P, t) 12)

+ b(P, t) 11) ]IP)dP,

(2)

where a(P, t) and b(P, t) are the probability amplitudes to find atom at time t with the momentum P in the excited and ground states, respectively. Schri:idinger equation for the probability amplitudes is written as ..

1

1

2[b(p + 1) + b(p -

za(p) = 2(w r P2

-

. 1 2 ib(p) = 2(w r P

+ L1)b(p) - 2[a(p + 1) + a(p -1)],

L1)a(p) -

1

1)],

(3)

where dot denotes differentiation with respect to dimensionless time T == [It, and the atomic momentum p is measured in units of the photon momentum fik f. The normalized recoil frequency, Wr == fikJ / rna [l, and the atom-field detuning, L1 == (w f - wa ) / [l, are the control parameters. We will interpret

Atomic de Broglie-Wave Chaos

371

Fig. 1. The resonant E~±) (.::1 = 0, dotted curve) and non-resonant E1±) (.::1 oF 0, solid curve) potentials of a two-level atom in the sbmding-,\vave field.

the wave-packet motion in the dressed-state basis 1+ ) L1 = sin812) + cos 811), 1- ) L1

cos 812) - sin 811).

(4)

The dressed states are eigcnstates of an atom at rest in a laser field with the eigenvalues of the quasienergy and the mixing angle 8

E~±) =

±J

LF 4

+ cos2 x, tan8 ==~. 2 cos:c

( -Ll2 cos:r

)2 +1.

(5)

The resonant Ef~~:) (.:1 = 0) and nOll-resonant E);:l (Ll 1= 0) potentials of the atom in the standing-wave field are drawn in Fig. 1. The wave function in the position space is connected to the wave function in the IIlOlnentum space by a Fourier transform. The probability amplitude to find the atom nearby a; == k f X in the potential E1-) in the dressed state basis is iXJ

c(;r;) = const /

dpcipx[a(p)eos8 - b(p)sin8].

(6)

-f:xJ

Let the atom be initially prepared in the ground state as a Gaussian wave packet in the momentum space with the normalized recoil frequency w,. = 10-:3 , the average initial momentum Po = 55 and the initial width Llp = 1, which corresponds to the position initial width Lh: = 1/2 or X = )...t!41f. The centroid of the wave packet is initially at the antinode Xo = O. The momentum probability at the time moment T is computed as vV(p, T) = la(p, T) 12 + Ihep, T) 12. The atomic wave packet, placed at the alltinode of the standing wave, is simultaneously at the top of the potential E(+) and at the bottom of the potential E(-) (see Fig. 1). Its I ± ! components move downwards (upwards) in the potentials E(±), respectively. The evolution of the wave packet depends, mainly, on the relation between the depth of the optical potential and the atomic kinetic energy. One can estimate the character of the bipotential motion as follows. The Hamiltonian of the internal degree of freedom in the basis I ±! = (11) ± 12))/ /2 has the form (7)

372

V. O. Vitkovsky, L. E. Konkov, and S. V. Prants

We linearize the potential in the vicinity of a node of the standing wave and estimate a small distance the atom makes when crossing the node as follows: Ox "':' Wr < Pnode > T, where < Pnode > is the mean atomic momentum near the node. We arrive now at the famous Landau-Zener problem to find a transition probability when the energy difference varies linearly in time. The asymptotic solution is well known

PLZ

=

exp

(-If

Wr

<

L12 Pnode

>

)

.

(8)

The quantity WrP is a measure of the atomic velocity. If WrP « 1, then an atom takes for the period of the Rabi oscillations a small distance as compared with the wavelength. There are three possibilities. l. L12 » Wr < Pnode >. The transition probability from one potential to another one is exponentially small. The wave-packet motion is, generally speaking, bipotential but without transitions between the nonresonant potentials. 2. L12 « Wr < Pnode >. The probability of a Landau-Zener transition is close to unity, and an atom changes the potential each time upon crossing any node. At L1 = 0, it is a bipotential motion in the resonant potentials. 3. L12 "':' Wr < Pnode >. The probabilities to change the potential or to remain in the same one upon crossing a node are of the same order. In this regime one may expect proliferation of components of the atomic wave packet and strong complexification of the atomic wave packets. Let the average initial atomic momentum and the recoil frequency be fixed, P = 55 and Wr = 10- 3 , respectively. So, Wr < Pnode >"':' 0.055 and we will change the detuning in order to investigate all the three cases. At L1 = 1, one gets L12 » Wr < Pnode >. This is the first case with negligibly small values of the probability of nonadiabatic transitions. At L1 = 0.2, we have the relation L12 "':' Wr < Pnode >, corresponding to the third case with nonadiabatic transitions at the nodes of the standing wave. To satisfy to the second condition, L12 «wr < Pnode >, one can work at exact resonance, L1 = O.

In the first case one expects a comparatively simple evolution of the atomic wave packet. Fig. 2a demonstrates that the initial wave packet splits from the beginning to a few components because the initial ground state is a superposition of the dressed states (4). The atomic kinetic energy is enough to perform a ballistic bipotential motion. The momentum changes in a comparatively small range, from 40 to 70 photon-momentum units. The packet does not split at the nodes but it, on the contrary, recollects in the momentum space at the nodes after initial spreading. However, this recollection smears out in course of time. To be sure that the wave-packet evolution is essentially nonadiabatic, we go to the position space and compute the probability Ic(x)12 to find the atom at the position x in the potential E~-) in the frame moving with the mean initial atomic velocity Po = 55. Fig. 2b demonstrates

jitomie de Broglie- Wave Chaos

373

Fig. 2. Adiabatic wave-packet motion in the nonresonant potentials at Ll. "'~ 1 in the momentum and position spaces. (a) Momentum probability tV(p,T) and (0) position probability Ic_(;:r)1 2 to find the atom in the potential E~). The color codes the corresponding probability values.

clearly the absence of nonadiabatic transitions in the position space at the llodes which are indicated by the slope lines in the moving frame.

Fig. 3 . 'Nave-packet motion at exact resonance L1 position spaces.

o in

t.he momentum and

Fig. 4 . Nonadiabat.ic wave-packet motion and transitions between the nonresonant potentials a.t L1 = 0.2 in the momentum a.nd position spaces.

Figure :3a shows evolution of the Gaussian waVG packet in the mornentum space at exact resonance (L1 = 0) when L12 «w,. < Pnode >. The color codes

374

V. O. Vitkovsk}; L. E. Konkov, and S. V. Prants

the values of the mornentmn probability. It follows from (5) that the initial ground atomic state is now an equal superposition of the dressed states. The initial wave packet splits into a mimber of components. The components with the momentum increasing from the beginning move ballistically in the potentia.! E~+). The components with the momentum decreasing from the beginning have not enough kinetic energy to overcome the barrier of the first potential well and they oscillate in the potential E6-) between the nodes at :r = 'if /2 and :r = - I I /2. Thus, we observe the quantnm phenomenon: atom moves in the positive direction over the hills of the optical potential and oscillates simultaneously in the well of this potential. It should be stressed that the period of the resonant potentials is twice as large as the period for the nonresonant ones. The resonant potentia.ls cross each other at the nodes (see Fig. 1). In the position space we observe Landau-Zener transitions at the nodes with the unit probability as it is given by the formula (8). The color in Fig. :)b codes the values ofthe probability (;r Wto find the atom in the potential E~-).

w

0.1

.--~--~---,

60.------------~

0.075

40 0.05 20

0.025

a)

o ...... o

_.L..-'-.;...:...:~___'___>_'

30

60

P

90

200

400

1: 600

Fig. 5. (a) Momentum probability distributions vV at the fixed time moment in the nonadiabatic (bold curve, Li. = 0.2), adiabatic (dashed curve. Li. = 1), and resonant (dotted curve, Li. = 0) c&ses. (b) Mean atomic momentum VB time in the nonadiabatie (bold lower curve, Li. = 0.2), adiabatic (bold upper curve, Li. = 1), and resonant (dotted curve, Li. = 0) eases.

At Ll = 0.2, the atomic ground state is the following superposition of the dressed states: 11) c:: 0.741 +).c, + 0.661- ).c,. The part of the initial wave packet, starting from the top of the potential E~+) , overcomes the barriers of that potential and moves in the positive direction of the axis :r proliferating at the nodes. The larger part of the initial packet, starting from the bottom of the potential E~-), will be trapped in the first well of that potential. Only its small part is capable to overcome the barrier. To illustrate that we show in Fig. 4 evolution of the initial Gaussian wave packet. As compared to Fig. 3a, we see an interference-like picture in the InomentulYl space and dear evidence of the Ilonadiabatic transitions between the nonresonant poten[,ials

Atomic de Broglie-Wave Chaos

375

in the position space. This results in a proliferation of the components of the atomic wave packet at the nodes of the standing wave. Figure 5a shows the momentum probability distributions W at the fixed time moment T = 30 in the nonadiabatic, adiabatic, and resonant cases. The distribution is narrower in the adiabatic case than in the other ones. The peaks of Ware more pronounced in the nonadiabatic case than in the resonant one. Figure 5b shows the dependence of the mean atomic momentum < P > on time for the three cases. This quantity oscillates in the adiabatic case in much nore narrow range than in the other cases where < p > is a wildly oscillating function.

3

Quantum-classical correspondence

In the semiclassical approximation, one treats the electronic degree of freedom quantum mechanically and the translational motion classically. The problem of motion of a point-like atom in a standing-wave laser field has been studied in detail in Refs. [6,1]. The five-dimensional Hamilton-Schrodinger equations of motion have been deduced. Along with two integrals of motion, they constitute a Hamiltonian autonomous system with two degrees of freedom moving in the phase space on a three-dimensional hypersurface with a given energy value. A number of nonlinear dynamical effects have been found in Refs. [6,1]: chaotic Rabi oscillations, Hamiltonian chaotic atomic transport and dynamical fractals, Levy flights and anomalous diffusion. One of the impressive effects is a random-like motion of an atom in a rigid optical lattice when the atom can fly through the optical potential in one direction, be trapped in its wells for a while and then continue to fly over its hills in the other direction [1]. This and the other effects are caused by local instability of the center-of-mass motion in a laser field. A set of atomic trajectories under certain conditions becomes exponentially sensitive to small variations in initial quantum internal and classical external states or/and in the control parameters. A detailed theory of Hamiltonian chaotic transport of atoms in a laser standing wave has been developed in [1]. The maximal Lyapunov exponent ,.\ characterizes the mean rate of the exponential divergence of initially close trajectories and serves as a quantitative measure of dynamical chaos. In Fig. 6a we show the dependence of ,.\ on the detuning Ll for an initially deexcited atom with the fixed value of the recoil frequency Wr = 10- 3 and initial conditions: Xo = 0 and Po = 55. As a measure of complexity of the wave packets, we take a number of local extrema N in the distribution function W(p) on a large time interval. Figure 6b plots the dependence of this quantity on the value of the detuning Ll. Comparing this quantum result with the semiclassical one in Fig. 6a, we see rather good correlation. It follows from the semiclassical simulation that the maximal Lyapunov exponent is zero nearby the resonance and when the detuning is large enough providing a regular point-like atomic motion.

376

V. O. Vitkovsky, L. E. Konkov, and S. V. Prants 0.04

A

,

"")\'

'(

O.Q3,

0.02

(

\

a) .~

\

""L", "' ~~"J

2---:O:-:.4---:0:-:.6---:0'"'.S-L\---'1 "O':----C0:-::.

Fig. 6. (a) Maximal Lyapunov exponent A with the classical atomic motion and (b) number of local extrema in the momentum distribution with the quantized atomic motion vs the detuning .:1.

Positive values of A mean chaotic atomic motion. Quantum evolution is rather regular in the same ranges of the detuning and it is complicated in that range of the detuning where the corresponding classical atomic motion is chaotic. We have shown that, varying the Rabi frequency or the detuning, it is possible to some extent to control propagation of de Broglie waves creating regular and chaotic wave packets. This work was supported by the RFBR (project no. 09-02-00358) and by the Program "Fundamental Problems of Nonlinear Dynamics" .

References l.V.Yu. Argonov and S.V. Prants. Theory of chaotic atomic transport in an optical lattice. Physical Review A, 75:art.no. 063428 , 2007. 2.R. Graham, M. Schlautmann, and P. Zoller. Dynamical localization of atomicbeam deflection by a modulated standing light wave. Physical Review A, 45:R19, 1992. 3.W.K. Hensinger, N.R. Heckenberg, G.J. Milburn, and H. Rubinsztein-Dunlop. Experimental tests of quantum nonlinear dynamics in atom optics. Journal of Optics B, 5:R83, 2003. 4.A. P. Kazantsev, G. 1. Surdutovich, and V. P. Yakovlev. World Scientific, Singapore, 1990. 5.D. Makarov, S. Prants, A. Virovlyansky, and G. Zaslavsky. World Scientific, Singapore, 2009 . 6.S. V. Prants. Chaos, fractals and flights of atoms in cavities. JETP Letters, 75:651, 2002.

377

Chaotic Behavior of Plasma Surface Interaction. A Table of Plasma Treatment Parameters Useful to the Restoration of Metallic Archaeological Objects C. L. Xaplanteris,·2 and E. Filippaki2 , Hellenic Military Academy, Vari Attika, Greece 2 Plasma Physics Lab, IMS, NCSR "Demokritos", Athens, Greece (e-mail: [email protected]) Abstract: In Plasma Physics laboratory of N.C.S.R. "Demokritos" the plasma chemistry method has been used for the restoration and conservation of metallic archaeological objects during the last decades. The obtained experience had led us to conclude that plasma parameters and different status of treated objects are so specific, so as to become unique. In the present paper the theoretical and experimental results of our laboratory are summarized. A treatment table of plasma parameters is given, which claims to be useful for the conservators. It is obvious that this treatment table needs to be completed and extended, so that it meets the uniqueness of each artifact. A theoretical study and the treatment of a variety of iron objects are presented. Keywords: plasma sheath potential, plasma restoration, corrosion, external potential, plasma parameters.

1. Introduction Three decades have passed since Daniels and coworkers used for first time the mechanical and chemical action of the plasma [1] to reduce the tarnish of silver on daguerreotypes. During the 80's, Veprek and coworkers systematized the plasma cleaning and restoring method [2-4], by creating the homonymous plasma reactor, where a low-pressure hydrogen plasma acts upon the artifacts. These procedures were very soon adopted by the Plasma laboratory of NCSR "Demokritos" due to the many archaeological artifacts, which are found at the Hellenic area. The plasma treatment processes are usually preferred, for the following reasons: the plasma temperature is low and consequently, the destruction risk of artifacts is avoided; the chemical reactions produce steady compounds on the objects (e.g. magnetite for iron objects), which consist the restorating behavior of plasma; finally, the plasma reactor volume allows the simultaneously treatment of many objects. Many experiments and studies have been carried out [5, 6], at the two plasma reactors of "Demokritos" since the early 90's. Recently, the utilization of an external d.c. electric current on the treated objects have showed very good results [7, 8]. The whole previous meditation leads to the conclusion that plasma cleaning proceedings are chaotic due to multi-parametric behavior of plasma and the complicated corrosion of metallic objects. Thus, the factors which affect the cleaning and restoring can be listed as below.

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•

The plasma production way, the nature of plasma gas, accompanied with its parameters (plasma density, temperature, pressure ect). • The kind of metal, and its metallurgy. • The type of corossion and the corossion rate which are depended on the burial environment of the objects and on the time of their burial. The purpose of the present paper is to classify the previous results, to add new ones from the last experimental results and to draw up a table of treatment methods useful to the restorators. The paper is organized as following: the new experimental studies are presented in Sec.2; the precedent results, many of which were studied in our laboratory [5-8], are briefly described in Sec. 3 and in SecA treatment tables are given, the main results are summarised and concluded.

2. New experimental results Beyond the previous work which has been done during the last five years, the plasma treatment was extended by the enforcement of the d.c potential on restorated objects. In this way, a d.c current passes both through objects and plasma, and proves that the Langmuir probe theory which basically is valid, must be modificated. It is known that the rf plasma production method gives to the electrons much more acceleration than the ions because of the very small electron mass. Furthermore, the electron energy loss by the collisions with ions or neutrals, is negligible due to the small value of the ratio me /

1m; . In this

way, the electron

temperature increases considerably (10.000-100.000 K O), while the ion and neutral temperature remains at low levels (about 500 K Ofor ions and 300 K Ofor neutrals). Consequently, plasma is not in thermodynamic equilibrium, while the high electron temperature makes plasma more effective for the oxide reduction. When none external potential is imposed on the treated objects, the plasma sheath theory is valid as it was proved in an extented previous work [7,8]. However, if an external d.c. potential is enforced on the objects, as in our experiment, the Langmuir probe theory is valid with the presupposition that the following thoughts are taken into consideration. Firstly, when a voltage is applied on the treated object with reference to the earth, the simple probe with very large surface is formed . From plasma sheath theory [7, 8] is known that the ion current density to the object surface, is ji = no ·e.c s (1) with no the plasma number density and C s is the ion sound speed. It is possible to approach, 2_

c,

Te + T;_Te_

2

= - - - = - = Vo

mi

mi

where Vo is the ion velocity at the sheath edge.

Chaotic Behavior of Plasma Surface Interaction

Similarly,

the

electron

current

density

to

the

object

surface,

379 IS

e¢(x)

je =

-~4 . n e . e . ce'

Taking into consideration that

ne = no' e

T,

and

the electron current density is JlJn e

je = -

L

no . e . ( STe

4

)

Yz 2.

e¢(x)

e

T,

(2)

JlJn e

The current I to the object (probe) is I = (Ji + j e)' S where S is the projected surface area into the plasma. By substitution of ion and electron current density from (1) and (2), the current I which is biased to voltage V is (3)

Vf is the floating potential, which is equal to the external enforcemed potential,

so that the current I vanishes. Since the object (probe) dimensions are large compared to the ion and electron gyro-radii and the Debye length, the area S is determined as following: S

f is .

= ~ . B.

When the probe is biased negatively enough, all ellectrons are repulsed while what remains is the ion current. This current is not affected by the voltage and is known as ion saturation current. The ion current is saturated because the bias potential gathers all ions across the surface sheath and the ion flux is determined by the flux of ions which enter into the sheath at the ion sound velocity. When the probe potential becomes larger than the plasma potential, there is no sheath and the electrons strike on the object surface with the thermal distribution of velocities. In this case the electron current is saturated by value I es =

~ noece . S, with Ce the electron thermal velocity. 4

Figure 1 shows the typical probe characteristic (I versus V), which presents three regions: the ion current saturation, the exponential electron distribution and the electron current saturation.

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C. L. Xaplanteris and E. Filippaki

sma

ion saturation io V

v

f

v p

p

e ponential region y-recorder

eo

electron saturation

Figure 1. The Langmuir probe and its characteristic.

If plasma is non-magnetized (as the plasma we use to treat corroded objects), it follows the above simple theory; the characteristic I - V draws the two saturation regions with the ratio of electron to ion saturation current is equal to ratio of electron thermal velocity to ion sound speed. However, in our experiment the probe's metallic surface into the plasma has been replaced with the treated objects, which have much more area than the probe's one. In spite of this big difference the Langmuir probe theory is valid in a very satisfactory way, and the measurements give currents proportional to the object area. Figure 2 shows the d.c. current, which passes through the objects and the plasma. As Figure 2 shows, a big ohmic resistance R is necessarely lined at the circuit to reduce the current I , which is measured on the value of some Amperes. The negatively charged object (cathode) attracts plasma ions, which strike on the surface with increased velocity; then, while the ions take free electrons from the cathode, reductive reactions take place as following:

3Fe203 +2H+ =? 2Fe304 +H 20

or

Fe203 +2H+ =? 2FeO+H 20

On the contrary, the positively charged object (anode) attracts the plasma electrons, which strike on the surface and reduce the oxides as the reactions show: 6Fe203 + 4e- =? 4Fe304 + 02

or

Fe203 + 2e- =? 2FeO +

h 02

Chaotic Behavior of Plasma SU/jace Interactioll

381

From the above chemical reactions it is obvious that the extemal d.c. potential makes the plasma much more active and reduce the object oxides to more stable compounds (magnetite) as the XRD examination can confirm.

plasma

Figure 2. The external dc potential circuit during plasma treatment.

3. Previous results Many inferences have been taked out from the early experimental work in our laboratory, as the plasma cleaning method is in operation during the last

two decades. An excavated nail fron the Holy Monastery of Simonopetra of Mount Athos before and after plasma treatment is shown in Photo 1.

Photo 1. An excavated nail before and after plasma treatment.

The factors and parameters, which affect the plasma cleaning, are too many and as a result, it is impossible to mentjon them without the risk of forgetting one. The present work has the ambition to start the drawing of some treatment tables in order to achieve the classification of the chaotic state. Firstly,

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the kind of the material used e.g. iron, copper, silver e.t.c. are presented; secondly, the corrosion depth, which is separated in three sizes for practical reason: lightly corroded, moderately corroded and heavy corroded; the corrosion depth depends on the time and the environment where the object is being corroded. Another parameter, which strongly affects the cleaning, is the plasma gas; gases such as H 2' N 2' CH 4 as combinations of various gases were used, except for N 2 and CH 4 due to identification of CN- on the pyrex surface. Along with the plasma gas, the other plasma parameters, as plasma pressure, the plasma density, the rf absorbed power and the ion temperature, are listed. Consequently, the treatment time is a remarkable factor; practically, three period of treatment times were used, periods of lOhours, 1O-20hours and a period that exceed 20 hours of treatment.

4. Summation-elaboration tables The factors and parameters, which were mentioned in Sec.3 have been tabulated on cleaning Tables I, II and III.

Table I. Plasma treatment table for iron objects

Light corroded

Iron obJects Medium corroded

Heavy corroded

Gas Pressure (Torr) Power (kW) Time Temperature

H2 0.8 0.7 8h-IOh 200°C to 300°C

H2 0.9 1 to 1.2 lOh-20h 250°C to 350°C

H2 0.7 0.7 more than 20h 200°C

Gas Pressure (Torr) Power (kW) Time Temperature

H2, N2 0.4,0.2 0.8 8h-10h 200°C to 300°C

H2,N2 0.8,0.4 1.1 to 1.2 lOh-20h 250°C to 350°C

H2, N2 0.4,0.2 0.7 more than 20h 200°C

Gas Pressure (Torr) Power (kW) Time Temperature

N2 0.7 0.8 more than 20h 200°C to 300°C

N2 0.9 1.0 10h-20h 250°C to 350°C

N2 0.5 0.5 more than 20h 200T

Chaotic Behavior of Plasma Surface Interaction

383

Table II. Plasma treatment table for bronze objects

Gas Pressure (Torr) Power (kW) Time Temperature

Light corroded

Bronze objects Medium corroded

H2 0.8 0.65 4h-6h 170' C-190'C

H2 1.2 1.0 2h-4h 230'C-240'C

Heavy corroded H2

Table III. Plasma treatment table for silver objects

.___________________________________________________________________~!!~~!:_~~J~~_~~______________________________________ _ Gas Pressure (Torr) Power (kW) Time Temperature

Light corroded

Medium corroded

Heavy corroded

H2 0.8 0.7 2h-4h 21O'C

H2 0.8 0.7 6h-8h 21O'C

H2 0.8 0.7 lOh-12h 21O' C

It is obvious, that the above tables need to be expanded not only for other kinds

of material s but also by taking into account other parameters which were mentioned in section 1, so that a useful plasma treatment manual for restorators will be completed. From the table I, which concerns the treatment of the iron objects we can see that the state of the corrosion of the object (light to medium or heavy corroded objects) is critical for the duration and for the temperature of the plasma treatment. So, as far as the light corroded objects concerns, when treated either with hydrogen or a combination of hydrogen and nitrogen, the temperature varies between 200-300 degrees of C and the duration 8-lOhours, for the medium corroded objects the temperature is between 250-300 degrees of C and the duration varies between 1O-20h. For the heavy corroded objects the temperature does not exceed 200'C and the duration is more than 20h. The lower temperature is essential in order to preserve the integrity of the object as well as the metallographic characteristics of the metal itself. If the treatment gas is nitrogen then the treatment duration is much longer for all the three categories of the corrosion state. The second table concerns bronze objects treated with hydrogen plasma. The duration of treatment varies between 2-4 hours and the temperature between 170 to 240' C. As far as silver objects concerns, the temperature is always the same 21O' C, while the duration varies between 2-12 hours depending on the corrosion state of the object.

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Evaluating the above results we assume that 1. Gas, temperature and time are the key parameters of the plasma process. 2. The extend of the corrosion layer is essential on the plasma duration and the temperature of plasma treatment. 3. The plasma treatment at all conditions is successful in decreasing the density of the corrosion products and therefore facilitates subsequent mechanical cleaning. The gradual elimination of the chlorine containing corrosion products in favor of the formation of more stable species and sometimes even the complete reduction back to metal is proportional to the duration of the plasma treatment. It should be taken into account that in case of heavy corroded objects a longer treatment in lower temperature is preferable to a shorter treatment in higher temperature as it does not make the object too brittle.

References [I] Daniels V., Plasma Reduction of Silver Tarnish on Daguerreotypes, Stud. In Conserv, 26,1981.

[2] Veprek, S., Patscheider, J. and Elmer, J., Restoration and Conservation of ancient artifacts: A new area of application of plasma chemistry, Plasma Chern. Plasma process, Vol. 5(2), 1985. [3] Patscheider, J. and Veprek, S. Application of low pressure Hydrogen plasma to the conservation of ancient iron artifacts, Stud. In Consern., Vol. 31,1986. [4] Veprek, S., Eckmann, Ch. ,and Elmer, 1. Th., Recent progress in the restoration of archaeological metallic artifacts by means of low-pressure plasma treatment Plasma Chern. Plasma process. Vol. 8 (4), 1988. [5] Kotzamanidi I., Sarris Em., Vassiliou P.,Kollia c., Kanias G.D., Varoufakis G.l. Filippakis S.E., Corrosion behavior of oxidized steel treated in reducing plasma environment-Chloride ions removal, British Corrosion Journal, 34(4), 285-29 I, ( 1999). [6] Kotzamanidi, I. Vassiliou, P. Sarris, Em. Anastasiadis, A., Filippaki, E., Filippakis, S.: Anti-corrosion methods and materials, Vol 49, No 4, 2002. [7] Xaplanteris, c., Filippaki, E. Plasma -surfaces interaction and improvement of plasma conservation system by application of a d.c. electrical potential. Topics on Chaotic Systems, Selected Papers from CHAOS 2008 International Conference, pp. 406-415, 2009. [8] Xaplanteris, C., Filippaki, E. Mechanical and chemical results in the plasmasurface contact. An extended study of sheath parameters. (to be published).

385

Simulation of Geochemical Self-Organization: Acid Infiltration and Mineral Deposition in a Porous Ferruginous Limestone Rock Farah Zaknoun, Houssam EI-Rassy, Mazen AI-Ghoul, Samia AI-Joubeily, Tharwat Mokalled, and Rabih Sultan Department of Chemistry, American University of Beirut P.O. Box 11-0236, 11072020 Riad EI-Solh, Beirut, Lebanon (e-mail: [email protected]) Abstract: The phenomenon of Liesegang banding finds its most striking natural similarity in the band scenery displayed in rock systems. Whereas theoretical modeling studies are extensive in the literature to simulate the reaction-transport dynamics of the medium, experimental simulations in-situ are very scarce. We carry out an empirical observation of the evolution of a deposition pattern embedded in a rock. A sulfuric acid solution is infiltrated into a porous ferruginous limestone rock by means of an infusion pump. This acidization causes the dissolution of the calcite (CaC0 3) mineral and deposition of calcium sulfate in the form of anhydrite (CaS04) and mainly gypsum (CaS04'2H20), behind the advancing dissolution front. We observe the growth of such a deposition pattern and investigate the extent of its resemblance with the well-known Liesegang patterns obtained in the laboratory. The composition of the seemingly banded zones is analyzed by powder X-ray diffraction, both qualitatively and quantitatively. The irregular shape of the band zone boundaries seems to be of fractal nature. Keywords: Liesegang, acidization, acid-infiltration, banding in rocks, ferruginous limestone, fractals.

1. Introduction In this paper, we shed light on the similarities between the band formation in rocks and the well-known Liesegang banding phenomenon [1,2]. Liesegang precipitation bands are obtained when co-precipitate ions interdiffuse in a gel medium due to a cycle of supersaturation, nucleation and depletion [3], coupled to the on-going diffusion process. Beautiful bands of precipitate appear as a result of this complex physico-chemical scenario. Representative specimens of colorful banded Liesegang patterns are displayed in Fig. 1, for a variety of precipitate systems. Liesegang systems share a great deal of similarity with natural systems displaying striped patterns and ripple morphology, notably in Biology and Geology. Biological rhythmic patterns are manifest in the stripes on tigers and zebras, belts of bacteria colonies [4,5], gallstones [6] and cysts. In Geology, agates and banded rocks present a typical example of rhythmic patterns [7-13], and are perhaps the closest widespread natural phenomenon to the Liesegang stratification, because they directly involve precipitation and

386

F. Zaknoun et al.

mineral deposition. Pictorial scenes of periodic banding in real natural geological systems are displayed in Fig. 2.

Figure 1. Liesegmlg precipitation patterns grown in gel for a variety of precipitate systems.

Between the rapid growth of bacterial colonies and the tremendously slow formation of rock patterns, we realize that self-organization can take place on a huge range of time scales (minutes to millions of years).

a.

b.

c.

Figure 2. Patterns of bands in natural stone systems (geology). a. Liesegang stone, a highly silicated banded rhyolite; b. an agate of malachite, hydrous copper carbonate; c. donut agate.

In the present study, we investigate the phenomenon of band formation in rocks by simulating one possible scenario that would take place in a real, rock medium. Instead of focusing on the study of existing, "ready made" banded patterns in rock specimens, we force and steer the band deposition inside a rock bed, by simulating the physico-chemical processes which lead to such deposition, thus exactly mimicking the real natural scenario. This is done by infiltrating a ferruginous limestone rock with sulfuric acid, causing the dissolution of the bare rock calcite (CaC0 3) medium, followed by the deposition of anhydrite (CaS04) or gypsum (CaS04'2H20), behind the dissolution front: Cae0 3 (s) + 2W (aq)

~ Ca 2+ (aq) + CO 2 (g) + H20 (l)

(1)

Simulation of Geochemical Self-Organization

Ca 2+ (aq) +

sol- (aq)

~ CaS04 (s)

387

(2)

Thus, we carry out our experiments in-situ, spanning long times as required by the appearance of the desired patterns. To our knowledge, no investigator tackled this approach to geochemical self-organization before. The study would rather routinely be based on the analysis of existing rock specimens displaying parallel bands, or essentially, on theoretical modeling and numerical simulations. The mechanism involves diffusion and percolation of the infiltrating solution, coupled to chemical reactions driving dissolution, porosity changes and precipitation. The mineral deposition requires a collective scenario of nucleation, as well as competitive growth of both nuclei and larger particles of precipitate. A representative schematic picture of the dynamical events is illustrated (in reduced form) in the following scheme, and portrayed in Fig. 3:

A

~---PAO

space Figure 3. Moving dissolution front of mineral A, infiltrated by aqueous species X, and producing the banded deposition of mineral B.

A + 2X ~ Y Y +Z k2) B

(3) (4)

where A respresents a solid mineral (here calcite, CaC0 3 ), Y the dissolved salt material (Ca2+), from the reaction with X (acidic water, H+), Z the counter-ion from the infiltrating solution (here SOl-) which reacts with Y to form precipitate B (CaS04). Thus this scheme (also illustrated in Fig. 3) is a simplified form of chemical reactions 1 and 2. kJ and k2 are rate coefficients used in the numerical solution of the mathematical model adopted in Ref. 14.

2. Experimental Section 2.1. Acid-Infiltration Experiments A red brown, porous, ferruginous limestone rock is used in the experiment. A relatively high porosity is an advantage in our experiments as less time is required for the acid percolation inside the rock. The acid was infiltrated into the rock by means of an infusion pump (M361 Multi-rate Infusion Pump).

388

F Zaknoun et al.

The pump syringe was filled with 50 mL of 4.3 M H2S04, which were delivered at the rate of 1.0 mL per hour, through a piece of teflon tubing fitted into the tiny orifice. As this type of rock is weak and brittle, it was neccessm'y that a small orifice, 3 mm diameter and 1 cm depth, be cm'efully drilled in a way to allow the insertion of the il~jection tube. With this setup, a well-controlled flow is attained. The pump syringe is refilled as soon as the acid is ·consumed. Figure 4 shows pictures of the treated rock at various stages of the infiltration-acidization evolution.

Figure 4. Ferruginous limestone rock acidized with H2S04 through the teflon tube shown. Time evolution of the rock treatment showing the gradual formation of the deposits. The numbers indicate the time in days.

2.2. Preliminary Tests A number of simple preliminary chemical tests are carried out to guide our analysis, while the final rock composition in various band domains is confirmed by X-ray analysis. 1.

A zone rich in CaC03 is indicated by an effervescence upon treatment with 6M HCI, due to the reaction: CaC03 (.I') + 2HCI (aq) ~ Ca2+ (aq) + 2Cl- (aq) + CO2 (g) + H20 (l)

2. 3.

A zone predominantly consisting of CaS04 would show no fizzing upon treatment with HCl. An inter-band zone void of CaS04 showing no reaction upon addition of HCl is an indication of the depletion of the bare rock medium of any carbonate, due to prior consumption by the originally infiltrating acid (H2S04), This would leave only the other minerals supporting the bare rock medium.

Simulation of Geochemical Se(f-O';r;anization

4.

5.

389

A small sample of white salt from a CaS04 ring dissolved in HCI, yields a yellow solution, attributed to the presence of ferric ions originating from the rock medium (ferruginous limestone). The presence of Fe3+ is confirmed by testing with KSCN solution. Upon addition of a few drops of SCN-, the solution turned blood-red indicating the formation of the FeSCN 2+ complex. Flame atomic absorption analysis of the precipitate (done on a Thermoelemental Solaar atomic absorption spectrophotometer), in CaS04-rich regions (after dissolution in HCI) was carried out to determine the elemental mass ratio of calcium to iron [14].

2.3. Composition Analysis of the Treated Rock

The treatment was fulfilled in a series of experiments, the latest of which was extended over a period of about one year and eight months. In the latter (most recent) expeliment, the H2S04 infusion started on October 5, 2005 and was stopped on May 23, 2007. The final pattern was examined to delineate the various regions wherein CaS04 bands may have formed. The pattern span area was divided into eight main textural regions, as highlighted in Fig. 5.

Figure 5. Final rock pattern at t:= 692 days. The rock was partitioned into eight "banded" zones for differentiation by X-ray analysis.

Powder samples from each region were taken and placed · in small tubes, which were stoppered thereafter. Each sample was analyzed by powder X-ray diffraction. Each diffractogram was mapped onto a Library of standard diffractograms for the calcite, gypsum and anhydrite minerals. Mter the peaks for each mineral were identified, the relative mass proportions of each mineral in the sample mixtures were determined by integrating the area under the peaks of each mineral [15]. The average composition of every region in each of the three minerals (over a representative number of samples) could then be calculated.

390

F Zilknoun et al.

3. Results and Discussion 3.1. Stage I At a first stage, the analysis of a treated ferruginous limestone rock was performed in a preliminary experiment wherein the H2S04 infiltration was stopped at time t = 161 days.

Figure 6. CaS04 deposition in the limetsone rock sample, infiltrated by H 2 S0 4 . The central region is a dense CaS04 precipitate zone. Then, beyond a brown depleted zone, a distinct belt of CaS04 suggests the formation of a first Liesegang ring. TIle photo is taken on day 161. A 1 em scale is placed on the picture.

The analysis of the white deposits in the central region and the ring shown in Fig. 6 confirmed the dominant presence of the gypsum mineral [14J. Photomicrographs of the deposit areas (notably in the ring region), before and after infiltration, depicted in Fig. 7, show the white crystals of the gypsum mineral embedded in the rock medium.

Figure 7.a. Rock texture before the acid injection. Magnification (lOx). The scale is indicated on the picture.

Fig. 7.b. Rock texture after acid injection, displaying the white crystals of CaS04 (labeled W) emhedded within the rock. Magnification (lOx).

Figure 7.c. Rock texture in the central region wherein the rock is entirely "stuffed" with CaS04.

Simulation (i Geochemical Self-Organization

391

3.2 Stage II The results of the experiment described in Sect. 2.1 (analysis procedure described in Sect. 2.3) are now presented. Samples from each of the regions shown in Fig. 5 were taken by drilling small holes in the desired area, collecting the resulting powder in small vials and storing them for later X-ray analysis. The diffractograms confirmed the formation of gypsum (CaS04'2H20), which was quite dominant over anhydrite (CaS04, anhydrous) as the main deposit product in reaction 2. The central region (domains 1, 2 and 3) consists almost entirely of gypsum. Then depletion starts in region 4 (almost exactly like in a real Liesegang pattern wherein bands start separating from a uniform zone that extends over a relatively wide spatial domain) . The % composition in gypsum (calculated for the relative mass proportions of gypsum and calcite exclusively) drops to 28.4 % (see Table 1, and notably Fig. 8). 120

Region

% gyps.

1

100

..,::I

2

97.5

Q)

3

98.9

E 100

~

80

~

60

4

28.4

40

5

85.8

20

6

17.6

7

5.4

0 0

2

3

4

5

6

7

8

Region

Table 1. Variation of % gypsum over the rock regions.

Figure 8. Variation of % gypsum over the different rock regions.

Although some gypsum is present, it has dropped significantly in favor of unreacted calcite, suggesting the residence of an inter-band zone, relatively depleted in the diffusing sol-, just as predicted by the Ostwald cycle [3]. Then comes a new gypsum band (region 5; 85.8% gypsum) and so on. Hence, the % composition in gypsum essentially alternates. The complete depletion of gypsum between the bands (no 0% zones) is rendered impossible by the complex random percolation of the infiltrating acid (not a simple diffusion process), given the broad distribution of the pore sizes. The unexpectedly low composition in the last point (instead of an increase; see Fig. 8) is due to the incomplete formation of the gypsum band at that stage.

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4. Fractal Reaction Front Contours We now focus on the shape of the front contours in the different regions. The shape of the acidization front is to a large extent governed by porosity variations [16,17]. As a result, irregularities emerge, and thus may give rise to fractal structures. We analyze the front contours using Matlab and the Image Processing Toolbox. After choosing a specific region (from domains 1-8), the RGB image is cropped, and the region of interest is retained. The interface between the two corresponds to the contour of the chosen region. We used color segmentation using the L*a*b color space whereby we select the color of the contour to a tolerance of 10%. The plate was then transformed into a binary image. Morphological operations were afterwards applied to fill any open holes. Contours of regions 8 (the outer front) and 4 delimited by this method are shown in Fig. 9.

Figure 9.a. Contour of region 8, of fractal dimension 1.87.

Figure 9.b. Contour of region 4, of fractal dimension 1.44.

The fractal dimension using the box count algorithm was computed and found to be 1.87 for region 8 (frame. 9.a) and 1.44 for region 4 (frame. 9.b). This study is under continuing investigation, wherein all the contours will be meticulously traced and analyzed.

5. Conculsions In this paper, we simulated a Liesegang banding expeirment in-situ, i.e. inside a real rock bed. After a long treatment with acid infiltrated from a central source, band formation was induced in a way similar to what we see in a naturally banded rock. X-ray analysis revealed the alternation of minerals, thus resembling a 2D Liesegang pattern. The contours of the acidization fronts were found to be fractal.

Simulation of Geochemical Self-Organization

393

Acknowledgements: This work was supported by a grant from the Lebanese National Council for Scientific Research (LNCSR). The authors thank Dr. Abdel-Fattah Abdel-Rahman, for his valuable and very helpful advice. All the X-ray and AA measurements were performed in the Central Research Science Lab (CRSL) of the American University of Beirut.

References [1] R. E. Liesegang, Chemische Fernwirkung, Lieseg. Photograph. Arch. 37: 305, 1896; continued in 37: 331,1896. [2] H. K. Henisch, Crystals in Gels and Liesegang Rings, Cambridge University Press, Cambridge, 1988. [3] Wi. Ostwald, Lehrbuch der Allgemeinen Chemie, 2. Aufl., Band 2, 2. Teil: Verwandtschaftslehre, Engelmann: Leipzig, 1896-1902, p. 778. [4] J. A. Shapiro and M. Dworkin, Eds., Bacteria as Multicellular Organisms, Oxford University Press, New York, 1997. [5] V. V. Kravchenko, A. B. Medvinskii, A. N. Reshetilov and G. R. Ivanitskii, Dokl. Akad. Nauk, 364: 114,1999; ibid, Dokl. Akad. Nauk" 364: 687,1999. [6] D. Xie, J. Wu, G. Xu, Q. Ouyang, R. D. Soloway, and T. Hu, 1. Phys.Chem. B, 103: 8602, 1999. [7] R. E. Liesegang, Geologische DijJusionen, Steinkopff, Dresden, 1913; R. E. Liesegang, Die Achate, Steinkopf, Dresden-Leipsig, 1915. [8] T. Moxon, Educ. Chem. July issue, 105,2001. [9] T. Moxon, Agate: Microstructure and Possible Origin, Terra Publications, Doncaster, 1996. [10] J. H. Kruhl, Ed., Fractals and Dynamic Systems in Geoscience, Springer-Verlag, Berlin, 1994. [II] P. Ortoleva, Geochemical Self-Organization, Oxford University Press, New York, 1994. [12] M. Ohkawa, Y. Yamashita, M. Inoue, R. Itagawa and S. Takeno, Mineralogy and Petrology, 70: 15,2000. [13] B. Jamtveit and P. Meakin (Eds.), Growth, Dissolution and Pattern Formation in Geosystems, Kluwer, Dordrecht, 1999. [14] M. Msharrafieh, M. AI-Ghoul and R. Sultan, in Complexus Mundi: Emergent Patterns in Nature, Proceedings of the Fractals 2006 Conference, Vienna, Austria; M. M. Novak, Editor; World Scientific, Singapore, 2006, pp. 225-236.

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[15] C. Suryanarayana and M. Grant Norton, X-Ray Diffraction, A Practical Approach, Plenum Press, New York, 1998; Experimental Module 7, Quantitative Analysis of Powder Mixtures, pp. 223-236. [16] K. Lund, H. S. Fogler and C. C. McCune, Chern. Eng. Sci. 28: 691-700,1973. [17] K. Lund and H. S. Fogler, Chern. Eng. Sci. 31: 373-380, 1976.

395

Author Index Akkaya E. E., 93

Hacinliyan A. S., 93

AI-Ghoul Mazen, 385

Haller George, 294

AI-Joubeily Samia, 385 Alomari Majdi M., 1 Aniszewska Dorota, 9

Iliopoulos A. c., 268

AwrejcewiczJan, 17, 139

Jevtic N., 101

Aybar O.

b., 93 Karakatsanis L. P., 268

Baehr Christophe, 27

Khots Boris, 111

Becerra Alonso David, 35

Khots Drnitriy, 111

Boutalis Yiannis, 346

Kolesnikov Alexander A., 119

Brianzoni Serena, 43

Kolesnikov Anatoly A., 128, 134

Byrne G., 147

Konkov L. E., 369

Chin Chou Hsin, 215

Kudra Grzegorz, 17

Christodoulou Manolis, 346

Kut;;beyzi

Krysko V. A., 139

Commendatore Pasquale, 59

t, 93

Kuzmenko Andrew A., 134 Kuznetsova E. S., 139

Daniels S., 147 De Campos Cristina, 51

Lan Boon Leong, 165

Dingwell Donald B., 51

Law V. J., 147

Dobrescu Loretti I., 67

Lawrance Anthony J., 155

Dodson C. T. J., 75

Liang Shiuan-Ni, 165

Dowling D. P., 147

Lucarini Valerio, 170

EI-Rassy Houssam, 385

Mammana Cristiana, 43

Ertel-Ingrisch Werner, 51

Manneville P., 185 Mello Pier A., 191

Filippaki E., 377

Mendez-Bermudez J. A., 191

Fraedrich Klaus, 170

Michetti Elisabetta, 43, 59 Mikishev Alexander B., 207

Gopar Victor A., 191

Mitskevich S., 139

Grigoras Carmen, 85

Mokalled Tharwat, 385

Grigoras Victor, 85

396

Author Index

Neamtu Mihaela, 67

Sapsis Themistoklis, 294

Nepomnyashchy Alexander A., 207

Schweitzer J. S., 101

Ni David C., 215

Skiadas Christos H., 302 Smorodin Boris L., 207

Oommen B. John, 284

Sokolov Valentin V., 309

Opri~

Soldatov V., 139

Dumitru, 67

Orman Gabriel V., 224

Sotiropouios Anastasios D., 320, 327 Sotiropoulos Vaggelis D., 320, 327

Pannekoucke Olivier, 27

Srisuchinwong Baniue, 338

Pantelidis Leonidas, 232

Sud K. K., 257

Papageorgiou C. D., 241 Papamarkou Theodore, 249

Temizer L, 93

Patidar Vinod, 257

Tereshko Valery, 35

Pavlos E. G., 268

Theodoridis Dimitris, 346

Pavlos George P., 268

Tsoutsouras V. G., 268

Perugini Diego, 51

Tynan 1., 147

Pinto Antonio, 59 Poli Giampero, 51

Umut Omiir, 362

Prants S. V., 369 Purohit G., 257

Vitkovsky V. 0., 369

Qin Ke, 284

Xapianteris C. L., 377

Rabih Sultan, 385

Zaknoun Farah, 385

Raptis T. E., 241

Zhigalov M., 139

Rybaczuk Marek, 9

Zhu Jian Gue, 1

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