WORLD SCIENTIFIC
ONLINEAR SCIENC Series Editor: Leon O. Chua
HiET HPPLICflTIONS TO LIVING SYSTEMS Erik Mosekilde, Yuri Maistrenko & Dmitry Postnov
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CHAOTIC SVNCHRONIZRTIOI APPLICATIONS TO LIVING SYSTEHS
WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.
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Series Editor; Leon O. Chua
Erik Mosekilde The Technical University of Denmark
Yuri Maistrenko National Academy of Sciences, Ukraine
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PREFACE The cooperative behavior of coupled nonlinear oscillators is of interest in connection with a wide variety of different phenomena in physics, engineering, biology, and economics. Networks of coupled nonlinear oscillators have served as models of spatio-temporal pattern formation and simple forms of turbulence. Systems of coupled nonlinear oscillators may be used to explain how different sectors of the economy adjust their individual commodity cycles relative to one another through the exchange of goods and capital units or via aggregate signals in the form of varying interest rates or raw materials prices. Similarly, in the biological sciences it is important to understand how a group of cells or functional units, each displaying complicated nonlinear dynamic phenomena, can interact with each other to produce a coordinated response on a higher organizational level. It is well-known, for instance, that waves of synchronized behavior that propagate across the surface of the heart are essential for the muscle cells to act in unison and produce a regular contraction. Waves of synchronized behavior can also be observed to propagate across the insulin producing beta-cells of the pancreas. In many cases the individual oscillators display chaotic dynamics. It has long been recognized, for instance, that the ability of the kidneys to compensate for variations in the arterial blood pressure partly rests with controls associated with the individual functional unit (the nephron). The main control is the so-called tubuloglomerular feedback that regulates the incoming blood flow in response to variations in the ionic composition of the fluid leaving the nephron. For rats with normal blood pressure, the individual nephron typically
vi
Preface
exhibits regular limit cycle oscillations in the incoming blood flow. For such rats, both in-phase and antiphase synchronization can be observed between adjacent nephrons. For spontaneously hypertensive rats, where the pressure variations for the individual nephron are highly irregular, signs of chaotic phase synchronization are observed. In the early 1980's, Fujisaka and Yamada showed how two identical chaotic oscillators under variation of the coupling strength can attain a state of complete synchronization in which the motion of the coupled system takes place on an invariant subspace of total phase space. This type of chaotic synchronization has subsequently been studied by a significant number of investigators, and a variety of applications for chaos suppression, for monitoring and control of dynamical systems, and for different communication purposes have been suggested. Important questions that arise in this connection concern the stability of the synchronized state to noise or to a small parameter mismatch between the interacting oscillators. Other questions relate to the form of the basin of attraction for the synchronized chaotic state and to the bifurcations through which this state loses its stability. Recent studies of these problems have led to the discovery of a large number of new phenomena, including riddled basins of attraction, attractor bubbling, blowout bifurcations, and on-off intermittency. In addition to various electronic systems, synchronization of interacting chaotic oscillators has been observed for laser systems, for coupled superconducting Josephson junctions, and for interacting electrochemical reactors. For systems of three or more coupled oscillators, one can observe the phenomenon of partial synchronization where some of the oscillators synchronize while others do not. This phenomenon is of interest in connection with the development of new types of communication systems where one mixes a message with a chaotic signal. Primarily through the works of Rosenblum and Pikovsky it has become clear that even systems that are quite different in nature (or oscillators that have different parameter settings) can exhibit a form of chaotic synchronization where the phases of the interacting oscillators are locked to move in synchrony whereas the amplitudes can develop quite differently. This phenomenon, referred to as chaotic phase synchronization, is of particular importance for living systems where the interacting functional units cannot be assumed to be identical. Kuramoto and Kaneko have initiated the study of clustering in large en-
Preface vii
sembles of interacting chaotic oscillators with a so-called global (i.e., all-to-all) coupling structure. This type of analyses is relevant for instance to economic sectors that interact via the above mentioned aggregate variations in interest rates and raw materials prices. However, biological systems also display many examples of globally coupled oscillators. The beta-cells in the pancreas, for instance, respond to variations in the blood glucose concentration, variations that at least partly are brought about by changes in the cells' aggregate release of insulin. Important questions that arise in this connection relate to the way in which the clusters are formed and break up as the coupling between the oscillators is varied. The purpose of the book is to present and analyze some of the many interesting new phenomena that arise in connection with the interaction of two or more chaotic oscillators. Among the subjects that we treat are periodic orbit threshold theory, weak stability of chaotic states, and the formation of riddled basins of attraction. In this connection we discuss local and global riddling, the roles of the absorbing and mixed absorbing areas, attractor bubbling, on-off intermittency, and the influence of a small parameter mismatch or of an asymmetry in the coupling structure. We also consider partial synchronization, transitions to chaotic phase synchronization, the role of multistability, coherence resonance, and clustering in ensembles of many noise induced oscillators. However, our aim is also to illustrate how all of these concepts can be applied to improve our understanding of systems of interacting biological oscillators. In-phase synchronization, for instance, where the nephrons of the kidney simultaneously perform the same regulatory adjustments of the incoming blood flow, is expected to produce fast and strong overall reactions to a change in the external conditions. In the absence of synchronization, on the other hand, the response of the system in the aggregate is likely to be slower and less pronounced. Hence, part of the regulation of the kidney may be associated with transitions between different states of synchronization among the functional units. Besides synchronization of interacting nephrons, the book also discusses chaotic synchronization and riddled basins of attractions for coupled pancreatic cells, homoclinic transitions to chaotic phase synchronization in coupled microbiological reactors, and clustering in systems of noise excited nerve cells. To a large extent the book is based on contributions that have been made over the last few years by the Chaos Group at the Technical University of Den-
viii
Preface
mark, by the Department of Mathematics, the National Academy of Sciences of Ukraine in Kiev, and by the Department of Physics, Saratov State University. We would like to thank our collaborators and students Brian Lading, Alexander Balanov, Tanya Vadivasova, Natasha Janson, Alexey Pavlov, Jacob Laugesen, Alexey Taborov, Vladimir Astakhov, Morten Dines Andersen, Niclas Carlsson, Christian Haxholdt, Christian Kampmann, and Carsten Knudsen for the many contributions they have made to the present work. Arkady Pikovsky, Jiirgen Kurths, Michael Rosenblum, Vladimir Belykh, Igor Belykh, Sergey Kuznetsov, Vadim Anishchenko, Morten Colding-J0rgensen, Jeppe Sturis, John D. Sterman, Laura Gardini, and Christian Mira are acknowledged for many helpful suggestions. We would also like to thank Niels-Henrik Holstein-Rathlau and Kay-Pong Yip who have made their experimented data on coupled nephrons available to us. Most of all, however, we would like to thank Vladimir Maistrenko, Oleksandr Popovych, Sergiy Yanchuk, and Olga Sosnovtseva who have been our closest collaborators in the study of chaotic synchronization. Without the enthusiastic help from these friend and colleagues, the book would never have been possible. The book has appeared at a time when research in chaotic synchronization is virtually exploding, and new concepts and ideas emerge from week to week. Hence, it is clear that we have not been able to cover all the relevant aspects of the field. We hope that the combination of mathematical theory, model formulation, computer simulations, and experimental results can inspire other researchers in this fascinating area. We have tried to make the book readable to students and young scientists without the highest expertise in chaos theory. On the other hand, the reader is assumed to have a good knowledge about the basic concepts and methods of nonlinear dynamics from previous studies. The book is dedicated to Lis Mosekilde. In her short scientific career she became the internationally most respected Danish expert in the fields of bone remodelling and osteoporosis. Lyngby, November 2001 Erik Mosekilde, Yuri Maistrenko and Dmitry Postnov
Contents
PREFACE
v
1
C O U P L E D N O N L I N E A R OSCILLATORS 1.1 The Role of Synchronization 1.2 Synchronization Measures 1.3 Mode-Locking of Endogenous Economic Cycles
1 1 7 13
2
T R A N S V E R S E STABILITY OF C O U P L E D M A P S 2.1 Riddling, Bubbling, and On-Off Intermittency 2.2 Weak Stability of the Synchronized Chaotic State 2.3 Formation of Riddled Basins of Attraction 2.4 Destabilization of Low-Periodic Orbits 2.5 Different Riddling Scenarios 2.6 Intermingled Basins of Attraction 2.7 Partial Synchronization for Three Coupled Maps
33 33 37 41 44 49 54 56
3
UNFOLDING THE RIDDLING BIFURCATION 3.1 Locally and Globally Riddled Basins of Attraction 3.2 Conditions for Soft and Hard Riddling 3.3 Example of a Soft Riddling Bifurcation 3.4 Example of a Hard Riddling Bifurcation 3.5 Destabilization Scenario for a — a,\ 3.6 Coupled Intermittency-III Maps 3.7 The Contact Bifurcation 3.8 Conclusions
75 75 80 88 93 95 104 109 116
4
TIME-CONTINUOUS SYSTEMS 4.1 Two Coupled Rossler Oscillators 4.2 Transverse Destabilization of Low-Periodic Orbits
123 123 125
ix
x
Contents
4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10
Riddled Basins Bifurcation Scenarios for Asynchronous Cycles The Role of a Small Parameter Mismatch Influence of Asymmetries in the Coupled System Transverse Stability of the Equilibrium Point Partial Synchronization of Coupled Oscillators Clustering in a System of Four Coupled Oscillators Arrays of Coupled Rossler Oscillators
130 134 140 145 147 154 162 166
5
C O U P L E D P A N C R E A T I C CELLS 5.1 The Insulin Producing Beta-Cells 5.2 The Bursting Cell Model 5.3 Bifurcation Diagrams for the Cell Model 5.4 Coupled Chaotically Spiking Cells 5.5 Locally Riddled Basins of Attraction 5.6 Globally Riddled Basins of Attraction 5.7 Effects of Cell Inhomogeneities
177 177 181 185 192 196 200 203
6
CHAOTIC PHASE SYNCHRONIZATION 6.1 Signatures of Phase Synchronization 6.2 Bifurcational Analysis 6.3 Role of Multistability 6.4 Mapping Approach to Multistability 6.5 Suppression of the Natural Dynamics 6.6 Chaotic Hierarchy in High Dimensions 6.7 A Route to High-Order Chaos
211 211 217 222 227 233 239 249
7
POPULATION DYNAMIC SYSTEMS 7.1 A System of Cascaded Microbiological Reactors 7.2 The Microbiological Oscillator 7.3 Nonautonomous Single-Pool System 7.4 Cascaded Two-Pool System 7.5 Homoclinic Synchronization Mechanism 7.6 One-Dimensional Array of Population Pools 7.7 Conclusions
259 259 262 265 270 274 280 284
Contents
xi
8
C L U S T E R I N G OF GLOBALLY C O U P L E D M A P S 8.1 Ensembles of Coupled Chaotic Oscillators 8.2 The Transcritical Riddling Bifurcation 8.3 Global Dynamics after a Transcritical Riddling 8.4 Riddling and Blowout Scenarios 8.5 Influence of a Parameter Mismatch 8.6 Stability of tf-Cluster States 8.7 Desynchronization of the Coherent Chaotic State 8.8 Formation of Nearly Symmetric Clusters 8.9 Transverse Stability of Chaotic Clusters 8.10 Strongly Asymmetric Two-Cluster Dynamics
291 291 296 302 307 313 318 321 326 329 334
9
INTERACTING NEPHRONS 9.1 Kidney Pressure and Flow Regulation 9.2 Single-Nephron Model 9.3 Bifurcation Structure of the Single-Nephron Model 9.4 Coupled Nephrons 9.5 Experimental Results 9.6 Phase Multistability 9.7 Transition to Synchronous Chaotic Behavior
349 349 354 359 365 370 375 382
10 C O H E R E N C E R E S O N A N C E OSCILLATORS 10.1 But What about the Noise? 10.2 Coherence Resonance 10.3 Mutual Synchronization 10.4 Forced Synchronization 10.5 Clustering of Noise-Induced Oscillations
395 395 400 404 408 412
INDEX
425
Chapter 1 COUPLED NONLINEAR OSCILLATORS 1.1
The Role of Synchronization
Synchronization occurs when oscillatory (or repetitive) systems via some kind of interaction adjust their behaviors relative to one another so as to attain a state where they work in unison. An essential aspect of many of the games we play as children is to teach us to coordinate our motions. We skip and learn to jump in synchrony with the swinging rope. We run along the beach and learn to avoid the waves that role ashore, and we take dancing lessons to learn to move in step with the music. One of the main problems in the swimming class is to learn to breath in synchrony with the strokes. Not necessarily one-to-one, as there are circumstances where it is advantageous to take two (or more) strokes per inhalation. However, the phase relations must be correct if not to drown. In much the same way, a horse has different forms of motion (such as walk, trot, and gallup), and each of these gaits corresponds to a particular rhythm in the movement of its legs [1,2]. At the trotting course, the jockey tries to keep the horse in trot to the highest possible speed. In its free motion, however, a horse is likely to choose the mode that is most comfortable to it (and, perhaps, least energy demanding). As the speed increases the horse will make transitions from walk to trot and from trot to gallup. Synchronization is a universal phenomenon in nonlinear systems [3]. Wellknown examples are the synchronization of two (pendulum) clocks hanging on a l
2
Chaotic Synchronization:
Applications
to Living
Systems
wall, and the synchronization of the moon's rotation with its orbital motion so that the moon always turns the same side towards the earth. A radio receiver functions by synchronizing its internal oscillator with the period of the radio wave so that the difference, i.e. the transmitted signal, can be detected and converted into sound. A microwave emitting diode is placed in a cavity of a specific form and size to make it synchronize with a particular resonance frequency of the cavity. In a previous book [4] we presented results on synchronization of coupled thermostatically controlled radiators and coupled household refrigerators. Synchronization can also be observed between coupled laser systems and coupled biochemical reactors, and it is clear that one can find thousands of other examples in engineering and physics. At the assembly line one has to ensure an effective synchronization of the various processes for the production to proceed in an efficient manner, and engineers and scientists over and over again exploit the technique of modulating (or chopping) a test signal in order to benefit from the increased sensitivity of phase detection. The history of synchronization dates back at least to Huygens' observations some 300 years ago [5], and both the history and the basic theory are recapitulated in a significant number of books and articles [6, 7]. For regular (e.g., limit cycle) oscillators, synchronization implies that the periodicities of the interacting systems precisely coincide and that differences in phase remain constant. In the presence of noise (or for chaotic systems) one can weaken the requirements such that the periodicities only have to coincide on average, and the phase differences are allowed to move within certain bounds. One may also accept occasional phase slips, provided that they do not occur too often [8]. One-to-one synchronization is only a simple manifestation of a much more general phenomenon, also known as entrainment, mode locking, or frequency locking. In nonlinear systems, a periodic motion is usually accompanied by a series of harmonics at frequencies of p times the fundamental frequency, where p is an integer. When two nonlinear oscillators interact, mode locking may occur whenever a harmonic frequency of one mode is close to a harmonic of the other. As a result, nonlinear oscillators tend to lock to one another so that one subsystem completes precisely p cycles each time the other subsystem completes q cycles, with p and q as integers [9, 10]. An early experience with this type of phenomenon is the way one excites a swing by forcing it at twice its characteristic frequency, i.e., we move the body through two cycles of a bending and stretching mode for each cycle of the swing. A similar phenomenon is
Coupled Nonlinear
Oscillators
3
utilized (in optics, electronics, etc.) in a wide range of so-called parametric devices. Contrary to the conventional assumption of homeostasis, many physiological systems are unstable and operate in a pulsatile or oscillatory mode [10, 11]. This s the case, for instance, for the production of luteinizing hormone and insulin hat are typically released in two-hour intervals [4]. In several cases it has been >bserved that the cellular response to a pulsatile hormonal signal is stronger han the response to a constant signal of the same magnitude, suggesting that he oscillatory dynamics plays a role in the control of the system [12]. Hormonal elease processes may also become synchronized, and it has been reported, or instance, that the so-called hot flashes that complicate the lives of many vomen during menopause are related to the synchronized release of 5-7 different lormones [13, 14]. The beating of the heart, the respiratory cycle, the circadian rhythm, and the avarian cycle are all examples of more or less regular self-sustained oscillations. The ventilatory signal is clearly visible in spectral analyses of the beat-to-beat variability of the heart signal, and in particular circumstances the two oscillators may lock together so that, for instance, the heart beats three or four times for each respiratory cycle [15]. The jet lag that we experience after a flight to a different time zone is related to the synchronization of our internal (circadian) rhythm to the local day-and-night cycles, and it is often said that women can synchronize their menstrual cycles via specific scents (pheromones) if they live close together. Rhythmic and pulsatile signals are also encountered in intercellular communication [16]. Besides neurons and muscle cells that communicate by trains of electric pulses, examples include the generation of cyclic AMP pulses in slime mold cultures of Dictostelium discoideum [17] and the newly discovered synchronization of the metabolic processes in suspensions of yeast cells [18]. Synchronization of the activity of the muscle cells in the heart is necessary for the cells to act in unison and produce a regular contraction. Similarly, groups of nerve cells must synchronize to produce the characteristic rhythms of the brain or to act as pacemakers for the glands of the hormonal systems [19]. On the other hand, it is well-known that synchronization of the electrical activity of larger groups of cells in the brain plays as essential role in the development of epileptic seizures [20]. However, nonlinear oscillators may also display more complicated forms of
4
Chaotic Synchronization:
Applications
to Living
Systems
dynamics, and an interesting question that arises over and over again in the biological sciences concerns the collective behavior of a group of cells or functional units that each display strongly nonlinear phenomena [21]. The human kidney, for instance, contains of the order of one million functional units, the nephrons. In order to protect its function, the individual nephron disposes of a negative feedback regulation by which it can control the incoming blood flow. However, because of the delay associated with the flow of fluid through the nephron, this regulation tends to be unstable and produce self-sustained oscillations in the various pressures and concentrations [22]. If the arterial blood pressure is high enough, the pressure oscillations in the nephron may become irregular and chaotic [23]. Neighboring nephrons interact with one another through signals that propagate along the afferent arterioles (incoming blood vessels) and, as experiments show, this interaction can lead to a synchronization of the regular pressure oscillations for adjacent nephrons [24].
Fig 1.1. Pressure variations in two neighboring nephrons for a hypertensive rat. Note that there is a certain degree of synchronization between the irregular (chaotic) signals. This synchronization is found to arise from interactions between the nephrons and not from common external influences.
'"*" 0
500 Time (sec)
1000
It is obviously of interest to examine to what extent similar synchronization phenomena are manifest in the irregular oscillations at higher blood pressures. Figure 1.1 shows an example of the chaotic pressure variations that one can observe (in the proximal tubule) for neighboring nephrons in a hypertensive rat. Although the two signals are strongly irregular, one is tempted to admit that there is a certain degree of synchronization: The most pronounced maxima and minima in the pressure variations occur almost simultaneously. Figure 1.2
Coupled Nonlinear Oscillators 5
shows a scanning electron microscope picture of the interaction structure for a couple of nephrons. Here, one can see how the common interlobular artery (IA) branches into separate afferent arterioles (af) for the two nephrons. The two ball-formed bundles are the capillary systems (the glomeruli) of the two nephrons. Here, blood constituents like water and salts are filtered into the tubular system of the nephrons and the remaining blood passes out through the efferent arterioles (e/). See Fig. 9.1 for a more detailed sketch of the structure of the nephron.
Fig 1.2. Scanning electron microscope picture of the arteriolar system for a couple of adjacent nephrons. The nephrons are assumed to interact with one another via muscular contractions that propagate along the afferent arterioles (af).
We would like to understand how the interaction between the nephrons influence the overall functioning of the kidney. Will there be circumstances, for instance, where the coupling produces a global synchronization of all the nephrons or will we see the formation of clusters of nephrons in different synchronization states? Will transitions between different states of synchronization play a role in the regulation of the kidney or will such transitions be related to the development of particular diseases? Similarly, each of the insulin producing /3-cells of the pancreas exhibits a complicated pattern of oscillations and bursts in its membrane potential [25]. Presumably through their relation to the exchange of calcium between the cell and its surroundings, these bursts control the release of insulin. The /3-cells are arranged in a spiral structure along capillaries and small veins. Via insulin receptors in the cell membrane, each cell can thus react to the release of insulin from cells that are upstream to it. At the same time, the /3-cells are coupled
6
Chaotic Synchronization:
Applications
to Living
Systems
via gap junctions through which ions and small molecules can pass from cell to cell. Again it is of interest to understand how the collective behavior of a group of cells is related to the dynamics of individual cells. Experiments indicate that there will be waves of synchronization moving across larger groups of cells in an islet of Langerhans [26]. In the economic realm, each individual production sector with its characteristic capital life time and inventory coverage parameters tends to exhibit an oscillatory response to changes in the external conditions [27]. Overreaction, time delays, and reinforcing positive feedback mechanisms may cause the behavior to become destabilized and lead to complicated nonlinear dynamic phenomena. The sectors interact via the exchange of goods and services and via the competition for labor and other resources. A basic problem for the establishment of a dynamic macroeconomic theory is therefore to describe how the various interactions lead to a more or less complete entrainment of the sectors [28]. The problems associated with chaotic synchronization have also attracted a considerable interest in the fields of electronics and radio engineering. Here, the attention centers around the possibilities of developing new types of communication systems that exploit the particular properties of deterministic chaos [29, 30, 31]. Important questions that arise in this connection pertain to the sensitivity of the synchronized state to noise or to a parameter mismatch between the interacting oscillators. Other questions relate to the behavior of the coupled system, once the synchronization breaks down, and to the initial conditions for which entrainment can be attained. It is a problem of considerable interest, whether or not one can mask a message by mixing it with a chaotic signal [32]. In order to discuss some of the problems that arise in connection with chaotic synchronization we shall apply a variety of different simple mathematical models. We start in Chapters 2 and 3 by considering a system of two (or three) coupled logistic maps. This leads us to a discussion of the conditions for synchronization in systems of coupled Rossler oscillators (Chapter 4) and in a system of two (nearly) identical /3-cells (Chapter 5). In this connection we show that a /3-cell has regions of chaotic dynamics between the different states of periodic bursting. Towards the end of the book the analysis will lead us to consider clustering in systems of many coupled chaotic oscillators (Chapter 8) and to examine interacting coherence resonance (i.e., stochastically excited) os-
Coupled Nonlinear
Oscillators
7
dilators (Chapter 10). On the way we shall discuss the characteristics of chaotic phase synchronization (Chapter 6) and use the obtained results to examine experimental data for the tubular pressure variations in neighboring nephrons (Chapter 9). Let us start, however, by discussing some of the characteristic signatures of synchronization in regular and chaotic systems. Thereafter, we shall use a model of two interacting capital producing sectors of the economy [28] to recall some of the basic concepts of the classical synchronization theory [33] and to illustrate the role of synchronization in macroeconomic systems.
1.2
Synchronization Measures
Let us consider some of the phenomena that one can observe in connection with chaotic phase synchronization [34, 35]. This is the type of synchronization that we expect to find between two coupled chaotic oscillators with different parameters such as, for instance, between neighboring nephrons in a hypertensive rat. The idea is to focus on the similarities between chaotic phase synchronization and the synchronization phenomena we know for regular oscillators. Among of the questions we would like to discuss are: What are the signatures of chaotic phase synchronization? Can we use similar diagnostic tools as we use for regular oscillators? What are the main bifurcation scenarios? First, however, we should perhaps recall some of the characteristics of the synchronization mechanism for regular oscillators [4, 9, 10]. From a mathematical point of view we understand the synchronization of two periodic oscillators as a transition from quasiperiodic motion to regular periodic behavior for the system as a whole. The quasiperiodic behavior is usually described as the motion on a torus. This motion is characterized by the presence of two incommensurate periods, asscociated with the motions of the individual oscillators. As coupling between the oscillators is introduced, both oscillators adjust their motions in response to the motion of the other, and when the coupling becomes strong enough a transition typically occurs where the two periods start to coincide. In the absence of coupling, the phase of each oscillator is a neutrally stable variable. There are no mechanisms that act to correct for a shift in phase. The amplitude, on the other hand, is controlled by a balance between instability and nonlinearity, and dissipation leads to a rapid decay of any pertubation of the amplitude. Hence, we conclude that mutual phase adjustments will be more significant than amplitude modulation [3]
8
Chaotic Synchronization:
Applications
to Living
Systems
At least for relatively small coupling strengths, the synchronization takes place via a saddle-node bifurcation [33]. On the surface of the torus a stable (node) and an unstable (saddle) cycle simultaneously emerge. Under variation of a control parameter (for instance, a parameter that controls the uncoupled period of one of the oscillators), the two cycles move away from one another along the torus surface to meet again and become annihilated on the opposite side. As a result, the synchronized state exists in a finite range of the control parameter. The typical situations where synchronization occurs are mutually coupled oscillators and periodically forced oscillators. Glass and Mackey [10] have discussed, for instance, how different forms of synchronization can be observed for periodically forced chicken heart cells. Sturis et al. [36] have described how the release of insulin from the pancreas in normal subjects can be synchronized to an external variation in the supply of glucose, and Bindschadler and Sneyd [37] have described how oscillations in the intracellular concentrations of calcium in biological cells can be synchronized via the exchange of ions through the gap junctions. Under such conditions, the relevant control parameters are the frequency mismatch Aw for the uncoupled oscillators and the coupling strength (or forcing amplitude) K. The phenomenon of synchronization can be described from different perspectives. If we look at the Fourier spectra of the oscillations, synchronization can be seen as a characteristic evolution of the amplitudes and frequencies of the spectral components. On the other hand, in terms of a phase space analysis, the synchronization mechanisms are the possible ways of transition from an ergodic (or nonresonant) two-dimensional torus (which, as mentioned above, is the image of quasiperiodic behavior) to a limit cycle, being the image of periodic oscillations. Figure 1.3 shows the typical structure of the 1:1 synchronization regime [38, 39]. To be specific, let us talk about the case of forced synchronization. The representative ways on the (Aw, .^-parameter plane are denoted with arrows: Route A: At weak coupling and a relatively small parameter mismatch the onset of resonance on the two-torus corresponds to the crossing of one of the saddle-node bifurcation lines SN. At this point, a pair of limit cycles (a stable and a saddle cycle) emerge on the torus surface. In terms of the Fourier spectra, this transition can be diagnosed from the approach and final merging
Coupled Nonlinear Oscillators 9
of the spectral peaks corresponding to the forcing signal and the self-sustained oscillations. Route B: In the case of a considerable frequency mismatch, increasing the forcing amplitude leads to the gradual suppression of the self-sustained oscillations of the forced system. The two-torus decreases in size and collapses into a limit cycle. Hence, the synchronized state arises at the curve of torus bifurcation. In the Fourier spectra, the spectral component of the self-sustained oscillations decreases in amplitude and disappears when the bifurcation line T is reached.
B
"*/\V.;- f i v
Fig 1.3. Typical structure of 1:1 synchronization region. SN is a curve of saddle-Eode bifurcation for a pair of stable and saddle limit cycles and SSN is a curve of saddlenode bifurcation for a pair of saddle and unstable limit cycles. BT is the so-called Bogdanov-Takens point and C denotes another codimension-2 bifurcation point. T is the torus bifurcation line, and H denotes a line of homoclinic bifurcation.
!:
Aco Route C: In the resonant parameter area, increasing the forcing amplitude does not lead to a qualitative change of the stable limit cycle. However, at the curve of saddle-saddle-node bifurcation SSN the saddle limit cycle and an unstable limit cycle (from the inside of the torus) are annihilated. Thus, the invariant torus surface (which is defined by the unstable manifolds of the saddle cycle) no longer exists above this line. This transition cannot be diagnosed by means of Fourier spectra for the synchronous oscillations. Route D: In some (usually narrow) parameter region a specific kind of transition can be detected in which a homoclinic bifurcation plays the key role.
10
Chaotic Synchronization:
Applications
to Living
Systems
Here, one observes a region of bistability, where the stable synchronous solution and the stable nonresonant torus coexist. Under variation of the control parameters, this bistability manifests itself in terms of a hysteretic behavior. Most of the published work on chaotic phase synchronization refers to the case where the chaotic dynamics has appeared through a cascade of perioddoubling bifurcations. This type of chaotic dynamics has a characteristic structure which manifests itself both in the rotation of the trajectory around some center point (Fig. 1.4(a)) and in the presence of a characteristic time scale which can be easily measured from the power spectrum (see Fig. 1.4(b)). These features of period-doubling chaos are important in relation to the problem of phase synchronization because they make it possible to introduce a simple measure of the instantaneous phase of the chaotic oscillations and to consider the mean return time to some Poincare secant as representative of the internal rhythm of the dynamics.
(b)
0
0.0
0.5
1.0
1.5
2.0
(0
Fig 1.4. The chaotic attractor of the Rossler model demonstrates the typical features of period doubling chaos. The phase trajectory rotates around some center in the phase space projection (a), and one can see the corresponding peak of the fundamental frequency in the power spectrum (b). A relatively small number of model dynamical systems have served an important role in the investigation of period-doubling chaos. The same is true for the problem of chaotic phase synchronization. Here, we can mention the Rossler model [40, 41], the Chua circuit [42, 43], and the Anishchenko-Astakhov generator [35, 44]. All of these models are three-dimensional oscillators demonstrating the period-doubling route to chaos, with the high-periodic and chaotic
Coupled Nonlinear
Oscillators
11
regimes located around a single unstable equilibrium point of saddle-focus type. During the period-doubling cascade, no additional complexity occurs, and these models thus describe a generic case. By way of example let us base our discussion on the Rossler model [45]:
x = —u>y — z, y =
UJX
+ ay,
z = (3 + z(x-n),
(1.1)
where a, (3, and \x are control parameters, and UJ defines the characteristic frequency of the oscillations. The chaotic dynamics of this model is well-studied. There are two possible types of chaotic attractor. With \i increasing, a cascade of period-doubling bifurcations leads to the emergence of chaos. This is called "spiral chaos". This type of chaos is illustrated in Fig. 1.4(a). If /x increases further, the more complicated chaotic motion referred to as "screw chaos" can be observed [44]. To study the synchronization mechanisms we rewrite the model (1.1) as a nonautonomous system: x = —ujy — z + y = UJX + ay, z = p + z{x-fi),
Ksinujft, (1.2)
where K is the amplitude of a external harmonic drive, and the forcing frequency ojj is fixed at 1.0. As illustrated in Fig. 1.4 (b) there is usually a well developed peak in the power spectrum of period-doubling chaos at a frequency close to the frequency of the period-one limit cycle from which the chaotic dynamics has originated. Following the changes in the peak position, one can find an interval for the control parameter UJ where the peak frequency UJQ is in a rational relation with the frequency of the forcing signal ujf. This method works well both for numerical simulations and for full scale experiments. The first observation of frequency locking for chaotic oscillations was made with a similar approach [34, 35]. Figure 1.5 illustrates the locking of the fundamental frequency of the chaotic oscillations for the nonautonomous Rossler model. For UJ G [0.922; 0.929], a 1:1 locking region is observed. Chaotic oscillations inside and outside this re-
12
Chaotic Synchronization:
Applications
to Living
Systems
Fig 1.5. Frequency locking region for the chaotic oscillations in the nonautonomous Rossler model with a = 0.2, /? = 0.2, // = 4.0, and K = 0.02.
""'6.920
0.925 CO
0.930
gion can be classified as synchronous and asynchronous, respectively. This is confirmed by Fig. 1.6 where the Poincare sections for both cases are plotted. It is interesting to note how Fig. 1.5 reproduces the variation that one observes in connection with the synchroniztion of a regular oscillator, forced by a periodic signal. In the interval of chaotic phase synchronization (0.922 < u> < 0.929), the characteristic frequency of the chaotic oscillations coincides precisely with the forcing frequency. This corresponds to the 1:1 step of the well-know devil's staircase for regular oscillations [4, 9, 10]. On both sides of the synchronization interval, the characteristic frequency of the chaotic oscillations shifts away from the forcing frequency. This region of asynchronous chaos replaces the quasiperiodic region for regular oscillators. Inspection of Fig. 1.6(b) shows how the structure of the Poincare section for the asynchronous chaotic dynamics reproduces the characteristic structure of the phase space projection for the autonomous Rossler oscillator (see Fig. 1.4). This implies that our stroboscopic measurements (the Poincare section for a forced system) catch the Rossler oscillator more or less at random in all different positions along its trajectory. In the synchronized chaotic case (Fig. 1.6(a)), on the other hand, although there is a certain scatter in the position of the Rossler system in the stroboscopic map, the system is always found in a relatively narrow region of phase space. Whenever the forcing signal has completed a full period, the synchronized Rossler oscillator is back to nearly the same position in phase space. In the following chapters (particularly in Chapter 6) we shall return to the problem of chaotic phase synchronization to discuss additional signatures of synchronization (e.g., the variation of the Lyapunov exponents) and to examine
Coupled Nonlinear
|
10.0
Oscillators
13
(a)
5.0
-5.0 • -10.0 -10.0
-10.0 -5.0
0.0
X
5.0
10.0
-10.0
-5.0
0.0
5.0
10.0
X
Fig 1.6. Phase projections of the Poincare section in the nonautonomous Rossler model for synchronous (a) and asynchronous (b) chaos at fj, = 4.0, a = 0.2, /? = 0.2, and K = 0.02. the bifurcations involved in the transition to chaotic phase synchronization. We shall also show that chaotic phase sychronization can develop along the same routes as we have illustrated for regular oscillations in Fig. 1.3. However, let us complete the present chapter with a discussion of the role of synchronization in economic systems. This is an area of research which appears so far to have attracted far too little attention.
1.3
Mode-Locking of Endogenous Economic Cycles
Macroeconomic models normally aggregate the individual firms of the economy into sectors with similar products, manufacturing processes and decision rules. Sometimes, only a single sector is considered (e.g., [46, 47]). This simplification is justified on pragmatic grounds by noting that it is impractical to portray separately all the firms in an industry or all the products on the market, and by arguing that the phenomena of interest are captured in sufficient detail by the aggregate formulation [48, 49]. Nevertheless, there are instances where aggregation is not justified. Economic sectors associated with the new information and communication technologies, for example, may show rapid growth while other, more traditional sectors show little increase or even decline. Here, disaggregation is required to provide a proper perspective of future developments. The oscillatory patterns that one can observe in many economic variables also display widely different periodicities. Commodity cycles in pork prices and slaughter rates typically exhibit a period of 3-4 years while similar cycles for
14
Chaotic Synchronization:
Applications
to Living
Systems
chicken and cattle have periodicities of about 30 months and 15 years, respectively [50]. These cycles are related to the feeding periods for the various animals. In other sectors, such as the auto industry or the tanker market, one can find periodicities that relate to the lifetime of the capital [4, 51]. Obviously, the various sectors interact with one another. However, the economic modeling literature is weak in providing guidelines for appropriate aggregation of dynamic systems, particularly when there are significant interactions between the individual entities. Models of capital investment, for instance, typically represent the average lead time and lifetime of the plant and equipment used by each firm. In reality there are many types of plant and equipment acquired from many vendors operating with a wide range of lead times. In response to changes in its external conditions, each firm will generate cyclic behaviors whose frequency, damping, and other properties are determined by the parameters characterizing the particular mix of lead times and lifetimes the firm faces. Because the individual firms are coupled to one another via the input-output structure of the economy, each acts as a source of perturbations on the others. How do the different lifetimes and lead times of plant and equipment affect the frequency, phase, amplitude, and coherence of economic cycles? How valid is aggregation of individual firms into single sectors for the purpose of studying macroeconomic fluctuations? Here, the issue of coherence or synchronization becomes important. The economy as a whole experiences aggregate business fluctuations of various frequencies from the short-term business cycle to the long-term Kondratieff cycle [4, 27]. Yet why should the oscillations of the individual firms move in phase so as to produce an aggregate cycle? Given the distribution of parameters among individual firms, why do we observe only a few distinct cycles rather than cycles at all frequencies - cycles which might cancel out at the aggregate level? A common approach to the question of synchronization in economics is to assume that fluctuations in aggregates such as gross domestic product or unemployment arise from external shocks, for example sudden changes in resource supply conditions or variations in fiscal or monetary policy [52]. Forrester [53] suggested instead that synchronization could arise from the endogenous interaction of multiple nonlinear oscillators, i.e., that the cycles generated by individual firms become reinforced and entrained with one another. Forrester also proposed that such entrainment could account for the uniqueness of the eco-
Coupled Nonlinear
Oscillators
15
nomic cycles. Oscillatory tendencies of similar periodicity in different parts of the economy would be drawn together to form a subset of distinct modes, such as business cycles, construction cycles, and long waves, and each of these modes would be separated from the next by a wide enough margin to avoid synchronization. Until recently, however, these suggestions have not been subjected to rigorous analysis. Roughly speaking, synchronization occurs because the nonlinear structure of the interacting parts of a system creates forces that " nudge" the parts of the system into phase with one another. As described by Huygens [5], two mechanical clocks, hanging on the same wall, are sometimes observed to synchronize their pendulum movements. Each clock has an escapement mechanisms, a highly nonlinear mechanical devise, that transfers power from the weights to the rod of the pendulum. When a pendulum is close to the position where the escapement releases, a small disturbance, such as the faint click from the release of the adjacent clock's escapement, may be enough to trigger the release. Hence, the weak coupling of the clocks, through vibrations in the wall, can bring individual oscillations into phase, provided that the two uncoupled frequences are not too different (see [3] for a more complete discussion of Huygens results). We have previously described how mode-locking and other nonlinear dynamic phenomena arise in a simple model of the economic long wave [4, 27, 54]. As described by Sterman [55], the model explains the long wave as a selfsustained oscillation arising from instabilities in the ordering and production of capital. An increase in the demand for capital leads to further increases through the investment accelerator or "capital self-ordering", because the aggregate capital-producing sector depends on its own output to build up its stock of productive capital. Once a capital expansion gets under way, self-reinforcing processes sustain it beyond its long-term equilibrium, until production catches up with orders. At this point, however, the economy has acquired considerable excess capital, forcing capital production to remain below the level needed for replacement until the excess has been fully depreciated, and room for a new expansion has been created. The concern of the present discussion is the model's aggregation of capital into a single type. The real economy consists of many sectors employing different kinds of capital in different amounts. Parameters, such as the average productive life of capital and the relative amounts of different capital components employed, may vary from sector to sector. In isolation, the buildings
16
Chaotic Synchronization: Applications to Living Systems
industry may show a temporal variation significantly different from that of, for instance, the machinery industry. An early study by Kampmann [56] took a first step in this direction by disaggregating the simple long-wave model into a system of several capital producing sectors with different characteristics. Kampmann showed that the multi-sector system could produce a range of different behaviors, at times quite different from the original one-sector model. The present analysis provides a more formal approach, using a two-sector model. One sector can be construed as producing buildings and infrastructure with very long lifetimes, while the other could represent the production of machines, transportation equipment, etc., with much shorter lifetimes. In isolation, each sector produces a self-sustained oscillation with a period and amplitude determined by the sector's parameter values. However, when the two sectors are coupled together through their dependence on each other's output, they tend to synchronize with a rational ratio between the two periods of oscillation. The extended long wave model [28] describes the flows of capital plant and equipment in two capital producing sectors. Each sector uses capital from itself and from the other sector as the only factors of production. Each sector receives orders for capital, from itself, from the other sector, and from the consumer goods sector. Production is made to order (no inventories are kept), and orders reside in a backlog until capital is produced and delivered. Each sector i —1,2 maintains a stock Kij of each capital type j = 1,2. The capital stock is increased by deliveries of new capital and reduced by physical depreciation. The stock of capital type j depreciates exponentially with and average lifetime of Tj. The difference in lifetime between the two sectors A T will be used as a bifurcation parameter to explore the robustness of the aggregated model. Output is distributed "fairly" between customers, i.e., the delivery of capital type j to sector i is the share of total output Xj from sector j , distributed according to how much sector i has on order with sector j , relative to sector j ' s total order backlog Bj. Hence,
Kit = x& Bj
and
- ^ Tj
(1.3)
Coupled Nonlinear
Oscillators
17
S-Sij = 0{j -Xj-j*-,
(1.4)
where a dot denotes time derivative. Sij represents the orders that sector i has placed with sector j but not yet recieved, and Oij represents the rate of sector i's new orders for capital from sector j . Each sector receives orders from itself On, from the other capital sector Oji, and from the consumer goods sector, y,. It accumulates these orders in a backlog S;, which it then depleted by the sector's deliveries of capital X{. Hence, Bi = {on + Oji - y{) - Xi, j ^ i.
(1.5)
Production capacity in each sector is determined by a constant-returns-to scale Cobb-Douglas function of the individual stocks of the two capital types, with a factor share a £ [0,1] of the other sector's capital type and a share 1 — a of the sector's own capital type, i.e., * = K?K}rKfj, j + i.
(1.6)
where the capital-output ratio K, is a constant. The parameter a determines the degree of coupling between the two sectors. In the simulation studies a is varied between 0, indicating no interdependence between the sectors, and 1, indicating the strongest possible coupling where each sector is completely dependent on capital from the other sector. A characteristic aspect of the Cobb-Douglas function is that it allows substitution between the two production factors, i.e., the same production capacity q can be achieved with different combinations of Ku and Kij. In this perspective, a is referred to as the elasticity of substitution. The output Xi from sector i depends on the sector's production capacity Q, compared to the sector's desired output x*. If desired output is much lower than capacity, production is cut back, ultimately to zero if no output is desired. Conversely, if desired output exceeds capacity, output can be increased beyond capacity, up to a certain limit. In our model, the sector's output is formulated as
xi = f(^jci
(1.7)
18
Chaotic Synchronization:
Applications
to Living
Systems
where the capacity-utilization function /(•) has the form
/(r)= 7 (l-(l^y), 7>1.
(1.8)
With this formulation /(0) = 0, / ( l ) = 1, and lim^oo / ( r ) = 7. Thus the parameter 7 determines the maximum production possible. In the present analysis we take 7 = 1.1. Note that / ( r ) > r, r G [0,1], implying that firms are reluctant to cut back their output when capacity exceeds demand. Instead, they deplete their backlogs and reduce their delivery delays. Sector i's desired orders o*j for new capital of type j consist of three components. First, all other things being equal, firms will order to replace depreciation of their existing capital stock, K^/TJ. Second, if their current capital stock is below (above) its desired level &*• firms will order more (less) capital in order to correct the discrepancy over time. Third, firms consider the current supply line Sij of capital and compare it to its desired level s*-. If the supply line is below (above) that desired, firms order more (less) in order to increase (decrease) the supply line over time. In total, our expression for the desired ordering rate becomes
dl. = ^L+KiZ^L
+^
i i
(1.9)
where the parameters rf and rf are the characteristic adjustment times for the capital stock and the supply line, respectively. This decision rule is supported by extensive empirical [57] and experimental [58, 59] work. The rule is based on a so-called anchoring-and-adjustment approach that is believed to capture the bounded rationality of real decision makers. Actual orders are constrained to be non-zero (cancellation of orders is not considered) and the fractional rate of expansion of the capital stock is also assumed to be limited by bottlenecks related to labor, market development, and other factors not represented in the production function. These constraints are accounted for through the expression
Coupled Nonlinear
Oscillators
19
where orders for new capital are expressed as a factor g(-) times the rate of capital depreciation. For the function g(-) we have assumed the form
9(u) =
r-TT
r^r
(1-11)
where the parameters have the following values /3 = 6, /ii = 27/7, \ii — 8/7, v\ = 2/3, and 1/2 = 3. These parameters are specified so that #(1) = 1, g'(l) = 1, and g"(l) = 0. Furthermore lim u ^ 0 0 g(u) = f3 and lim^-oo g(u) = 0. Note that g(u) has a neutral interval around the equilibrium point (u = 1) where actual orders equal desired orders. The desired capital stock fc*- is proportional to the desired production rate x* with a constant capital-output ratio. Thus, it is implicitly assumed that the relative prices of the two types of capital are constant, so there is no variation in desired factor proportions. Hence,
tyj ~
K x
ij i
(.I'l^,)
where Ky is the capital-output ratio of capital type j in sector i. In calculating the desired supply line s*-, firms are assumed to account for the delivery delay for each type of capital. The target supply line is taken to be the level at which the deliveries of capital, given the current delivery delay, would equal the current depreciation of the capital stock. The current delivery delay of capital from a sector is the sector's backlog divided by its output. Thus,
4 = ^
,
(1-13)
Finally, orders from the consumer goods sector j/j are assumed to be exogenous, constant, and equal for both sectors. The last assumption is not without consequence, since the relative size of the demands for the two types of capital can change the dynamics of the model considerably [56]. The capital-output ratios and average capital lifetimes are formulated in such a way that the aggregate equilibrium values of these parameters for the model economy as a whole remain constant and equal to the values in Sterman's original model. Specifically, the average capital lifetimes in the two sectors are
20
Chaotic Synchronization: Applications to Living Systems AT , AT TI = T + — a n d T2 = T - —
(1.14)
and the capital-output ratios are T-
T' 1
KU = (l-a)K-,
Kij^an- ,
100
i ^ j,
and nt = K}raKfj.
(1.15)
Fig 1.7. Simulation of the onesector model. The steady-state behavior is a limit cycle with a period of approximately 47 years. The plot shows production capacity, production, and desired production of capital equipment, respectively. All variables are shown on the same scale. Maxima are reached in the order: Desired production, actual production, capacity.
150
Time (Years)
The average lifetime of capital r is taken to be 20 years and the average capitaloutput ratio K = 3 years. The above formulation assures that capacity equals desired output when both capital stocks equal their desired levels and that the equilibrium aggregate lifetime of capital and equilibrium aggregate capital-output ratio equal the corresponding original parameters in the one-sector model. Furthermore, parameters in the decision rules were scaled to the average lifetime of capital produced by that sector. Thus, when there is no coupling between the sectors (a = 0), one sector is simply a time-scaled version of the other. Hence, the parameters are rf
= r*!*, r f = ^Zl T
T
and
j . = ,jZi
(Ll6)
T
where (as in the original model) TK - 1.5 y, r 5 = 1.5 y, and S = 1.5 y. Figure 1.7 shows a simulation of the limit cycle of the one-sector model (a = 0, AT = 0). Even with the modifications we have introduced, the behavior
Coupled Nonlinear Oscillators 21
of our model is virtually indistinguishable from that of the original model [55]. With the above parameters, the equilibrium point is unstable and the system quickly settles into a limit cycle with a period of approximately 47 years. Each new cycle begins with a period of rapid growth, where desired output exceeds capacity. The capital sector is thereby induced to order more capital, which, by further swelling order books, fuels the upturn in a self-reinforcing process. Eventually, capacity catches up with demand, but at this point it far exceeds the equilibrium level. The self-ordering process is now reversed, as falling orders from the capital sector lead to falling demand, which collapses to the point where only the exogenous goods sector places new orders. A long period of depression follows, during which the excess capital is gradually depleted, until capacity reaches demand. At this point, the capital sector finally raises enough orders to offset its own discards, increasing orders above capacity and initiating the next cycle. To explore the robustness of the single-sector model to differences in the parameters governing the individual sectors, we now simulate the model when some parameters differ between the two sectors. In spite of its simplicity, the model contains a considerable number of parameters which may differ from sector to sector. In the present analysis, we vary the difference A T in capital lifetimes for different values of the coupling parameter a. As described above, we have scaled all other parameters with the capital-lifetime parameters in such a way that, when a = 0, each sector is simply a time-scaled version of the original one-sector model.
Fig 1.8. Synchronization (1:1 mode-locking) in the coupled twosector model. The figure shows the capacity in each of the two sectors as a function of time in the steady state. The difference in capital lifetimes AT is 6 years. The coupling parameter a is 0.25. The machinery sector leads the oscillations. 200
300
Time (Years)
22 Chaotic Synchronization: Applications to Living Systems In the simulations that follow, sector 1 is always the sector with the longer lifetime of its capital output, corresponding to such industries as housing and infrastructure, while sector 2 has the shorter lifetime parameter, corresponding to the machine and equipment sector. Introducing a coupling between the sectors will not only link the behaviors together, but also change the stability properties of the individual sectors. A high value of the coupling parameter a implies that the strength of the capital self-ordering loop in any sector is small. In the extreme case a = 1, each sector will not order any capital from itself. If the delivery delay for capital from the other sector is taken as exogenous and constant, the behavior of an individual sector changes to a highly damped oscillation. Indeed, a linear stability analysis around the steady-state equilibrium of an individual sector shows that the equilibrium becomes stable for sufficiently high values of a. As will become evident below, this stability at high values of the coupling parameter has significant effects on the mode-locking behavior of the coupled system.
Fig 1.9. 2:2 mode-locking resulting from a period-doubling bifurcation. As the difference AT is increased to 9 years, the 1:1 mode is replaced by an alternating pattern of smaller and larger swings, so that the total period is doubled. As in the previous figure, a = 0.25. 0
100
200
300
400
500
Time (Years)
As long as the parameters of the two sectors are close enough, we expect synchronization (or 1:1 frequency locking) to occur, i.e., we expect that the different cycles generated by the individual sectors will adjust to one another and exhibit a single aggregate economic long wave with the same period for both sectors. The stronger the coupling a, the stronger the forces of synchronization are expected to be. As an example of such synchronization, Fig. 1.8 shows the outcome of a simulation performed with a difference in capital lifetimes between the two sectors of A T = 6 years and a coupling parameter a = 0.25. The two sectors, although not quite in phase, have identical periods of oscillations. The
Coupled Nonlinear
Oscillators
23
larger excursions in production capacity are found for sector 2 (the "machinery" sector), which is also the sector that leads in phase. The lifetime difference of 6 years corresponds to a lifetime for machinery capital of 17 years and a lifetime of buildings and infrastructure of 23 years. If, with the same coupling parameter, the difference in capital lifetimes is increased to A r = 9 years, we observe a doubling of the period. The two sectors now alternate between high and low maxima for their production capacities. This type of behavior is referred to as a 2:2 mode. It has developed out of the synchronous 1:1 mode through a period-doubling bifurcation [60]. The 2:2 solution is illustrated in Fig. 1.9. As the difference in lifetimes is further increased, the model passes through a Feigenbaum cascade of period-doubling bifurcations (4:4:, 8:8, etc.) and becomes chaotic at approximately A T = 10.4. years. Figure 1.10 shows the chaotic solution generated when A r = 10.7 years. Calculation of the largest Lyapunov exponent confirms that the solution in Fig. 1.10 is chaotic. We conclude that deterministic chaos can arise in a macroeconomic model that in its aggregated form supports self-sustained oscillations, if the various sectors (because of differences in parameter values) fail to synchronize in a regular motion.
Fig 1.10. Synchronized chaotic behavior. As the difference in capital lifetimes is increased further, the behavior becomes chaotic. For Ar = 10.7 years, the model shows irregular behavior, and initial conditions close to each other quickly diverge. Nonetheless, the two sectors remain locked with a ratio of unity between their average periods. 200 300 Time (Years)
A more detailed illustration of the route to chaos is provided by the bifurcation diagram in Fig. 1.11. Here we have plotted the maximum production capacity attained in sector 1 over each cycle as a function of the lifetime difference A r . The difference in capital lifetimes spans the interval 6 y < AT < 30 y. When A r = 30 y, the lifetime of the short-lived capital is just five years while the lifetime of the long lived capital is 35 years. The coupling parameter is kept
24
Chaotic Synchronization; Applications to Living Systems
Fig 1.11. Bifurcation diagram for increasing lifetime difference A r and constant a. The figure shows the local maxima attained for the capacity of the long-lived capital producer in the steady state for varying values of the lifetime difference. From left to right the main regions of periodic behavior correspond to the 1:1, 1:2, 1:3, and 1:4 synchronization regions. 0.0 -tromtmim
6
8
n
10
iii,|«
12
14
prmm,
16
18
m»)
20
22
|
24
26
28 30
Lifetime difference (years)
constant equal to a = 0.2. Inspection of the figure shows that the 1:1 frequencylocking, in which the production capacity of sector 1 reaches the same maximum in each long-wave upswing, is maintained up to A r « 6.4 years, where the first period-doubling bifurcation occurs. (Identification of the various periodic modes cannot be made from the bifurcation diagram alone, but involves the time and phase plots as well.) In the interval 6.4 y < AT < 7.9 y , the longwave upswings alternate between a high and a low maximum. Hereafter follows an interval up to approximately A r = 8.1 years with 4:4 locking, an interval of 8:8 locking, etc. Within the interval approximately 8.2 y < AT < 11.8 y small windows of periodic motion are visible between regions of chaos. In the region around 12.4 y < AT < 13.0 y chaos gives way to the 2:3 mode-locked solution and the associated period-doubling cascade 4:6, 8:12, etc. Another region of chaotic behavior follows until about A r « 15.2 years, where the system locks into 1:2 motion. Similarly the regions of 1:3 and 1:4 entrainment are clearly visible as A r continues to increase. Note that the 1:4 region bifurcates into 2:8 at around A rft*27.6 years, but then returns to 1:4 motion at A r as 28.3 years, rather than cascading through further doublings to chaos. The phase diagram in Fig. 1.12 gives an overview of the dominant modes for different combinations of the lifetime difference A r and the coupling parameter or. The zones of mode-locked (i.e., periodic) solutions in this diagram are the well-known ArnoPd tongues [33, 38, 39]. Besides the 1:1 tongue, the figure shows a series of l:n tongues, i.e., regions in parameter space where the buildings industry completes precisely one long-wave oscillation each time the machinery
Coupled Nonlinear Oscillators 25
industry completes n oscillations. Between these tongues, regions with other commensurate wave periods may be observed. An example is the 2:3 tongue found in the area around a = 0.15 and A r = 12 years. Similar to the 2:2 period-doubled solution on the right-hand side of the 1:1 tongue, there is a 2:4 period-doubled solution along part of the right-hand edge of the 1:2 tongue. 1.0 0.9 0.8
SacMr I Ua*tnt4 capital produce) it ruble la ieiaUioa hr alpha, ta
Seao. 2 Mori-livid capital producer) • ruble kraWoa Ac aloha abova Hue too.
0.7 0.6
IIH|llllimi|lllllllll|lllllllll|lllllllll|lllllllll|lllllllll|Tllllllll|lllllllll|lllllllll
12
14
16
18
20
22
24
26
28
30
Lifetime difference (Years)
Fig 1.12. Parameter phase diagram. The figure summarizes the steady-state behavior of the two-sector model for different combinations of the coupling parameter a and the lifetime difference Ar. A region labeled p : q indicates the area in parameter space where the model shows periodic mode-locked behavior of p cycles for sector 1 and q cycles for sector 2. The dashed curves across the diagram indicate the value of a above which each sector in isolation becomes stable. Above these lines synchronous 1:1 behavior prevails.
The phase diagram in Fig. 1.12 also reveals that the synchronous 1:1 solution extends to the full range of the lifetime differences A r for sufficiently high values of the coupling parameter a. When a is large enough, the individual sectors become stable, if the delivery delay and demand from the other sector are taken as exogenous. For reference, two curves have been drawn in Fig. 1.12, defining the regions in which one or both of the individual sectors are stable. As a increases, the overall behavior is increasingly derived from the coupling between the sectors and less and less from the autonomous self-ordering
26
Chaotic Synchronization:
Applications
to Living
Systems
mechanism in each individual sector. Thus, for high values of a, there is less competition between the two individual, autonomous oscillations and stronger synchronization. For large differences in capital lifetimes and low values of the coupling parameter a, the short-lived capital sector (sector 2) completes several cycles for each oscillation of the long-lived sector (sector 1). However, as a is increased, the short-term cycle is reduced in amplitude and, for sufficiently high values of a, it disappears altogether, resulting in a synchronous 1:1 solution. The locally stabilizing effect of high values of a creates an interesting distortion of the Arnol'd tongues in Fig. 1.12. For instance, the figure reveals that both the 1:1 region and the 2:2 region stretch above and around the other regions for high values of a. By employing only a single capital-producing sector, the original long-wave model [55] represents a simplification of the structure of capital and production. In reality, "capital" is composed of diverse components with different characteristics. We have focused on the difference in the average lifetime of capital and it is clear from our analysis that a disaggregated system with diverse capital lifetimes exhibits a much wider variety of fluctuations. For moderate differences in parameters between the sectors, the coupling between sectors has the effect of merging distinct individual cycles into a more uniform aggregate cycle. The period of the cycle remains in the 50-year range, although the amplitude may vary greatly form one cycle to the next. The behavior of the two-sector model thus retains the essential features of the simple model and is robust to the aggregation of all firms into a single sector. Entrainment in the disaggregated model arises only via the coupling introduced by the input-output structure of capital production. Other sources of coupling were ignored. The most obvious links are created by the price system. If, for instance, one type of capital is in short supply, one would expect the relative price of that factor to rise. To the extent that sectors can substitute one type of capital for another, one would expect demand for the relatively cheaper capital components to rise. In this way, the price system will cause local imbalances between order and capacity across the sectors to equalize, thus helping to bring the individual sectors into phase. (We have performed a few preliminary simulations of a version of the model that includes a price system and these simulations show an increased tendency for synchronization). The degree of substitution between capital types in the production function may well be an important factor: One would expect high elasticities of substitution to yield
Coupled Nonlinear
Oscillators
27
stronger synchronization. A next step in the study of coupled economic oscillators could therefore involve introducing relative prices and differing degrees of substitution. Another, more immediate extension of the above discussion would involve looking at more than two sectors. On the one hand, a wider variety of capital producers would introduce more variability in the behavior and, hence, less uniformity. On the other hand, as the system is disaggregated further, the strength of the individual self-ordering loops is reduced to near zero, and overall dynamics will more and more be determined by the interaction between sectors. It would also be interesting to consider the influence of other (more global) macroeconomic linkages, such as the Keynesian consumption multiplier. Our preliminary results demonstrate the importance of studying non-linear entrapment in the economy. The intricacies of such phenomena suggest that there is a vast unexplored domain of research in the area of economic cycles. We suppose that nonlinear interactions could play as large a role in shaping economic cycles as do the external random shocks on which much of mainstream business cycle theory relies. At the same time, our discussion points to the similarities in nature between the problems we meet in macroeconomic systems and in the biologically oriented problems discribed in other chapters of this book.
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Chaotic Synchronization:
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Systems
[5] C. Hugenii, Horologium Oscillatorium (Parisiis, France, 1673). English translation: The Pendulum Clock (Iowa State University Press, Ames, 1986). [6] C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New-York, 1964). [7] I. Blekhman, Synchronization in Science and Technology (ASME Press, New York, 1988). [8] R.L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963). [9] J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos (Wiley and Sons, Chichester, 1986). 10] L. Glass and M.C. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, 1988). 11] G. Leng, Pulsatility in Neuroendocrine Systems (CRC Press, Boca Raton, 1988). 12] Y.-X. Li and A. Goldbeter, Frequency Specificity in Intercellular nication, Biophys. J. 55, 125-145 (1989).
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[18] S. Dan0, P.G. S0rensen, and F. Hynne, Sustained Oscillations in Living Cells, Nature (London) 402, 320-322 (1999). [19] N. Kopell, G.B. Ermentrout, M.A. Whittington, and R.D. Traub, Gamma Rhythms and Beta Rhythms Have Different Synchronization Properties, Proc. Nat. Acad. Sci. 97, 1867-1872 (2000). [20] F. Mormann, K. Lehnertz, P. David, and C.E. Elger, Mean Phase Coherence as a Measure of Phase Synchronization and Its Application to the EEG of Epilepsy Patients, Physica D 144, 358-369 (2000). [21] K. Kaneko, Relevance of Dynamic Clustering to Biological Networks, Physica D 75, 55-73 (1994). [22] P.P. Leyssac and L. Baumbach, An Oscillating Intratubular Pressure Response to Alterations in Henle Loop Flow in the Rat Kidney, Acta Physiol. Scand. 117, 415-419 (1983). [23] K.S. Jensen, E. Mosekilde, and N.-H. Holstein-Rathlou, Self-Sustained Oscillations and Chaotic Behaviour in Kidney Pressure Regulation, Mondes en Develop. 5 4 / 5 5 , 91-109 (1986). [24] N.-H. Holstein-Rathlou, Synchronization of Proximal Intratubular Pressure Oscillations: Evidence for Interaction between Nephrons, Pfliigers Archiv 408, 438-443 (1987). [25] H.P. Meissner and M. Preissler, Ionic Mechanisms of the Glucose-Induced Membrane Potential Changes in /3-Cells, Horm. and Metab. Res. (Suppl.) 10, 91-99 (1980). [26] E. Gylfe, E. Grapengiesser, and B. Hellman, Propagation of Cytoplasmic Ca2+ Oscillations in Clusters of Pancreatic B-Cells Exposed to Glucose, Cell Calcium 12, 229-240 (1991). [27] J.D. Sterman and E. Mosekilde, Business Cycles and Long Waves: A Behavioral Disequilibrium Perspective. In Business Cycles: Theory and Empirical Methods, ed. W. Semmler (Kluwer Academic Publishers, Dordrecht, 1994). [28] C. Haxholdt, C. Kampmann, E. Mosekilde, and J.D. Sterman, ModeLocking and Entrainment of Endogenous Economic Cycles, System Dyn. Rev. 11, 177-198 (1995).
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[29] N.F. Rulkov, Images of Synchronized Chaos: Experiments with Circuits, Chaos 6, 262-279 (1996). [30] L.M. Pecora, T.L. Carroll, G.A. Johnson, D. J. Mar, and J.F. Heagy, Fundamentals of Synchronization in Chaotic Systems, Concepts, and Applications, Chaos 7, 520-543 (1997). [31] 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 CS-44, 927-935 (1997). [32] K.M. Short, Steps Toward Unmasking Secure Communications, Int. J. Bifurcation and Chaos 4, 959-977 (1994). [33] V.I. Arnol'd, Small Denominators. I. Mappings of the Circumference onto Itself, Am. Math. Soc. Transl., Ser. 2, 46, 213-284 (1965). [34] G.I. Dykman, P.S. Landa, and Yu.I. Neymark, Synchronizing the Chaotic Oscillations by External Force, Chaos, Solitons and Fractals 1, 339-353 (1991). [35] V.S. Anishchenko, T.E. Vadivasova, D.E. Postnov, and M.A. Safonova, Synchronization of Chaos, Int. J. Bifurcation and Chaos 2, 633-644 (1992). [36] J. Sturis, E. Van Cauter, J.D. Blackman, and K.S. Polonsky, Entrainment of Pulsatile Insulin Secretion by Oscillatory Glucose Infusion, J. Clin. Invest. 87, 439-445 (1991). [37] M. Bindschadler and J. Sneyd, A Bifurcation Analysis of Two Coupled Calcium Oscillators, Chaos 11, 237-246 (2001). [38] M.A. Taylor and I.G. Kevrekidis, Some Common Dynamic Features of Coupled Reacting Systems, Physica D 5 1 , 274-292 (1991). [39] C. Knudsen, J. Sturis, and J.S. Thomsen, Generic Bifurcation Structures of Arnol'd Tongues in Forced Oscillators, Phys. Rev. A 44, 3503-3510 (1991). [40] M. Rosenblum, A. Pikovsky, J. Kurths, Phase Synchronization Oscillators, Phys. Rev. Lett. 76, 1804-1807 (1996).
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[41] A. Pikovsky, G. Osipov, M. Rosenblum, M. Zaks, and J. Kurths, AttractorRepeller Collision and Eyelet Intermittency at the Transition to Phase Synchronization, Phys. Rev. Lett. 79, 47-50 (1997). [42] V.S. Anishchenko, T.E. Vadivasova, V.V. Astakhov, O.V. Sosnovtseva, C.W. Wu, and L.O. Chua, Dynamics of Two Coupled Chua's Circuits, Int. J. Bifurcation and Chaos 5, 1677-1699 (1995). [43] L. Chua, M. Itoh., L. Kocarev, and K. Eckert, Chaos Synchronization in Chua's Circuit. In Chua's Circuits: A Paradigmafor Chaos, edited by R.N. Madan (World Scientific, Singapore, 1993). [44] V.S. Anishchenko, Dynamical Chaos - Models and Experiments. Appearance Routes and Structure of Chaos in Simple Dynamical Systems (World Scientific, Singapore, 1995). [45] O.E. Rossler, An Equation for Continuous Chaos, Phys. Lett. A 57, 397398 (1976). [46] P. A. Samuelson, Interactions Between the Multiplier Analysis and the Principle of Acceleration, The Review of Economic Statistics 2 1 , 75-78 (1939). [47] R. Goodwin, The Nonlinear Accelerator and the Persistence of Business Cycles, Econometrica 19, 1-17 (1951). [48] J.W. Forrester, Industrial Dynamics (MIT Press, Cambridge, 1961). [49] H. Simon, The Sciences of the Artificial (MIT Press, Cambridge, 1969). [50] D.L. Meadows, Dynamics of Commodity Production Cycles (Wright-Allen Press, Cambridge, 1970). [51] Z. Zannitos, The Theory of Oil Tanker Rates (MIT Press, Cambridge, 1966). [52] V. Zarnowitz, Recent Work on Business Cycles in Historical Perspective: A Review of Theories and Evidence, The Journal of Economic Literature 23, 523-580 (1985). [53] J.W. Forrester, Growth Cycles, De Economist 125, 525-543 (1977).
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[54] E. Mosekilde, E.R. Larsen, J.D. Sterman, and J.S. Thomsen, Nonlinear Mode-Interaction in the Macroeconomy, Annals of Operations Research 37, 185-215 (1992). [55] J.D. Sterman, A Behavioral Model of the Economic Long Wave, Journal of Economic Behavior and Organization 6, 17-53 (1985). [56] C. E. Kampmann, Disaggregating a Simple Model of the Economic Long Wave (Working Paper no. D-3641, Sloan School of Management, M.I.T., Cambridge, Mass., U.S.A. 1984). [57] P.M. Senge, A System Dynamics Approach to Investment-Function Formulation and Testing, Socio-Economic Planning Science 14, 269-280 (1980). [58] J.D. Sterman, Misperceptions of Feedback in Dynamic Decision Making, Organizational Behavior and Human Decision Processes 43, 301-335 (1989). [59] J.D. Sterman, Modeling Managerial Behavior: Misperceptions of Feedback in a Dynamic Decision Making Experiment, Management Science 35, 321339 (1989). [60] M. Feigenbaum, Quantitative Universality for a Class of Nonlinear Transformations, J. Stat. Phys. 19, 669-706 (1978).
Chapter 2 TRANSVERSE STABILITY OF COUPLED MAPS 2.1
Riddling, Bubbling, and On-Off Intermittency
In the early 1980's, Fujisaka and Yamada [1] showed how two identical chaotic systems under variation of the coupling strength can attain a state of full synchronization where the motion of the coupled system takes place on an invariant subspace of total phase space. In spite of the fact that the systems are chaotic, their interaction allow them to move precisely in step. For two coupled identical, one-dimensional maps, for instance, the synchronous motion is one-dimensional and occurs along the main diagonal in the phase plane. The transverse Lyapunov exponent Ax provides a measure of the stability of the chaotic attractor perpendicular to this direction. As long as Ax is negative, a trajectory moving in the neighborhood of the synchronized chaotic state will, on average, be attracted towards this state. Chaotic synchronization has subsequently been studied by a large number of investigators [2, 3, 4], and a variety of applications for chaos suppression, for monitoring and control of dynamical systems, and for different communication purposes have been suggested [5, 6, 7]. Important questions that arise in this connection concern the form of the basin of attraction for the synchronized chaotic state and the bifurcations through which this basin, or the attractor itself, undergoes qualitative changes as a parameter is varied. Under what conditions will the interacting chaotic oscillators be able to synchronize if they are 33
34
Chaotic Synchronization:
Applications
to Living
Systems
started out of synchrony? What is the response of the synchronized state to noise or to a small parameter mismatch, and what happens when the synchronization breaks down? Recent studies of these and related problems have led to the discovery of a variety of new phenomena, including riddled basins of attraction [8, 9, 10, 11, 12] and on-off intermittency [13] that can be observed on either side of the so-called blowout bifurcation [10, 11], where the transverse Lyapunov exponent Aj. changes sign. On-off intermittency is an extreme form of intermittent bursting [14, 15] that can occur in the presence of a small positive value of Ax. In this case the chaotic set on the invariant subspace is no longer transversely attracting on average. However, immediately above the blowout bifurcation, where Aj_ is still small, a trajectory may spend a relatively long time in the neighborhood of the invariant subspace. From time to time, the repulsive character of the chaotic set manifests itself, and the trajectory exhibits a burst in which it moves far away from the invariant subspace, to be reinjected again into the proximity of this subspace. Besides a reinjection mechanism, the occurrence of on-off intermittency hinges on the fact that the positive value of the transverse Lyapunov exponent applies on average over long periods of time. For shorter time intervals, the net contribution to Aj. may be negative, and the trajectory is attracted to the chaotic set [13]. If there is no reinjection mechanism, a trajectory started near the invariant subspace may exhibit a superpersistent chaotic transient in which the behavior initially resembles the chaotic motion before the blowout bifurcation. Eventually, however, almost all trajectories will move away from this region and approach some other attractor (or go to infinity). When on-off intermittency occurs, the blowout bifurcation leads to a dynamics that is confined by nonlinear mechanisms to a region of phase space (the trapping zone or absorbing area) situated inside the original basin of attraction [16, 17, 18, 19], and this basin remains essentially unaffected by the blowout bifurcation. Variation of a parameter that causes the attractor to grow (or the basin to shrink) may then produce a crisis in which the attractor abruptly disappears as it makes contact with the basin boundary [20, 21, 22]. Riddled basins of attraction denote a characteristic form of fractal domain of attraction that can be observed on the other side of the blowout bifurcation, where the transverse Lyapunov exponent is negative [23, 24, 25, 26]. Even though the chaotic set is now attractive on average, particular orbits (usually of
Transverse Stability of Coupled Maps
35
low periodicity) embedded in the chaotic attractor may be transversely unstable. In this case we talk about weak attraction or attraction in the Milnor sense [27]. The chaotic set in the invariant subspace then attracts a set of points of positive Lebesgue measure in phase space. However, arbitrarily close to any such point one may find a (small but) positive Lebesgue measure set of points that are repelled by the chaotic attractor. The emergence of riddled basins of attraction occurs through a so-called riddling bifurcation (sometimes referred to as a bubbling transition [23]) in which the first orbit embedded in the synchronized chaotic attractor loses its transverse stability. This phenomenon was described in more detail by Lai et al. [28] and by Venkataramani et al. [29, 30]. They suggested that the riddling bifurcation takes place as two repelling orbits located symmetrically on either side of the invariant subspace approach the chaotic attractor and collide with a saddle cycle embedded in this attractor (in a so-called subcritical pitchfork bifurcation). Alternatively, a point cycle embedded in the synchronized state can lose its transverse stability through a period-doubling bifurcation [26]. In both cases, i.e., for the pitchfork as well as for the period-doubling bifurcation, the transition may be subcritical or supercritical depending on the sign of a certain quantifier (the so-called Lyapunov value) [31]. After a supercritical bifurcation, the transversely destabilized orbit will be surrounded by saddle cycles with their unstable manifolds stretching along the invariant subspace of the synchronized state. This leads to attractor bubbling and to the phenomenon of local riddling [23, 25]. In this case, the trajectories cannot escape a region around the synchronized state until one or two additional local [32] or global [33] bifurcations have occurred. The riddling bifurcation causes tongues of finite width to open up along the transversely unstable directions from each point on the desynchronized periodic orbit [28]. In these tongues trajectories move away from the chaotic attractor. Similar tongues open up from each preimage of the points on the repelling cycle, and since these preimages are dense in the invariant subspace an infinite number of tongues emerge, creating the characteristic riddled structure in which the basin of attraction locally becomes a fat fractal. Maistrenko et al. [34, 35] have studied chaotic synchronization and the formation of riddled basins of attraction for a system of two coupled piecewise linear maps. They related the various phenomena to the different types of instability for the chaotic set in the invariant subspace. Also investigating cou-
36
Chaotic Synchronization: Applications to Living Systems
pled piecewise linear maps, Pikovsky and Grassberger [36] discovered that even when the coupled system exhibits a stable synchronized behavior as indicated by a negative value of the transverse Lyapunov exponent, the basin of attraction may have full measure and yet be densely filled with unstable periodic orbits. Hence, the synchronized attractor is surrounded by a strange invariant set which is dense in a region around the attractor, and the synchronized state is not asymptotically stable. Pikovsky and Grassberger [36] also observed a bifurcation in which a synchronized one-dimensional attractor explodes into a two-dimensional attractor that contains the strange invariant set. Gardini et al. [37] investigated a one-parameter family of twisted, logistic maps. Particularly interesting in the present context is their description of the contact bifurcations in which the boundary of the absorbing area for a chaotic set touches the basin boundary. As we shall see in Chapter 3, global bifurcations of this type are often involved in the transition from local to global riddling [33]. The purpose of the present chapter is to provide a first, phenomenological introduction to some of the interesting phenomena that are involved in the loss of synchronization and the emergence of riddled basins of attraction. With this purpose we shall study a simple two-dimensional map F = Fafi : R2 —> R2 given by the recurrence formula x
n+l = J/n+1 =
Ja\xn) + £\Vn ~ xn) fa(yn)+e{Xn-yn)-
M.\
Here, fa is the one-dimensional logistic map fa:x>-Kix(l-x),
a; €[0,1],
o£[0,4],
(2.2)
and e € R is the coupling parameter. We first determine the regions of weak stability in the (a, e) plane for the synchronized chaotic attractor and calculate the transverse Lyapunov exponent Ax as a function of e for values of the nonlinearity parameter a where the individual map displays a homoclinic bifurcation. These are the parameter values at which fa{x) is known to produce a chaotic attractor with an absolutely continuous invariant measure. The formation of riddled basins of attraction is discussed in Sec. 2.3, and in Sec. 2.4 we determine the bifurcation curves for the transverse destabilization of low-periodic orbits embedded in the chaotic attractor. For one-band, two-band, and four-band chaotic dynamics we hereafter follow the changes that take place in the attractor and its basin of attraction under passage of the riddling and
Transverse Stability of Coupled Maps
37
blowout bifurcations [26]. It is shown that the loss of weak stability does not necessarily affect the basin of attraction. Instead, the chaotic attractor may spontaneously break the symmetry and spread into the two-dimensional phase space. It is also shown that the emergence of transversely unstable trajectories, while being a necessary condition for basin riddling, is not sufficient for global riddling to occur [33]. In Sec. 2.6 we illustrate the phenomenon of intermingled basins of attraction [8, 9] for a situation where the system has two coexisting four-band attractors, each displaying a riddled basin structure. Finally, in Sec. 2.7, we consider the phenomenon of partial synchronization in a system of three coupled logistic maps [38]. Here, we shall see how two of the chaotic oscillators can synchronize with one another, while the third oscillator remains unsynchronized. As described in Chapter 1, this problem is of particular interest in connection with the development of new types of communication systems.
2.2
Weak Stability of the Synchronized Chaotic State
The metric and topological properties of the logistic map have been studied extensively over the last two or three decades [39, 40]. It is known, for instance, that for any a G [0,4], fa has no more than a single attractor, which may be a cycle of points or of chaotic intervals. The parameter set C = {a G [0,4] : fa has an attracting cycle} is open and everywhere dense. At the same time, the parameter set K, = {a G [0,4] : fa has an absolutely continuous invariant measure} is nowhere dense (i.e., it has a Cantor-like structure), and the measure n{IC) > 0 [41]. Figure 2.1 shows a part of the bifurcation diagram for the logistic map with the well-known period-doubling cascade of attracting cycles 72". Increasing the parameter a beyond the accumulation point a* = 3.569... for this cascade, a reverse cascade of homoclinic bifurcations of the cycles ^2" takes place at the parameter values a„. At ao, for instance, the unstable fixed point XQ = 1 — 1/a undergoes its first homoclinic bifurcation as the critical point xc = 1/2 is mapped into XQ in three interations of fa. The bifurcation points an can easily be determined numerically (oo can also be calculated explicitly). The first four are given by a0 = 3.67857351042832..., ai = 3.59257218410697...,
38
Chaotic Synchronization: Applications to Living Systems
i.O
-====5Sl X
0.5
-c:^^8l -===
00
a'
3.54
a,
«2 , 3.59
an 3.64
3.69
Fig 2.1. Bifurcation diagram for the individual map fa:x^ ax(l - x) for 3.54 < o < 3.89. At each of the homoclinic bifurcation points o n , the map has an absolutely continuous invariant measure. a2 -
3.57480493875920...,
a 3 = 3.57098594034161....
In Fig. 2.1, a = ao is the parameter point at which the two bands of the chaotic attractor merge into a single band. At this point, /„ has an attracting interval To = [/ 0 (oo/4); (OQ/4)] consisting of two subintervals TQ = [/o(oo/4); XQ] U [XQ\ ( a o/4)] that are mapped one into the other under the action of /„. Moreover, fa has an absolutely continuous invariant measure in IV Hence the map is chaotic [39, 40]. Similarly, for a — an, fa has an attracting cycle F^n of 2 n intervals consisting of 2" + 1 subintervals that are pairwisely mapped one into another under the action of /„". Moreover, having an absolutely continuous invariant measure, /„ is chaotic in Fan. The main diagonal {x = y} is a one-dimensional invariant manifold of the two-dimensional map F. This implies that a point on the diagonal will be mapped into another point on this line, or, in other words, F({x = y}) C {x — y}. The existence of such a one-dimensional invariant manifold is clearly a consequence of the restrictions imposed by the symmetric coupling of two identical one-dimensional maps. Any small mismatch between the maps leads, in general, to the disappearance of the one-dimensional manifold, with the result that the dynamics becomes two dimensional. By subtracting the two one-dimensional maps in Eq. (2.1), one finds a
Transverse Stability of Coupled Maps 39
transverse line {x + y = 1 — 2e/a} that also maps into the main diagonal under the action of F [42]. This line is the preimage of the main diagonal, and part of the line (which maps into the attractive interval on the main diagonal) will belong to the basin of attraction for stable solutions on the diagonal. By adding the two one-dimensional maps in Eq. (2.1), one can show that the preimage of the transverse line is a circle centered in (a;, y) = (1/2,1/2) and with a radius {a2-2a + 4eY^/aV2. For any point on the main diagonal, F 0j£ has an eigendirection ui = (1,1) along the diagonal, and an eigendirection u-i — (1, —1) perpendicular to it. The corresponding eigenvalues are i/i = f'a{x) = a(l - 2x)
(2.3)
and z,2 = fa(x)
- 2e = a ( l - 2s) - 2e.
(2.4)
Along the diagonal the coupling vanishes, and the dynamics coincides with that of the one-dimensional map fa{x). Let I C M. be a chaotic attractor for fa{x), then A — {x = y G / } will be a one-dimensional invariant chaotic set for the coupled map system. A attracts points from its one-dimensional neighborhood along the diagonal. Does it also attract points from its two-dimensional neighborhood Us(A)? In other words, is A an attractor in the plane? The answer to this question clearly depends on the values of the transverse eigenvalues v
N
Ax= U m - £ > !£(*„)-2e|,
(2.5)
n=l
where {xn = fa(x)}^Li is a typical itinerary on A. If the set A has an absolutely continuous invariant measure, e.g., for the above mentioned parameter values a„, the value of Ax will be the same for almost all trajectories on A. If Ax is negative, we expect that A is attracting on average in a two-dimensional neighborhood. As previously mentioned, this type of stability is referred to as weak stability or stability in the Milnor sense [27], and the transition in which Ax changes sign is referred to as the blowout bifurcation [12]. Indeed, as shown by Alexander et al. [8, 9], in the case that A is a finite union of intervals and the invariant measure of fa on A is absolutely continuous, the
40
Chaotic Synchronization:
Applications
to Living
Systems
condition Ax < 0 guarantees that A attracts a positive Lebesgue measure set of points from any two-dimensional neighborhood Us(A). Moreover, weak stability in the Milnor sense implies that the measure fj, of points that are attracted to A approaches the whole measure of U$(A) as the width of the neighborhood 5 ->• 0, i.e., ( g ( AA\) nn[ /rrj , ( AAW )) ,,(R( limM L (2.6) <s-+o (J.{US(A)) Here B(A) denotes the basin of attraction of A, i.e., the set of points (x,y) £ K2 for which the stationary state (or UJ limit) is contained in A. When Ax becomes positive, the synchronized chaotic state loses its average attractivity. 4.0
-
w
I a
T> -2.0
1
R. 11 V V
-
/
2.0
Fig 2.2. Regions of parameter plane where the transverse Lyapunov exponent Ax < 0. Figure 2.2 shows the regions of parameter space in which Ax < 0, so that the synchronized attractor is (at least) weakly stable. The figure was obtained by performing 1000 scans of X±(e) for different values of a with a similar resolution along the e axis [26]. The stability regions clearly reflect the complexity of the bifurcation structure. In particular, we notice the irregular variation with a in the chaotic regime. For values of a below the accumulation point a* = 3.569..., the individual map displays an attracting cycle, and the synchronized behavior is also periodic. In this case there is no distinction between weak and asymptotic (or strong) stability, and our results are similar to those obtained by Schult et at. [43]. In each of the periodic windows in the region a > a*, the distinction between weak and strong stability likewise disappears. Moreover, for an ./V-periodic
Transverse Stability of Coupled Maps
41
synchronous state 7 ^ = {x\, £2, •••, %N}, the criterion for transverse stability N
J ] |/'K) - 2e\ < 1,
(2.7)
n=l
is satisfied if and only if the coupling parameter e belongs to the union of N (possibly overlapping) intervals, each including one point from 7jy. Some of these intervals correspond to positive values for e, and some of them correspond to e < 0. The stability intervals are clearly seen in Fig. 2.2. We notice, for instance, the stability regions for the period-6 solution around a — 3.63, for the period-5 solution around a = 3.74, and for the period-3 solution in the interval around a = 3.84. For each of these windows we also observe the signature of the period-doubling cascade in which they end. In a similar manner, the two-band chaotic attractor existing for a = a\, may give rise to a two e-intervals with Ax < 0, and the four-band chaotic attractor existing for a = 0,2 may have four (partly overlapping) intervals for e with weak stability.
2.3
Formation of Riddled Basins of Attraction
The condition A^ < 0 guarantees that almost all trajectories on A are transversely attracting. However, there can still be an infinite set of trajectories (having positive Lebesgue measure) in the neighborhood of A that are repelled from it. As we shall see, this situation is quite generic for our system of coupled logistic maps. To obtain an attractor in the usual (topological) sense, we must ascertain that all trajectories on A are transversely attracting [44]. Consider, for example, the fixed point P(xo,xo). Inserting XQ = 1 — 1/a into Eq. (2.4) and requiring that the magnitude of vi be less than 1, we find a coupling interval a— 1 a—3 in which the fixed point is transversely stable. If a > 3 and e falls outside this interval, both v\ and v^ will be numerically larger than 1, and P becomes a repelling node. Under the action of Fafi, a trajectory starting close to such a repelling node moves away from it along an integral curve. Under general conditions (yi 7^ v^ and v\ ^ 1/2), these integral curves can be obtained in a neighborhood of P by
42
Chaotic Synchronization: Applications to Living Systems
the smooth transformation £ = XQ + 6?j2 + higher order terms of the curves |^| = q ^ M / M " ^
(2.8)
where f = a; + y — 2XQ and r) = y — x are new coordinates defined along and perpendicular to the diagonal, respectively. C is an arbitrary constant that depends on the initial conditions, and b = af{v\ — j/f). Expression (2.8) gives the integral curves of the linearized map DF around the fixed point. It follows from this expression that the integral curves are tangential to the »?-axis for I^21 < \v\\ and tangential to the £-axis for |i^| < \v\\. These two situations are illustrated in Fig. 2.3. Formulae for the invariant curves of the nonlinear system (2.1) for a cycle Pn of any period n in the diagonal will be derived in Chapter 3.
Fig 2.3. The two generic forms for the repelling tongues that develop from each point on a transversely unstable periodic cycle. Note that these tongues do not have sharp boundaries. They are made up by bundles of trajectories that happen to fall in the neighborhood of P.
Let us now suppose that the fixed point P{XQ,XQ) belongs to the chaotic attractor A on the main diagonal, and that A has an (ergodic) absolutely continuous invariant measure on it. The fixed point P will then have infinitely many preimages, and the set of these preimages will be dense in A. Hence, we expect that the set U$ of preimages of the tongues TlM a n d 11^ will be
Transverse Stability of Coupled Maps 43
dense in some neighborhood of A (except in some half-neighborhoods of the end points for the intervals that span the attractor).
Fig 2.4. The riddling bifurcation produces a dense set of repelling tongues.
This situation is schematically illustrated in Fig. 2.4. The region Iff (shown in gray) is the locally repelling set of the attractor A. Each point from U#\A leaves the neighborhood U$(A) in a finite number of iterations. It is important to note that this repelling property has a local character. Our analysis says nothing about the fate of the trajectories once they have left the neighborhood Us. This depends on the global dynamics of F, and two different scenarios may occur: Scenario 1 Having left the locally repelling region U$(A), the trajectories wander around in phase space. However, they are restricted by nonlinear mechanisms to moving within an absorbing area [18, 19, 33, 37] that lies strictly inside the basin of attraction. The trajectories can never escape the absorbing area. Hence, they cannot diverge or approach an attractor outside the absorbing area. If there is no other attractor, apart from A, inside the absorbing area, then, sooner or later, most of the trajectories will return to Us(A). Some of them may again be mapped into U%(A) and, hence, again leave the neighborhood of A. This
44
Chaotic Synchronization:
Applications
to Living
Systems
type of dynamics gives rise to the temporal bursting characteristic of on-off intermittency. As long as the transverse Lyapunov exponent is positive, the bursts will never stop, and in essence we have an attractor in two-dimensional phase space with a more or less pronounced maximum of its invariant density in the neighborhood of the original one-dimensional attractor. As Aj_ becomes negative, the bursts tend to stop. However, this transition may not be very sharp, and, particularly in the presence of noise, one may still see some bursting even for negative values of Ax (attractor bubbling). (If another attractor exists inside the absorbing area, one may observe a global riddling of the basin of attraction for the weakly stable synchronized chaotic state with initial conditions that lead to the other attractor. We shall return to this situation in more detail in Chapter 3. Examples of the types of basins of attraction that one can observe in this case are shown in Figs. 2.13 and 2.19. The absorbing area itself will be defined in more detail in the following sections.) Scenario 2 If the nonlinear mechanisms are too weak to restrict the motion to an absorbing area inside the basin of attraction for the synchronized state, the alternative is that most points leaving U%{A) diverge or go to another attractor. This can be an attracting point cycle, an invariant torus, or a chaotic attractor. On the other hand, provided that Ax < 0 the measure of the complementary set f/| C U$\U]j of points that are attracted to A and never leave Us normally approaches the measure of Us for 5 —> 0 [8, 9]. Under these conditions we expect the basin of attraction to attain a globally riddled structure with holes that belong to the basin of another attractor. Mathematically expressed, this implies that, for any 8 > 0, (i) the complementary set Us(A)\B(A) will be everywhere dense in Us, and (ii) 0 < fi[B(A) C\ Us(A)] < /i[Us(A)]. (In contrast to this, the case of asymptotic stability of A is characterized by the existence of a 5 > 0 such that B{A) C Us — U\. This requires that all trajectories on A are transversely stable.)
2.4
Destabilization of Low-Periodic Orbits
In order to determine the boundaries in parameter space for the regions of absolute stability for the synchronized chaotic attractor A, we can start by considering the transverse stability of the various point cycles embedded in this attractor. We have already seen that the conditions for transverse stability for
Transverse Stability of Coupled Maps 45
the fixed point are — (a — l ) / 2 < e < — (a — 3)/2. The slanting line denoted "fixed point" in the middle of the stability diagram of Fig. 2.5 represents the lower edge of this zone. The upper end falls outside the range of coupling strengths considered in the figure. Hence, the fixed point is transversely stable to the right of the fixed point line. Along this line, destabilization of the fixed point happens in a subcritical pitchfork bifurcation. This implies that two repelling fixpoints (period-1 cycles) existing symmetrically on either side of the synchronous fixed point before the bifurcation, meet and are annihilated while at the same time destabilizing the synchronous fixed point in the transverse direction. 3.75
3.55
Fig 2.5. Bifurcation curves for the transverse destabilization of various low-periodic cycles embedded in the synchronized chaotic attractor. Regions in which the synchronized chaotic state is asymptotically stable are hatched. If more than one cycle of a given period occur, the cycle arising at the lowest value of o was considered.
The stability intervals for a point cycle Pn of period n are formally given by Eq. (2.7). The period-2 cycle P2 arises at a = 3, and thereafter alternates between the points a + l ± V ( o + l)(a-3) (2.9) 2a Evaluating the conditions for transverse stability for this cycle for a > 1 + \JZ Z l , 2 = 2/1,2
46
Chaotic Synchronization:
Applications
to Living
Systems
gives two intervals for e of which the interval to the left of e = —1/2 is given by - ^ [ 1 + \ / ( a + l)(a - 3) + 1] < e < ~[1
+ y/(a + l)(a - 3) - 1].
(2.10)
The borderlines of this interval are represented by the two curves denoted "period 2" in Fig. 2.5. The period-2 cycle is transversely stable between these curves. To the right the destabilization happens through a supercritical perioddoubling bifurcation, and to the left through a subcritical pitchfork bifurcation. The subcritical pitchfork bifurcation is qualitatively similar to the bifurcation we have just described for the synchronous fixed point. After the supercritical transverse period-doubling bifurcation, the points of the now repelling, symmetric period-2 orbit are each surrounded by a pair of points of an asymmetric period-4 saddle cycle with its unstable manifolds stretching along the synchronization manifold. Similarly, the curves denoted "period 4", "period 6", and "period 8" in Fig. 2.5 bound the regions of transverse stability for these cycles [26]. The minimum of the curve for the period-6 cycle falls at a = 3.626, where this cycle first arises. With the values of a considered in the figure, the period-3 cycle has not yet appeared in the individual map. Hence synchronized behavior with this periodicity cannot occur, and the figure delineates the regions of transverse stability for the most important cycles with a period less than or equal to n = 8. There are reasons to believe that transverse destabilization of cycles of higher periodicity generally play a less important role [47], and this is strongly supported by our numerical calculations. From the information in Fig. 2.5 we can therefore determine the regions of asymptotic stability for the synchronized chaotic attractor in each of the intervals ai < a < ao and ao < a. The first of these regions is bounded on both sides by the transverse destabilization of the period-2 cycle which occurs before destabilization of the period-8 and period-4 cycles. For higher values of a, destabilization of the period-6 cycle limits the region of asymptotic stability for the synchronized chaotic attractor, and there is also a region in parameter space where destabilization of the period-4 cycle is the first to take place. Below the homoclinic bifurcation point ao, destabilization of the fixed point is not significant because the two-band chaotic attractor existing in this range does not contain the fixed point. For a > ao, however, the synchronized chaotic state is no longer asymptotically stable to the left of the fixed point curve.
Transverse Stability of Coupled Maps 47
2
1
0
-1 -2
-1
0
1
2
Fig 2.6. Variation of the transverse Lyapunov exponent Ax with the coupling parameter e for o = a0. The synchronized chaotic attractor is absolutely stable for (approximately) —1.31 < e < -1.24.
Figure 2.6 shows a scan of the transverse Lyapunov exponent X± as a function of the coupling parameter for a = ao- For this value of a, the individual map exhibits a one-band chaotic attractor consisting of two subintervals at the moment when they merge. The points e = —1.544 and e = —0.478, where Aj_ changes sign, are the blowout bifurcation points. In the interval between these points, the synchronized chaotic state is at least weakly stable. Outside this interval, A is a so-called chaotic saddle [23, 45]. Also indicated in Fig. 2.6 is the interval from e = —1.31 to e = —1.24 in which the chaotic attractor is asymptotically stable. Figures 2.7(a)-(e) portray the basins of attraction for the synchronized chaotic state Ao at various values of the coupling parameter e. In each of these figures initial conditions leading to the chaotic attractor are plotted as gray points, and initial conditions leading to another attractor (or infinity) are left blank. Figure 2.7(a) shows the basin of attraction for e = —1.4, i.e., a little to the left of the region of asymptotic stability. In this region the synchronized period-6 cycle is transversely unstable, and the figure reveals the characteristic appearance of a globally riddled basin with a dense set of tongues with points that are repelled from the attractor emanating from the period-6 cycle and its preimages. As previously noted, the basin of attraction includes a section of a transverse line that maps into the attracting interval of the main diagonal, as well as sections of the circle that is a preimage of the transverse line. From each
48
Chaotic Synchronization.
Applications
125 a = a 0 . E = - l 4
to Living
Systems
125 a == a 0 , s = - l 2
(a)
y
-0 051 -0 05 125 a = ao,&=-l 1
tS
nos
125
125
-0 05
125 a = a o , E = - l 0
(c)
(b)
"(d)
^M y
-0 051 _ -0 05
-0 05t -0 05
125 125 a = a(,,s=-0
5
~^
•^1
1v
y
(e)
. ^ H KI T
S
^""'-^1
yF% \
^r -.* t * i
I:. *
0 05
-0.05
Fig 2.7. Basins of attraction for the one-band chaotic attractor AQ with different values of the coupling constant for a = a0 :(a) global riddling for e = —1.4, (b) local riddling for s = - 1 . 2 , (c) global riddling for e — —1.1, (d) global riddling for s = —1.0, and (e) global riddling close to the blowout bifurcation for e = - 0 . 5 . The basin of attraction is plotted gray in (a)-(d) and black in (e).
Transverse Stability of Coupled Maps
49
of these structures we have a similar dense set of repelling tongues, contributing altogether to the complexity of the basin. Figure 2.7(b) shows the basin of attraction for a coupling parameter immediately to the right of the region of asymptotic stability (e = —1.2). Here the in-phase period-2 cycle is transversely unstable. However, while the basin has a fractal boundary, there are no tongues in it belonging to the basin of another attractor. This is characteristic of a locally riddled basin of attraction where trajectories repelled from the synchronized chaotic state never reach the basin boundary and sooner or later return to the neighborhood of the attractor. In this case the synchronized state may be referred to as a Milnor attractor, i.e., it attracts all points from its neighborhood except a set of zero measure. As e is further increased, a transition occurs in which trajectories repelled from the main diagonal start to make contact with the basin boundary. As illustrated in Fig. 2.7(c), we then recover the globally riddled structure. Here e = —1.1, and the synchronized state is a weak attractor in the Milnor sense, i.e., there is a finite measure of points in its neighborhood that are repelled from it. For e — —1.0 (Fig. 2.7(d)), a new structure in the basin of attraction becomes manifest. Here we observe two lines parallel to the main axes and crossing the diagonal at a period-2 point. Each of these lines are invariant with respect to the second iterate of Fafi. A similar set of curves are found to cross the diagonal at the other point of the in-phase period-2 orbit. Finally, in Fig. 2.7(e), e = —0.5, and we are close to the blowout bifurcation at e = —0.478. Here the measure of the points that are attracted to the synchronized chaotic state becomes very small, and the majority of initial conditions lead to diverging orbits.
2.5
Different Riddling Scenarios
The above scenario for a — CLQ was characterized by a direct transition from asymptotic stability to global riddling at the left bifurcation point (e = —1.31). where the period-6 orbit becomes transversely unstable, and by a transition involving an interval with local riddling at the right bifurcation point (e = — 1.24), where the period-2 cycle undergoes a supercritical transverse perioddoubling bifurcation. Let us now consider what happens at other values of the nonlinearity parameter [26]. Figure 2.8 shows a scan of the transverse Lyapunov exponent Ax for a = a\
50
Chaotic Synchronization: Applications to Living Systems 2
1
0
-1
-
2
-
1
0
1
2
8 Fig 2.8. Variation of the transverse Lyapunov exponent Ax with the coupling parameter e for a = ai. Here we have two intervals of weak stability. The synchronized attractor is absolutely stable for (approximately) —1.46 < e < —1.16.
where the individual map fa exhibits a two-band chaotic attractor. Inspection of the figure shows that we now have two regions of weak stability for the synchronized chaotic state, one for positive values and one for negative values of e. In the region of negative coupling constants, blowout bifurcations occur at e = -1.472 and e = —1.0385. The region of absolute stability extends from e = -1.464 to e = —1.156. At both ends of this region, the in-phase period-2 cycle becomes transversely unstable. Figures 2.9(a)-(c) show typical examples of the basins of attraction observed in this region. At the same time, they illustrate an interesting change in the chaotic dynamics. In Fig. 2.9(a) (e = —1.3) we have an absolutely stable two-band attractor A\ on the main diagonal. There are holes in the basin of attraction. However, these holes do not emanate from points embedded in the attractor. Hence the basin is not riddled, but has a fractal boundary. Figure 2.9(b) was obtained for e = —1.1, i.e., immediately after the loss of asymptotic stability in the riddling bifurcation at e = —1.156. This provides a new example of a locally riddled basin. Trajectories that are repelled from the main diagonal do not reach the basin boundary, but sooner or later most of them return to the neighborhood of the synchronized state. As e is increased a little more, we observe a spontaneous breaking of the symmetry as the chaotic attractor spreads into two-dimensional phase space. The basin of attraction, on the other hand, changes only slightly. This is shown in Fig. 2.9(c) for e = —1.03.
Transverse Stability of Coupled Maps 51
1251 a
= a, ,e =-1 3
(a)
125f a
=a,,e=-l.l
(b)
y
-0 051 _ -0 05
-OOSL
-0 05 1 25
ra~=a,,s=-l,03
-0 051 -0 05
(c)
125
Fig 2,9. Basins of attraction for different values of the coupling parameter for a = oi : (a) fractal basin boundary for absolutely stable attractor, (b) locally riddled basin for weakly stable attractor, and (c) two- dimensional attractor restricted to the absorbing area.
Hence, in this case there is no interval with global riddling, but we observe a transition from local riddling to on-off intermittency when Ax becomes positive. The two-dimensional attractor A[ is bounded in phase space to an absorbing area defined by the iterates of the critical curves for Fafi [37, 46]. In this way the absorbing area plays a similar role for a two-dimensional non-invertible map as does the iterates of the critical point for the one-dimensional logistic map. The absorbing area lies fully within the basin of attraction, and as long as there is no contact between the two boundaries, the two-dimensional attractor continues to exist. With further increase of e (approximately at e = —0.95) a crisis takes place m which the borderline of the absorbing area touches the basin boundary,
52
Chaotic Synchronization:
Applications
to Living
Systems
and the two-dimensional attractor suddenly disappears. a = ai, s=0.33
-0.15
_
1.1
Fig 2.10. Basin of attraction for o = Oi and e = 0.33.
Figure 2.10 shows an example of the basins of attraction that one can observe in the other parameter window of weak stability for a = a%. Here e = 0.33, and the density of points in phase space from which the trajectory is attracted by the synchronized chaotic state is quite low. Only globally riddled basins of attraction are observed in this parameter window. For a = 02, a scan of the transverse Lyapunov exponent shows three regions of weak stability. This is illustrated in Fig. 2.11. The largest region extends from the blowout bifurcation at e = —1.462 to the blowout bifurcation at e == —1.0134. Within this region we find a region of asymptotic stability delineated by the riddling bifurcations at e = -1.461 and e = —1.015. In both of these bifurcations, the in-phase period-8 cycle becomes transversely unstable. (Transverse destabilization of the period-2 cycle is not significant since the points of this cycle do not belong to the chaotic set.) Figure 2.12(a) shows the locally riddled basin of attraction with fractal boundaries that one can observe for e = —1.14. In Fig. 2.12(b) the coupling parameter is increased to e = —1.01. This is immediately after the blowout bifurcation. The basin of attraction remains practically unaffected by this change. However, the dynamics of the coupled map system spontaneously breaks the
Transverse Stability of Coupled Maps 53
a = ai
J -1.462L
f-1.013...
V
0
1
8 Fig 2.11. Variation of the transverse Lyapunov exponent X± with the coupling parameter e for 125n7=" a=
-oos -0.05
a 2 ,s=-i.i4
(a)
125
12j
Fig 2.12. Basins of aitiaction for a = 02: (a) locally riddled basin with fractal basin boundary, and (b) two coexisting eight-band chaotic attractors.
symmetry, and the synchronized four-band attractor is replaced by two mutually symmetric two-dimensional eight-band attractors. The eight-band chaotic attractors have developed via a sequence of bifurcations from two mutually symmetric period-4 stable cycles produced in a saddle-node bifurcation away from the diagonal. Finally, Pig. 2.13 shows the basin of attraction for the synchronized four-band attractor in the second parameter window of weak stability (a = 02, e = —0.2). For these parameter values the synchronized chaotic state coexists with a stable
54
Chaotic Synchronization: Applications to Living Systems
1.25
a = a2,s =-0.2
y
0 05
-0.05
1.25
Fig 2.13. The basin of attraction for the synchronous chaotic state is riddled with holes that belong to the basin of a coexisting periodic cycle. Note that in this case the repelling tongues have sharp and well-defined edges.
period-2 point cycle shown by crosses and the basin of attraction for the chaotic state is riddled with holes that belong to the basin of attraction of the point cycle. Note that in this case the repelling tongues have sharp and well-defined edges, defined by the stable manifolds of asynchronous saddle cycles situated along these edges and produced in the stabilization of the asynchronous period2 cycle. As we shall see in Chapter 3, such asymmetric point cycles play a fundamental role in the global dynamics of our two-dimensional system (2.1).
2.6
Intermingled Basins of Attraction
Consider now the map Fa<e for e = — 1:
F 1
- AJ*U(i-*)-(*-v)J-
( }
e = — 1 is the value of the coupling parameter for which we observed the particular basin structure depicted in Fig. 2.7(d). It is easy to show that for any a the two mutually perpendicular straight lines x = X2 and y = x% both remain invariant under two iterations of F_i, i.e., Ftx{{x = X2}) C {x = X2}
Transverse Stability of Coupled Maps 55
and F*i({y = £2}) C {y = ^2}. Here #2, as given by Eq. (2.9), is the larger of the two amplitudes 0:1,2 for the period-2 cycle 72. Let g : x \-~¥ g(x), x G l denote the one-dimensional map that Flt induces along these lines, i.e., (g(x),x2) = Fl1(x,x2) (2.12)
Fig 2.14. Chaotic attractors off the main diagonal: (a) sketch of the one-dimensional map g(x), (b) its bifurcation diagram, (c) the corresponding longitudinal Lyapunov exponent, and (d) the transverse Lyapunov exponent [26].
Figure 2.14(a) portrays the function g(x) for a = 3.6, and Fig. 2.14(b) shows the bifurcation diagram obtained by varying a over the interval from 3.4 to 3.74. Figures 2.14(c) and (d) provide scans of the corresponding longitudinal and transverse Lyapunov exponents Ay and Ax, respectively. g(x) has a finite interval over which it is unimodal, and Fig. 2.14(b) demonstrates that it produces a transition to chaos in accordance with the usual Feigenbaum scenario.
56
Chaotic Synchronization:
Applications
to Living
Systems
Magnifications of Figs. 2.14(c) and (d) show that Ay is positive for a = 3.6, and that Ax is negative. Hence, for this value of a, the numerical calculations indicate that g{x) displays a two-band chaotic attractor. By virtue of its symmetry with respect to the main diagonal, the map F-\ exhibits two one-dimensional four-band invariant chaotic sets A4 and A4. Figure 2.15 displays the basins of attraction for each of these sets. For a = 3.6 and e = — 1, the synchronized state on the main diagonal is a chaotic saddle, and almost all trajectories that remain finite approach either A4 or A4. However, for both of these sets the basin of attraction is riddled with holes that belong to the basin of the other set. Hence, the basins of attraction exhibit an intermingled structure [9] with one half measure set of points that are attracted to A4, and one half measure set that is attracted to A4. Figure 2.15(b) shows a magnification of the intermingled basin structure in a region around (x2, X2).
2.7
Partial Synchronization for Three Coupled Maps
The purpose of the present section is to illustrate the phenomenon of partial synchronization by considering a system of three nonlinearly coupled logistic maps [38]. For a specific form of asymmetric coupling, we determine the region in parameter space in which two of the maps can attain a state of chaotic synchronization while the third map exhibits independent chaotic behavior. We also study the bifurcations through which this state of partial synchronization arises and breaks down, and we follow the associated transformations of the basins of attraction. Problems of this nature arise, for instance, in connection with the development of new principles for secure communication based on the so-called masterslave configuration [5, 48, 49, 50] Master:
\ *n+i = f(xn) + e (f(zn){ Zn+1 = f(zn) + £ (f(xn) (2.13)
siave- { y»+i
=
+e
z
f(y*) (f( ")-
\ un+i = f(zn) + e (f(yn)
-
As before, f(x) is here a one-dimensional, chaotic map. When transmitting the signal zn from the master system we want to be able to read the message
Transverse Stability of Coupled Maps 57
(a)
i
(b) ••-"•••^••'•lA.-^g^'""
'-.-^
Fig 2.15. Intermingled basins of attraction for two coexisting four-band chaotic attractors: (a) overview, and (b) detail around one of the points of the period-2 cycle. Initial conditions from which the trajectory approaches one of the chaotic attractors are marked black, while initial conditions leading to the other attractor are left blank [26].
58
Chaotic Synchronization:
Applications
to Living
Systems
xn from the slave system. This will be possible if the parameters of the coupled system can be adjusted such that yn synchronizes with xn. On the other hand, to mask xn during the transmission process, synchronization between xn and zn should be avoided. Whenever xn and yn synchronize, un and z„ will also synchronize, and by comparing the variables of the slave system, one can immediately detect whether synchronization has occurred or not. Hence, the fourth equation in (2.14) can be omitted, and we arrive at a system of three coupled maps: x„+i = f{xn) + e(f(zn) - f(xn)) J/„+i = f(yn)+e(f(zn)-f(yn)) (2.14) Zn+l
=
f(z„) + S(f(xn)
-
f(zn))
with / = fa : R —> K. The essential aspects of the above problem may be formulated as follows: We would like to find a set of parameters {a, e} for which the chaotic variables x and y synchronize with one another while the variable z remains unsynchronized. Hence, the asymptotic state (x, y, z) should belong to the synchronization plane Uxy = {(x, y, z) \x = y } without becoming restricted to the main diagonal D = {(x,y,z)\x = y = z}. Since we want this situation to be realized for a massive set of initial conditions, the following criteria must be fulfilled [49, 50]: 1. The synchronization plane n ^ must be invariant under the action of F. 2. In order to avoid complete synchronization, the main diagonal must be transversely repelling in the plane Hxy (or it should attract only a zero Lebesgue measure set of points). 3. The two-dimensional chaotic attractor (or attractors) that under these conditions can exist in Uxy must be stable in the direction perpendicular to this plane. Figure 2.16 illustrates the situation for a set of three coupled logistic maps with a = ao and e = 0.98. Here, we have three different two-dimensional chaotic attractors in the synchronization plane. In each of these attractors the variables x and y are synchronized while z is not. Actually, as we shall see below, the occurrence of partial synchronization in the system (2.14) is restricted to an interval around e = 1. Moreover, by contrast to the original two-dimensional coupled maps system (2.1), F is not symmetric with respect to permutation of the variables. The planes Uxy and Iixz = {(x,y,z)\ x — z) are invariant under
Transverse Stability of Coupled Maps 59
y
"
Fig 2.16. Partial synchronization in a system of three coupled one-dimensional maps. The limiting states are three two-dimensional chaotic attractors situated in the synchronization plane {x — y, z}. The figure was obtained for the coopled map system (2.14) with fa(x) ~ oar(l - x), a = a0 = 3.678573... and e = 0.98.
F whereas the plane Uyz = {(x,y,z) \y = z} is not. The main diagonal D is also invariant. Prom a stability point of view, the partial synchronization may be strong or weak, depending on whether the chaotic set embedded in the synchronization manifold 11^ is asymptotically stable (in the Lyapunov sense) or not. In the case of weak stability as defined by Milnor [27], the synchronized chaotic state is stable on average (its transverse Lyapunov exponent is negative), but particular orbits embedded in this state are unstable in the direction perpendicular to IX^. The simplest form of asymptotic dynamics that can occur in the coupled map system (2.14) is the fully synchronized (or coherent) behavior in which all three oscillators display the same temporal variation, xn = yn = zn, so that the motion is restricted to the main diagonal in the three-dimensional phase space. Along this diagonal the dynamics is specified by the one-dimensional map f(x), and the Lyapunov exponent is 1
N
(2.15) n=l
where f(x) denotes the derivative of f(x) and xn, n = 1,2... represents a typical itinerary. Obviously, Ar> > 0 if the dynamics of f(x) is chaotic.
60
Chaotic Synchronization:
Applications
to Living
Systems
Let us first consider the stability of the fully synchronized motion. In a point (x, x, x) of D, the Jacobian matrix of the coupled map system (2.14) is [49, 50]
DF = f{x)
/ l - e 0 \ e
0 l-e 0
e \ e . l-e)
(2.16)
Besides the direction ao = (1,1,1) along the diagonal, the Jacobian matrix DF has the eigenvectors axy — (1,1, —1) and axz = (0,1,0) which lie, respectively, in the plane Uxy and in the plane Uxz. The corresponding eigenvalues — f'{x)(l — 2e) and vxz = f'(x)(l — e), and the associated Lyapunov exponents are [51] AaJ!/ = Azj + l n | l - 2 e | (2.17) for deviations away from the main diagonal in the plane IL^y, and Axz = A£, + l n | l - e |
(2.18)
for deviations away from D in the plane Hxz. If Xxy < 0, the fully synchronized state is at least weakly stable in the plane Il^j,. This implies that D attracts a positive Lebesgue measure set of points from its two-dimensional neighborhood in the synchronization plane. For the fully synchronized state to be asymptotically stable, all trajectories embedded in this state must be transversely stable. As we shall see below, regions of parameter space may exist in which Xxy < 0 while at the same time orbits embedded in the synchronized chaotic state are transversely unstable. As before, the transition in which A^ changes sign from negative to positive is referred to as the blowout bifurcation [8, 9, 10, 11]. In the partially synchronized state, the motion is restricted to the plane II xy , and the three-dimensional map system (2.14) reduces to F\
•
X
The expression (2.18) for the Lyapunov exponent with respect to perturbations transverse to Il^y maintains its general form, but \o must be replaced by
^=^^Elni/'wi
( 2 - 2 °)
n=l
where xn, n = 1,2... now denotes the first coordinate of the points {(xn, zn)™=1\ of a typical trajectory on the partially synchronized chaotic set A embedded
Transverse Stability of Coupled Maps 61
in the synchronization plane TLxy. As illustrated in Pig. 2.16, several partially synchronized attractors may coexist for the same parameter values {a, e). Each attractor Am, rn= 1,2... will then be characterized by its own value of Xxz, A™ = \cz(An)- For the partially synchronized chaotic state to be at least weakly stable we must have A™ < 0. To proceed with the analysis let us now assume that the one-dimensional map fa(x) = ax(l — x), 3 < a < 4.
-0.5
0
0.5
1
1.5
e Fig 2.17. Overview of the main stability regioas in {a,e} parameter plane [38]. The hatched area is the region of asymptotic stability for the fully synchronized chaotic state. At the edge of this region, the fixed point becomes transversely unstable, and as we move further away from the line e = | , additional low periodic cycles become transversely unstable. The blowout bifurcation at the outer borderline of the black domain delineates the region of weak stability for the fully synchronized chaotic state. In the dark gray region around e = 1, partial synchronization may occur.
The two-dimensional phase diagram in Fig. 2.17 provides an overview of the main stability regions in {a, e} parameter plane for dynamics that is restricted to the synchronization plane Uxy [38]. The hatched area represents the region in which the fully synchronized motion on the main diagonal is asymptotically stable. In this area, all trajectories embedded in the fully synchronized state are transversely stable. At the right hand edge of the hatched region, the fixed point PQ — (XQ, XQ, XQ) , XQ = 1 — 1/a undergoes a supercritical pitchfork bifurcation and two mutually symmetric period-1 cycles (fixed points) are produced on either side of D. This occurs when the transverse eigenvalue J%(XQ) = f'(xo)(l - 2e) becomes equal to 1, which happens at e = |(o — l)/(a — 2). For a = ao, the pitchfork bifurcation takes place at e = 0.7979. At the left
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hand edge of the hatched region, the fixed point PQ undergoes a supercritical period-doubling bifurcation as the transverse eigenvalue vxy (XQ) becomes equal to —1, or for e = \{a — 3)/(a — 2). For a = a0, this occurs at e = 0.2021. Situated in the synchronization plane Hxy but off the main diagonal D these cycles represent various forms of partially synchronized behavior. To differentiate these cycles from the fully synchronized cycles, we shall refer to them as asynchronous. As we move out of the light gray region, the period-2 cycle a+l±V(q+l)(a-3) *l,2 =
(2 Ya
-21)
also turns transversely unstable. This occurs in period-doubling bifurcations at e = ^[l±((a+l)(o-3)-l)s].
(2.22)
For a = ao, the bifurcation points are e = 0.1610 and e = 0.8391. Both of these period-doubling bifurcations are supercritical and produce an asynchronous period-4 cycle off the main diagonal. As we move farther away from the line e = \ where vxy = 0, the synchronized period-4 cycle also undergoes a supercritical transverse period-doubling to give birth to an asynchronous period-8 cycle. To the right hand side of the phase diagram this occurs for e = 0.8451 for a = ao. For slightly higher values of e (e = 0.851), the synchronized period-8 cycle likewise undergoes a supercritical transverse period-doubling giving rise to an asynchronous period-16 cycle. Figure 2.18 illustrates the situation in the Uxy phase plane for a = ao and £ = 0.81, i.e., immediately after the transverse destabilization of the fixed point and before the transverse destabilization of the synchronous period-2 cycle has occurred [33]. Here, the synchronization manifold {x = z] represents the main diagonal D in K3. The repelling fixed point PQ on this diagonal is indicated by a full circle, and the two mutually symmetric period-1 saddle cycles produced in the initial pitchfork bifurcation are shown as small triangles. Pi represents one of these saddle cycles with its stable and unstable manifolds drawn as fine lines. The arrows on these manifolds indicate their forward direction. Also indicated in Fig. 2.18 are the points of the synchronous period-2 cycle (small squares). The curves denoted L\ and L2 are arcs of so-called critical curves [18, 19]. L\ and Z/2 are obtained from the first and the second iterates for the map F\x=y of the points [37] L0 = {(x,z)eR2 : \DF\x=y\=0} (2.23)
Transverse Stability of Coupled Maps 63
Fig 2.18. Situation in phase space after the transverse destabilization of the fixed point Pa has occurred. The small squares represent the points of the synchronized period-2 cycle. Also shown in this figure is the mixed absorbing area A' as defined by the outer loops of the unstable manifolds to the asynchronous period-1 saddle cycles and by sections of the critical curves L\ and Li. The absorbing area A is bounded by sections of the critical curves [33]. a = a0 and e = 0.81.
at which the Jacobian determinant \DF\ of F\x=y vanishes. The critical curves define the absorbing area A, and, as previously noted, they play the same important role for restricting the dynamics of two-dimensional endomorphisms as the iterates of the critical point do for one-dimensional non-invertible maps. Returning to Fig. 2.17, the fractal curve represents the moments of blowout bifurcation where Xxy becomes positive, and the fully synchronized state loses its average transverse stability. Finally, the dark gray domain delineates the region of partial synchronization. Since partial synchronization requires \xy > 0 and Axz < 0, this phenomenon can only occur to the right hand side of the diagram where e « 1. For a = ao, the blowout bifurcation at which \xy becomes positive and the fully synchronized chaotic state loses its weak stability occurs at E = 0.8563. At the rightmost edge of the dark gray region, the partially synchronized chaotic attractor disappears in a boundary crisis [52]. For a = ao, this occurs at s = 1.089. Consider now the further development of the dynamics for a = ao- At e = 0.8486, the two asynchronous period-1 saddle cycles stabilize in an inverse,
64
Chaotic Synchronization:
Applications
™^**>*—
1
z
to Living
_
,
,' ',
Systems
\
"'
«
4
-'«-v
' , ':
)
' 4-:
Fig 2.19. Basins of attraction for the synchronized chaotic state (white), for the coexisting asynchronousfixedpoints (light and dark gray), and for the asynchronous period-4 cycle (scattered black points). Also indicated on this figure (black circles) are the asynchronous period-2 saddle cycles whose stable manifolds define the basin boundaries for the asynchronous fixed points, o = oo and e = 0.8804. subcritical period-doubling bifurcation in the direction of their unstable manifolds. This produces a pair of mutually symmetric period-1 stable nodes, each surrounded by an asynchronous period-2 saddle cycle. The stable manifolds of these saddle cycles delineate the basins of attraction for the stable period-1 orbits. Right after the stabilization, the immediate basins of attraction stretch as narrow tongues all the way down to the repelling fixed point PQ. With further increase of the coupling parameter, the points of the period-2 saddle cycles move farther apart, and the basins of attraction for the asynchronous period-1 nodes become wider. This situation is illustrated in Fig. 2.19 for e = 0.8804. Here, the basin of attraction for the fully synchronized chaotic state is shown in white. The mutually symmetric stable period-1 orbits are indicated by crosses. The basins of attraction for these two solutions are shown in light and dark gray, respectively. In addition to the immediate basins issuing from the repelling fixed point (light gray cross), these basins consist of all the preimages of the immediate basins. On the boundaries of the immediate basins, the mutually symmetric
Transverse Stability of Coupled Maps 65
period-2 saddle cycles are indicated as black circles. Also shown in Fig. 2.19 are the transversely unstable synchronous period-2 cycle (light gray squares) surrounded by the points of the asynchronous period-4 cycle (crossed circles), and the transversely unstable synchronous period-4 cycle (dark gray circles).
0.92
s
1-089
0.92
S
1.089
Fig 2.20. Bifurcation diagram for solutions of family A\ that develop from one of the asynchronous period-1 cycles produced by stabilization of the saddle cycles arising in the transverse pitchfork bifurcation of the fixed point (a). Corresponding variation of the Lyapunov exponents (b). a = a0-
Each of the stable asynchronous solutions corresponds to a particular limiting state for the coupled map system. Under variation of the coupling parameter these states may develop into more complicated attractors. Let us consider the following different families of attractors in the synchronization plane Ylxy: A\: attractors arising from the asynchronous period-1 cycles produced in the initial pitchfork bifurcation of the fixed point PQ = {x$, XQ, XQ) AI'. attractors arising from the asynchronous period-4 cycle born in the transverse period-doubling bifurcation of the synchronous period-2 cycle A3: two-dimensional chaotic attractors born after the blowout bifurcation of the synchronous chaotic attractor on the diagonal. Let us start by considering the family A\. This family begins with the stabilization of the asynchronous period-1 cycles at e = 0.8486. The bifurcation diagram in Fig. 2.20(a) illustrates the development that takes place within this family as the coupling parameter is increased from 0.92 to 1.089, and Fig. 2.20(b) shows the corresponding variations of the Lyapunov exponents Ai,
66
Chaotic Synchronization: Applications to Living Systems
Fig 2.21. Phase portrait of the two mutually symmetric fourpiece chaotic attractors of family Ai that exist for a = oo and e = 0.965. Also shown is the two-dimensional attractor produced after the blowout bifurcation of the fully synchronized chaotic set. Each of these three attractors represents a state of partial synchronization [38].
and A2, in the synchronization plane 11^ and Xxz in a direction transverse to this plane. We immediately observe that A^ is negative and numerically fairly large in the interval around e = 1. On the other hand, both Ai and A2 are positive in most of the interval above e = 0.96. Hence, we have a two-dimensional chaotic attractor in the synchronization manifold, and this attractor is at least weakly stable in the direction transverse to Uxy. Inspection of the bifurcation diagram in Fig. 2.20(a) shows that the asynchronous period-1 cycle first undergoes a Hopf bifurcation of a map (also referred to as a Neimark bifurcation) at e = 0.9305. This produces a quasiperiodic behavior on a non-resonant torus. Further developments result in the appearance of a resonant torusfollowedby torus destruction and a transition to chaos. Figure 2.21 displays a phase portrait of the two mutually symmetric fourpiece chaotic attractors that exist for e = 0.965 [38]. The dark and light gray regions represent parts of their basins of attraction, and the white region is part of the basin of attraction for the two-dimensional chaotic attractor of family A3 that has developed after a blowout bifurcation of the fully synchronized chaotic attractor. As e is further increased, the four pieces of the partially synchronized chaotic states merge with one another, and for e = 1 the three partially synchronized chaotic attractors merge into a single large attractor in the synchronization plane. This explains the dramatic change in the size of the attractor that can be seen in Fig. 2.20(a) for this value of the coupling
Transverse Stability of Coupled Maps 67
Fig 2.22. Large partially synchronized chaotic attractor that exist for a = O,Q and e = 1.07. For slightly larger values of the coupling parameter, this attractor disappears in a boundary crisis.
parameter. Figure 2.22 shows the partially synchronized chaotic attractor that exists for e — 1.07, i.e., after the merging of the three two-dimensional attractors displayed in Fig. 2.21 and before the boundary crisis.
Let us finally consider the process through which the large two dimensional attractor around the main diagonal (shown in black in Figs. 2.16 and 2.21) is born. This process is illustrated in Figs. 2.23(a)-(c) where the coupling parameter attains the values e — 0.9250,0.9255, and 0.9300, respectively. As before, the dark and light gray areas represent the basins of attraction for the mutually symmetric attractors of family A\. After the blowout bifurcation of the fully synchronized state at e = 0.8563, the four-band chaotic attractor of family A% controls the white basin of attraction. In Figs. 2.23(a)-(c), this attractor is colored black. In Fig. 2.23(a) the points of the repelling period-2 cycle on the main diagonal are indicated by squares. As the coupling parameter is increased we observe how the four-band chaotic attractor of Fig. 2.23(a) expands and merges in a process that first generates a set of so-called rare points [53J. It is also worth noticing how the final attractor in Fig. 2.23(c)
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Chaotic Synchronization: Applications to Living Systems
Fig 2.23. Formation of the large two-dimensional, partially synchronized attractor around the diagonal through merging of the branches of the four-band chaotic attractor: e = 0.9250 (a), 0.9255 (b), and 0.9300 (c). a = a0. remains b o u n d e d b y a set of critical curves.
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Transverse Stability of Coupled Maps
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A. S. Pikovsky, On the Interaction of Strange Attractors, Z. Phys. B 55, 149-154 (1984). L. M. Pecora and T. L. Carroll, Synchronization in Chaotic Systems, Phys. Rev. Lett. 64, 821-824 (1991). C. W. Wu and L. O. Chua, A Unified Framework for Synchronization and Control of Dynamical Systems , Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 979-998 (1994). N. F. Rulkov, Images of Synchronized Chaos: Experiments with Circuits, Chaos 6, 262-279 (1996). L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar, and J. F. Heagy, Fundamentals of Synchronization in Chaotic Systems, Concepts, and Applications, Chaos 7, 520-543 (1997). G. Kolumban, M. P. Kennedy, and L. O. Chua, The Role of Synchronization in Digital Communications - Part I: Fundamentals of Digital Communications, IEEE Trans. Circuits and Systems CS-44, 927-935 (1997). J. C. Alexander, J. A. Yorke, Z. You, and I. Kan, Riddled Basins, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 795-813 (1992). J. C. Alexander, B. R. Hunt, I. Kan, and J. A. Yorke, Intermingled Basins for the Triangle Map , Erg. Theor. Dyn. Syst. 16, 651-662 (1996). J. C. Sommerer and E. Ott, A Physical System with Qualitatively Uncertain Dynamics, Nature (London) 365, 138-140 (1993). E. Ott and J. C. Sommerer, Blowout Bifurcations: The Occurrence of Riddled Basins and On-Off Intermittency, Phys. Lett. A 188, 39-47 (1994). E. Ott, J. C. Sommerer, J. C. Alexander, I. Kan, and J. A. Yorke, Scaling Behavior of Chaotic Systems with Riddled Basins, Phys. Rev. Lett. 7 1 , 4134-4137 (1993). N. Piatt, E. A. Spiegel, and C. Tresser, On-Off Intermittency: nism for Bursting, Phys. Rev. Lett. 70, 279-282 (1993).
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[15] L. Yu, E. Ott, and Q. Chen, Fractal Distribution of Floaters on a Fluid Surface and the Transition to Chaos for Random Maps , Physica D 53, 102-124 (1991). [16] C. Mira and F. Roubellat, Cas ou le domaine de stabilite d'un ensemble limite attractif d'une recurrence du deuxieme ordre n'est pas simplement connexe, C.R. Acad. Sci. Paris 268, Serie A, 1675-1678 (1969). [17] I. Gumowski and C. Mira, Recurrences and Discrete Dynamical Systems (Springer Verlag, New York, 1980). [18] C. Mira, Chaotic Dynamics (World Scientific, Singapore, 1987). [19] C. Mira and T. Narayaninsamy, On Two Behaviors of Two-Dimensional Endomorphisms. Role of the Critical Curves, Int. J. Bifurcation Chaos Appl. Sci. Eng. 3, 187-194 (1993). [20] C. Grebogi, E. Ott, and J. A. Yorke, Chaotic Attractors in Crisis, Phys. Rev. Lett. 48, 1507-1510 (1982). [21] C. Grebogi, E. Ott, and J. A. Yorke, Basin Boundary Metamorphoses: Changes in Accessible Boundary Orbits , Physica D 24, 243-262 (1987). [22] C. Mira, D. Fournier-Prunaret, L. Gardini, H. Kawakami, and J.C. Cathala, Basin Bifurcations of Two-Dimensional Noninvertible Maps: Fractalization of Basins, Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 343381 (1994). [23] P. Ashwin, J. Buescu, and I. Stewart, Bubbling of Attractors and Synchronization of Chaotic Oscillators, Phys. Lett. A 193, 126-139 (1994). [24] J. F. Heagy, T. L. Carroll, and L. M. Pecora, Desynchronization Orbits, Phys. Rev. E 52, R1253-R1256 (1995).
by Periodic
[25] P. Ashwin, J. Buescu, and I. Stewart, From Attractor to Chaotic Saddle: A Tale of Transverse Instability, Nonlinearity 9, 703-737 (1996). [26] Yu.L. Maistrenko, V.L. Maistrenko, A. Popovich, and E. Mosekilde, Transverse Instability and Riddled Basins in a System of Two Coupled Logistic Maps, Phys. Rev E 57, 2713-2724 (1998).
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[27] J. Milnor, On the Concept of Attractor , Commun. Math. Phys. 99, 177195 (1985). [28] Y.-C. Lai, C. Grebogi, J. A. Yorke, and S. C. Venkataramani, Riddling Bifurcation in Chaotic Dynamical Systems, Phys. Rev. Lett. 77, 55-58 (1996). [29] S.C. Venkataramani, B.R. Hunt, and E. Ott, Bubbling Transition, Phys. Rev. E 54, 1346-1360 (1996). [30] S.C. Venkataramani, B.R. Hunt, E. Ott, D.J. Gauthier, and J.C. Bienfang, Transitions to Bubbling of Chaotic Systems, Phys. Rev. Lett. 77, 5361-5364 (1996). [31] Yu. Maistrenko, V. Maistrenko, O. Popovych, and E. Mosekilde, Desynchronization of Chaos in Coupled Logistic Maps, Phys. Rev. E 60, 28172830 (1999). [32] V. Astakhov, A. Shabunin, T. Kapitaniak, and V. Anishchenko, Loss of Chaos Synchronization through the Sequence of Bifurcations of Saddle Periodic Orbits, Phys. Rev. Lett. 79, 1014-1017 (1997). [33] Yu. L. Maistrenko, V. L. Maistrenko, A. Popovich, and E. Mosekilde, Role of the Absorbing Area in Chaotic Synchronization, Phys. Rev. Lett. 80, 1638-1641 (1998). [34] Yu. Maistrenko and T. Kapitaniak, Different Types of Chaos Synchronization in Two Coupled Piecewise Linear Maps, Phys. Rev. E 54, 3285-3292 (1996). [35] Yu. Maistrenko, T. Kapitaniak, and P. Szuminski, Locally and Globally Riddled Basins in Two Coupled Piecewise-Linear Maps, Phys. Rev. E 56, 6393-6399 (1997). [36] A. S. Pikovsky and P. Grassberger, Symmetry Breaking Bifurcation for Coupled Chaotic Attractors , J. Phys. A 24, 4587-4597 (1991). [37] L. Gardini, R. Abraham, R. J. Record, and D. Fournier-Prunaret, A Double Logistic Map , Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 145-176 (1994).
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[38] A.V. Taborov, Yu.L. Maistrenko, and E. Mosekilde, Partial Synchronization in a System of Coupled Logistic Maps, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 1051-1066 (2000). [39] P. Collet and J.-P. Eckmann, Iterated Maps of the Interval as Dynamical Systems (Berkhauser, Boston, 1980). [40] W. De Melo and S. van Strien, One-Dimensional Dynamics (Springer, New York, 1993). [41] M. Jakobson, Absolutely Continuous Invariant Measure for One Parameter Families of One Dimensional Maps , Commun. Math. Phys. 8 1 , 39-88 (1981). [42] J. Fr0yland, Introduction to Chaos and Coherence (Institute of Physics, Bristol, 1992). [43] R.L. Schult, D.B. Creamer, F.S. Henyey, and J.A. Wright, Symmetric and Nonsymmetric Coupled Logistic Maps, Phys. Rev. A 35, 3115-3118 (1987). [44] M.M. Sushchik, N.F. Rulkov, and H.D.I. Abarbanel, Robustness and Stability of Synchronized Chaos: An Illustrative Model, IEEE Trans. Circ. Syst. 44, 867-873 (1997). [45] H. E. Nusse and J. A. Yorke, Analysis of a Procedure for Finding Numerical Trajectories Close to Chaotic Saddle Hyperbolic Sets , Erg. Theoret. Dyn. Syst. 11, 189-208 (1991). [46] C. Mira, L. Gardini, A. Barugola, and J.-C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps (World Scientific, Singapore, 1996). [47] B. R. Hunt and E. Ott, Optimal Periodic Orbits of Chaotic Systems, Phys. Rev. Lett. 76, 2254-2257 (1996). [48] A.S. Dmitriev, M. Shirokov, and S.O. Starkov, Chaotic Synchronization in Ensembles of Coupled Maps, IEEE Trans. Circuits and Systems 44, 918926 (1997). [49] M. Hasler, Yu. Maistrenko, and O. Popovych, Simple Example of Partial Synchronization of Chaotic Systems, Phys. Rev. E 58, 6843-6846 (1998).
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[50] Yu. Maistrenko, 0 . Popovych, and M. Hasler, On Strong and Weak Chaotic Partial Synchronization, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 179203 (2000). [51] M. de S. Vieira, and A.J. Lichtenberg, Nonuniversality of Weak Synchronization in Chaotic Systems, Phys. Rev. E 56, R3741-R3744 (1997). [52] C. Grebogi, E. Ott, and J.A. Yorke, Crises, Sudden Changes in Chaotic Attractors and Transient Chaos, Physica D 7, 181-200 (1983). [53] Yu. Maistrenko, I. Sushko, and L. Gardini, About Two Mechanisms of Reunion of Chaotic Attractors, Chaos, Solitons and Fractals 9, 1373-1390 (1998).
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Chapter 3 UNFOLDING THE RIDDLING BIFURCATION 3.1
Locally and Globally Riddled Basins of Attraction
In the previous chapter we made a first presentation of some of the phenomena that one can observe in connection with the synchronization of two (or more) identical chaotic oscillators. Under variation of the coupling parameter we showed how a state of complete (or partial) synchronization can be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state itself remains attracting on the average. The purpose of the present chapter is to continue our analysis of coupled non-invertible maps to show that the transition to riddling can be soft or hard depending on whether the first orbit to lose its transverse stability undergoes a super- or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to a locally riddled basin. As an illustration of this analysis we present three different scenarios for the onset of riddling and for the subsequent transformations of 75
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the basins of attraction. However, let us start the discussion by recalling some of the basic concepts and ideas that we introduced in Chapter 2. Riddled basins of attraction may be observed in regions of parameter space where the synchronized chaotic state is attracting on average (the typical transverse Lyapunov exponents are negative) while at the same time particular orbits embedded in the chaotic set are transversely unstable (the corresponding eigenvalues are numerically larger than one) [1, 2, 3, 4, 5]. The basin of attraction for the synchronized chaotic state may then become a fat fractal, riddled with initial conditions from which the trajectories diverge towards infinity or approach other asymptotic states. The transition in which the first orbit on the chaotic set becomes transversely unstable is referred to as the riddling bifurcation. For a system of two symmetrically coupled one-dimensional maps, this bifurcation can be either a pitchfork bifurcation (eigenvalue +1) [6] or a period-doubling bifurcation (eigenvalue —1) [7, 8]. However, transverse destabilization of orbits embedded in the chaotic set is not sufficient for an observable riddling to arise. This will depend on the global dynamics of the system. Having left the locally repelling regions in the neighborhood of the chaotic set, the trajectories may wander around in phase space without ever approaching another attractor (or escaping to infinity). Sooner or later most of them will return to the neighborhood of the synchronization manifold. Some may again be mapped into repelling tongues, while others will be attracted by the chaotic set, and eventually almost all trajectories starting close to the invariant manifold will end up in the synchronized state. This produces the phenomenon that has been called local riddling [3, 4, 5, 7, 8, 9, 10]. In the presence of noise, a locally riddled basin of attraction will manifest itself in the form of attractor bubbling [3, 4, 5, 11, 12], where intervals of desynchronized bursting behavior occur. Denoting the synchronized chaotic state by A, its 5-neighborhood by Us(A), and its basin of attraction by B(A), the basin is said to be locally riddled if there exists a 6 > 0 such that A attracts almost all trajectories originating from points of Us(A), i.e., fi {B{A) D Us (A)} = fx {Us (A)}, where fi {•} denotes Lebesgue measure. In contrast to the case of asymptotic stability, the transverse repulsive character of orbits embedded in the synchronized chaotic set implies that the neighborhood of any point of A will contain a positive measure set of points that leave Us(A) in a finite number of iterations. These points still belong to B(A), however, since the trajectories will sooner or later return to
Unfolding the Riddling Bifurcation
77
the neighborhood of A. Alternatively, the global dynamics of the system may be such that it allows direct access for trajectories repelled from the neighborhood of the transversely destabilized orbit (as well as from the neighborhoods of its dense set of preimages) to go to some other attractor (or infinity). This is the case of global riddling. A then attracts a positive Lebesgue measure set of points from U$(A), but not the full measure, i.e., 0 < n{B(A)r\Us(A)} < n{Us{A)}. We have previously described how the distinction between these two types of riddling depends on the existence of an absorbing area [13, 14, 15] that controls the global dynamics of the system and restrains trajectories starting near the synchronized chaotic set from reaching other limiting states. A number of definitions relating to the concepts of locally or globally riddled basins of attraction, weak attractors and absorbing areas have recently been proposed by Ashwin and Terry [16]. Qualitative theoretical results for the scaling behavior of chaotic systems near the riddling transition were first reported by Ott et al. [17]. The structure of the globally riddled basins of attraction is clearly distinguishable between the cases where the trajectories that are repelled from the synchronized chaotic state approach some asynchronous state inside the absorbing area for A or diverge towards infinity. In the former case, illustrated by Fig. 2.19, the repelling tongues generally have sharp and well-defined edges that follow the stable manifolds of saddle cycles produced in the stabilization of the attracting asynchronous state. In the latter case, where the trajectories diverge towards infinity, the repelling tongues consist of bundles of trajectories issuing from the neighborhood of the unstable synchronous cycles and from their dense set of preimages. This is illustrated in Fig. 2.4. The riddling (or bubbling) transition itself may be characterized as being either soft or hard. This distinction was introduced by Venkataramani et al. [11, 12] to describe two different situations that can be observed immediately after the first orbit has lost its transverse stability. After a soft transition, trajectories starting in the neighborhood of the synchronization manifold will remain close to this manifold. After a hard transition, on the other hand, trajectories starting close to the synchronization manifold can immediately move far away in phase space, and some may approach other attractors. In Chapter 2 it was suggested that the distinction between a soft and a hard riddling bifurcation is related to the super- or subcritical nature of the transverse bifurcation in which the first orbit embedded in the synchronized chaotic
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set loses its stability. In this chapter, we shall establish general analytical conditions for the transverse bifurcation of a periodic orbit to be either super- or subcritical. This derivation is based on the construction of an asymptotical one-dimensional map acting along the transverse invariant manifolds of the orbit that first loses its stability. In a supercritical transverse destabilization of a periodic orbit, the unstable manifolds of the asynchronous saddle cycle(s) born in the bifurcation (together with elements of the boundary of the absorbing area) will form a so-called mixed absorbing area that stretches along the synchronized chaotic set and from which trajectories starting near the chaotic set cannot escape. As the asynchronous saddle cycle(s) under variation of a control parameter move(s) away from the synchronization manifold, the width of the mixed absorbing area will grow. This leads to a synchronization error that increases as yj\e — ec\, with |e — e c | < < 1 denoting the distance of the control parameter from the bifurcation point. As opposed to the distinction between locally and globally riddled basins of attraction, the distinction between soft and hard riddling bifurcations only involves local conditions close to the synchronization manifold. A hard riddling bifurcation may lead to locally or globally riddled basins of attraction, depending on the conditions far from the synchronization manifold. As we shall show, however, immediately after a soft riddling bifurcation, the basin of attraction can be locally riddled only. The purpose of the subsequent sections is to illustrate these concepts in more detail by presenting three different scenarios for the onset of riddling and for the subsequent development of the basin of attraction for a system of two coupled logistic maps. We follow the location of the relevant asynchronous cycles and determine their stable and unstable invariant manifolds. We also determine both the absorbing and the mixed absorbing areas and discuss their significance for the observed dynamics. The first scenario illustrates how the transition from locally to globally riddled basins of attraction can occur via a contact bifurcation between the basin of attraction for the synchronized chaotic state and its absorbing area [13, 14, 18, 19]. The second scenario involves a direct transition to global riddling following a subcritical transverse bifurcation of a synchronized periodic orbit. In this case, the mixed absorbing area exists before the riddling bifurcation and disappears at the moment of bifurcation. Finally, in the third scenario, we follow a long and interesting sequence of bifurcations after the destabilization of the synchronous period-2 cycle in a
Unfolding the Riddling Bifurcation
79
supercritical transverse period-doubling. In this case, the asynchronous period4 saddle produced in the riddling bifurcation stabilizes in an inverse subcritical period-doubling bifurcation before the contact bifurcation between the basin of attraction and the absorbing area takes place. This gives rise to the emergence of a new attracting state inside the absorbing area for the synchronized chaotic set. In this connection, we also present a phase diagram delineating the regions in parameter space where the various solutions exist. At the end of the chapter we consider a system of two coupled one-dimensional maps that each displays type-Ill intermittency [20]. This implies that the transition to chaos for the individual map takes place via a subcritical perioddoubling bifurcation rather than via the usual Feigenbaum cascade of supercritical period-doubling bifurcations. We determine the regions in parameter space where the transverse Lyapunov exponent is negative so that the synchronized chaotic state is a least weakly stable. The bifurcation curves for the transverse destabilization of low-periodic orbits embedded in the synchronized chaotic state are obtained, and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations. Our purpose is again to illustrate the role of the absorbing area in restraining the dynamics, once the transverse destabilization of the synchronized chaotic state has occurred. By following the variations in the boundaries of the absorbing area and the basin of attraction under variation of the coupling parameter we provide an alternative illustration of the contact bifurcation that causes the transition from local to global riddling. At the same time, the system of the coupled type-Ill intermittency maps exhibits a new example of the phenomenon of intermingled basins of attraction that we observed for a system of two coupled logistic maps in Chapter 2. Chaotic synchronization and riddled basins of attraction in a system of two coupled type-I intermittency maps were studied by Manscher et al. [21]. In this case, the transition to chaos in the individual map takes place via a tangent (or saddle-node) bifurcation. Strictly speaking the weak attractor formed after a riddling bifurcation does not exist in the presence of noise or a small asymmetry (parameter mismatch). Hence, to observe experimentally the weakly attracting state the time that a trajectory on average spends near this state must be long compared with the temporal resolution of the experiment. To examine this problem, Venkataramani et al. [11, 12] constructed a particular two-dimensional map in which the stability transverse to a symmetric direction with chaotic dynamics could
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be explicitly controlled. This approach also underlines the fact that, besides in coupled identical oscillators, riddled basins of attraction can also arise in systems with other types of symmetry, provided that this symmetry allows for a chaotic dynamics to take place in an invariant manifold of lower dimension than the full phase space. Venkataramani et al. derived scaling relations for the time that a trajectory on average spends in the neighborhood of the invariant manifold as a function of the asymmetry parameter and the amplitude of the applied noise. As emphasised above, the critical curves and the so-called absorbing area play an important role in limiting the dynamics in phase space for two-dimensional endomorphisms (non-invertible maps). An absorbing area A for a twodimensional map F is a closed subset of the plane, bounded by a finite number of arcs of critical curves, which is trapping in the sense that points within A can never escape, i.e., F(A) C A, and for which a neighborhood exists in which the points are mapped into A in a finite number of iterations. The absorbing area is invariant if F(A) = A. The concept of an absorbing area has been worked out in considerable detail in a series of publications by Mira and coworkers [13, 14, 22, 23]. Here, one can also find a set of more precise definitions. (Note, however, that the French tradition with respect to the designations of the critical curves is a little different from ours which, we believe, coincide with common usage.) In a detailed investigation of a one-parameter family of twisted logistic maps, Gardini et al. [24] have illustrated how the stationary dynamics is bounded by the critical curves, and how the global bifurcations that this dynamics undergoes can be explained in terms of collisions between the boundary of the absorbing area and the basin boundary. Finally, Celka [25] has used the concept of critical curves to estimate the synchronization regions for two unidirectionally coupled skew tent maps.
3.2
Conditions for Soft and Hard Riddling
Let us again consider the system 2-n+l
=
Ja\xn)
T £\})n ~ xn)
Vn+1
=
fa{Vn)
+ £{x„ ~ Vn)
/o -i\
of two symmetrically coupled logistic maps fa{x) = ax(l — x) with 3 < a < 4 and - 2 < e < 2. At a = a0 = 3.678573..., the fixed point x0 = 1 - 1/a
Unfolding the Riddling Bifurcation
81
undergoes its first homoclinic bifurcation, and two chaotic bands merge into a single band. Likewise, for a — a^ = 3.592572..., the period-2 cycle undergoes its first homoclinic bifurcation, and four chaotic bands merge into two. At each of these bifurcation points, fa(x) has an absolutely continuous invariant measure and the dynamics of fa(x) is chaotic. In order to delineate the regions of parameter space where the synchronized chaotic state is asymptotically stable we have previously considered the transverse stability for each of the most important low-periodic point cycles 7„ (see Sec. 2.4). For a = ao we have found that the interval of asymptotic stability is bounded by a transverse period-doubling bifurcation of the period-6 cycle at e = —1.31 and by a transverse period-doubling of the period-2 cycle at e = —1.24. For a = oi, the interval of asymptotic stability for the synchronized chaotic state stretches from e = —1.46 to e = —1.16, with both the upper and the lower ends being associated with a destabilization of the period-2 cycle. Let us now examine these bifurcations in more detail in order to establish the conditions for the transverse bifurcations to be either sub- or supercritical. The transverse destabilization of the fixed point is not associated with any riddling bifurcation in our model. However, to illustrate our method let us first examine the bifurcation in which the fixed point PQ = (XQ, XO),XQ = 1 — 1/a loses its transverse stability. The map F : M2 —> IR2 is non-invertible, and it is easy to see that the determinant \DF\ of the Jacobian vanishes along two branches of a hyperbola (the critical curves [13, 14])
L0= v):y =
{^
+
^ ^(f-^)}
^
that cross the diagonal {x = y} in the points xc\ = 1/2 and xC2 = 1/2 — e/a. It follows that F is a diffeomorphism in some neighborhood of PQ, provided that a ^ 2 and a ^ 2(1 — e). The same provisions imply that v\\ ^ 0 and v± 7^ 0. Moreover, the first non-resonant condition j/y ^ v±_ will be satisfied for non-vanishing values of the coupling parameter. Here, v\\ = f'(xo) = 2 — a and V]_ — f'(xo) — 2s = 2 — a — 2e are the longitudinal and transverse eigenvalues of F in PQ as given by Eqs. (2.3) and (2.4), respectively. There will then exist a neighborhood of the fixed point where F has two one-dimensional invariant manifolds W\\ and W± of Po- As illustrated in Fig. 3.1, the longitudinal manifold W\\ coincides with the synchronization manifold {x = y}, and
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Chaotic Synchronization: Applications to Living Systems
the transverse manifold Wj_ intersects it perpendicularly. The transverse manifold will be unique as long as the fixed point is a saddle, and the manifolds form a one-parameter family when PQ becomes an unstable node [26]. The first non-resonant condition (i/\\ ^ v±) guarantees C 1 -smoothness of W± in some neighborhood Us(Po) of PQ. In the interval of interest the fixed point is unstable in the direction along W\\. As discussed in Sec. 2.3, PQ loses its transverse stability when \v±_\ = 1, either in a pitchfork bifurcation for e = (1 — a)/2 or in a period-doubling bifurcation for e = (3 — a)/2.
Fig 3.1. Longitudinal and transverse invariant manifolds W\\ and W± for the map F in the
fixed point P0. US(PQ) denotes a small neighborhood around Poll! our analysis we approach the point of transverse bifurcation from the side where the fixed point is transversely stable. To examine how our map acts along the manifold W± we rewrite F in terms of the variables £ = (x + j/)/2 and ri = (yx)/2,
where, as before, / = fa, and / ' denotes its derivative. In a neighborhood of PQ the transverse manifold may be expanded as W± = {(£, rj) : £ =
(3.4)
Unfolding the Riddling Bifurcation
83
Linear contributions to ip(rj) are absent because W± is parallel to the r\axis for r) = 0. To determine the coefficient B we substitute (3.4) into (3.3) and use the invariance of W±_ with respect to the action of F. This gives B = a/ (y\\ — v2^) from which B can be determined provided that the second non-resonant condition v\\ ^ v\ is satisfied. This condition guarantees C2smoothness of W±. In a neighborhood of PQ, the action of F along W± can now be represented by a one-dimensional map h : W± ->• W± with the asymptotic expression h : i] ->• v\j) + Crf + higher order terms.
(3.5)
Quadratic terms in this expansion vanish by virtue of the symmetry of the system. Inserting finally Eq. (3.5) into Eq. (3.3) and using our result for B we obtain C = —2a2/ (y\\ — u\). It is easy to see that both the first and the second non-resonant conditions are satisfied for s j^ 0 and a > 3. Moreover, since un < 0 for a > 3, C is positive at both bifurcation points. Hence, the bifurcations of the one-dimensional map h in T] = 0 are sub- or supercritical depending on the sign of the product u±C. If V\C < 0, the bifurcation is supercritical and if v±C > 0, it is subcritical. In this way, we conclude that the transverse pitchfork bifurcation of the fixed point that occurs at e = (1 — a)/2 is subcritical and that the transverse perioddoubling bifurcation at e = (3 — a)/2 is supercritical. Let us now examine the bifurcations in which the period-2 cycle a+l±V/(q+l)(a-3) £1,2 = 2/1,2 = *—z (3-6) la loses its transverse stability. It follows from the above analysis that F is a diffeomorphism in a neighborhood of both points of the symmetric period-2 cycle, provided that a ^ 1 + \/5 and o ^ l + y/A + (1 + 2e)2. These provisions also imply that i>\\ ^ 0 and v± ^ 0 where "|| = /'(si)/'(*2) = 1 - (a + l)(a - 3)
(3.7)
and
" i = (/'(zi) - 2e)(f'(x2)
- 2e) = (1 + 2ef - (a + l)(a - 3)
(3.8)
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are the longitudinal and transverse eigenvalues for the period-2 cycle. f'{x) denotes the derivative of f{x). Moreover, for e ^ 0 and e ^ —1, the first nonresonant condition v\\ ^ vL will be satisfied. For each of two period-2 points, a neighborhood will then exist in which the transverse invariant manifolds W±j are at least C 2 -smooth. Invariance in this case obviously applies with respect to the iterated map F2. In the parameter interval of interest, the symmetric period-2 cycle is unstable in the direction of the longitudinal manifold W\\ = {x = y}. As discussed in Sec. 2.4, the cycle loses its transverse stability either in a pitchfork bifurcation (i/ x = 1) for 1 1 ± y/{a + l)(a-3) + l (3.9) £ = ~2 or in a period-doubling bifurcation (v± = — 1) for £ =
1 1 ± V(o + l ) ( o - 3 ) - l "2
.
(3.10)
To investigate how the map (3.1) acts along the transverse manifolds W±j we again rewrite F in the form (3.3). The term "transverse manifold" is meant to denote the manifold in which the asynchronous cycle(s) involved in the bifurcation is (are) situated. In the neighborhood of each period-2 cycle point P, we can expand the one-dimensional manifolds W±j = {(£, r]) : £ =
higher order terms.
(3-11)
The coefficients Bi can be obtained by inserting Eq. (3.11) into Eq. (3.3) and using the invariance of W±i. This gives Bt = a [%i+1 + i / y / foi - v\)
(3.12)
where v\\ti = f'fa), v_L,i = f'{xi) - 2e, and xi+1 = f(xi). From Eq. (3.12), Bi can be calculated provided that the second non-resonant condition v\\ / v\ is satisfied. This condition will always be fulfilled since v« < 0 in the chaotic regime a > a*. The one-dimensional mapping h, : Wj_j ->• W±j of F2 along the transverse manifolds of the period-2 cycle takes the asymptotic form
Unfolding the Riddling Bifurcation
hi : 77 -» 1^77 + Ciif
+ higher order terms.
85
(3.13)
As before, quadratic terms do not arise in this expansion because of the symmetry of the system. Inserting Eq. (3.13) into Eq. (3.3) and using our results (3.12) for B; we obtain [18] Ci = —la
{vl,i + u\\,i+i) ("-M+1 + "II ~
-fi
v
l
"11,^-M)
"x,i
(3.14)
The bifurcations of the symmetric one-dimensional map hi will again be super- or subcritical depending on the sign of the product u±Ci. (Obviously, Ci must have the same sign for the two transverse manifolds). If v±Ci < 0, the bifurcation is supercritical, and it is subcritical for v\Ci > 0. Direct calculation shows that Cj is positive in the relevant parameter interval. Hence, we conclude that the transverse pitchfork bifurcation of the symmetric period-2 cycle is subcritical and that the transverse period-doubling is supercritical. As an illustration to this discussion, Fig. 3.2 shows the region in parameter space in which the symmetric period-2 cycle is transversely stable. This region is delineated by the curves (3.9) and (3.10) along which the subcritical pitchfork and the supercritical period-doubling bifurcations take place. Figure 3.3 shows the results of a numerical evaluation of the coefficients Ci for the upper point (x2, £2) of the period-2 cycle. The two curves denoted C[F and CPD refer to the transverse destabilization via a pitchfork and a period-doubling bifurcation, respectively. (It is a simple matter to obtain analytic expressions for these curves using the form (3.13). However, the expressions are somewhat complicated, and we omit them here). The curve for CPF passes through zero for a = 3 where the symmetric period-2 cycle first arises. The curve for CPD diverges for a = 1 + y/Z = 3.236 where the longitudinal eigenvalue i/y vanishes. This is the point where the individual map fa{x) displays a superstable period-2 orbit. It is also the minimal value of a for which a transverse period-doubling bifurcation can occur (see Fig. 3.2). The above approach can be extended to the case of an n-periodic symmetric cycle [27]. Then, the transverse manifolds W±ti asymptotically take the form (3.11) with Bi = -^Ri{n).
(3.15)
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Chaotic Synchronization: Applications to Living Systems 4.0
a 3.5
3.0 -2.0
-0.5
1.0
Fig 3.2. Region of transverse stability for the symmetric period-2 cycle. Destabilization occurs via a subcritical pitchfork bifurcation (lower curve) or via a supercritical period-doubling bifurcation (upper curve). For our discussion of chaotic synchronization, only the region a> a* ss 3.567 is of interest. Part of this stability regime can also be seen in Fig. 2.5.
40.0
C
0.0
3.0
1+V5
Q
4.0
Fig 3.3. Variation of the coefficients CfF and CfD associated, respectively, with the transverse pitchfork and the transverse period-doubling bifurcations of the symmetric period-2 cycle.
Unfolding the Riddling Bifurcation
87
Here, n
^11 = 1 1 ^ )
and
n
"J-= II tf'fc) - 2e) 8=1
2= 1
are the longitudinal and transverse eigenvalues of the n-periodic cycle, respectively. The functions Ri(n) can be obtained from the following recurrence relations that express the results for an n-periodic cycle in terms of the results for cycles of lower periodicity [18]: k
Ri(k + l) = vli+kRi(k)
+ J ] uli+j_,
(3.16)
J=I
with Ri(l) — 1 and k = 1,2,3...n — 1. Here, vy = / ' (x;) and I/_L,( = / ' fai) — 2e. The points x\ are supposed to be ordered in such a way that xi = fJ (a:;) with I = i + j , modulus n. The one-dimensional mapping /i, : W±j -» W±j of F " along Wi,i takes the asymptotical form (3.13) with n
Q = -aV-^^Qi(fc)
(3.17)
where the functions Qi(k) can be obtained recurrently from jfe-i
Qi(k + 1) = v\\,i+k-iQi{k) with
<9J(1)
- 2a Y,
rt,i+j-i
(3.18)
= 2Bj, (5,(2) = 21/p.Bj - 2a, and fc = 2, 3, ...n - 1.
Obviously, for a given cycle, all d must have the same sign at the moment of bifurcation \v±\ = 1- Depending on whether or not this sign agrees with the sign of u±, the transverse bifurcation will be sub- or supercritical. With this result in mind we show below that the riddling bifurcation will be soft if it arises from a supercritical bifurcation of some cycle. If it is associated with a subcritical bifurcation of a cycle, the riddling bifurcation will be hard. In the latter case the bifurcation may lead directly to a global riddling of the basin of attraction. The soft transition, on the other hand, only leads to a locally riddled basin.
88
3.3
Chaotic Synchronization:
Applications
to Living
Systems
Example of a Soft Riddling Bifurcation
Figure 3.4 illustrates the situation in the phase plane of our coupled map system for a = ao and s = —1.234. With this value of a the transverse period-doubling of the period-2 cycle occurs at e = —1.2373. The full line along the diagonal represents the synchronized chaotic state, and the two points on this line indicated by open circles are the points of the period-2 cycle that has just undergone a transverse period-doubling. In the vicinity of these points, the four points indicated by circles with crosses through are the points of the asynchronous period-4 cycle that has appeared in the bifurcation. The period-4 cycle is a saddle cycle with a stable manifold that connects it with the period-2 repeller. The unstable manifolds of the period-4 saddle stretch along the synchronization manifold. The arrows on the various stable and unstable manifolds denote their forward directions. It should be noted that since we are dealing with a non-invertible map, a (stable or unstable) manifold is allowed to intersect itself. Note also how the two point pairs of the period-4 cycle, in accordance with the above discussion, have moved different distances away from the period-2 cycle. Also shown in Fig. 3.4 is the fractal boundary of the basin of attraction for the synchronized chaotic state. In order to demonstrate a number of important details, Fig. 3.5 displays a magnification of the upper part of the phase plane. Here e = —1.225. A denotes (part of) the synchronized chaotic set, and Pi and P4 are points on the period-2 repeller and the period-4 saddle cycle, respectively. Finally, on the boundary of the basin of attraction (and indicated by small triangles in Fig. 3.5) we find the points P$ of a period-8 repeller. Any neighborhood of this repeller contains a positive measure set of initial conditions from which the trajectories diverge towards infinity. As bounded by the outmost loops of the unstable manifolds for the period4 cycle (drawn as thin lines) and by segments of the critical curves L\ — L& (drawn as heavier lines), the mixed absorbing area A [13, 14, 28] completely surrounds the synchronized chaotic set. With the situation depicted in Fig. 3.5, the mixed absorbing area is a closed invariant set, i.e., F(A') = A', and trajectories starting inside the mixed absorbing area (or at its boundary) cannot leave it. A trajectory starting near the synchronization manifold {a; = y} will move along this manifold until it comes close enough to the point P2 of the unstable period-2 cycle. The trajectory will then move out towards P\. How-
Unfolding the Riddling Bifurcation
89
1.25
y
-0.05 -0.05
x
1.25
Fig 3.4. Situation in phase plane immediately after the supercritical transverse period-doubling of the symmetric period-2 cycle. The figure shows the period-4 saddle cycle with its stable and unstable manifolds. Also indicated are the absorbing area as bounded by segments of critical curves L^ and the basin of attraction with its fractal boundary, e = —1.234.
ever, as it approaches this point it will be guided along the unstable manifold of Pi back towards the synchronization manifold. Moreover, most trajectories starting from a neighborhood outside A' will follow the unstable manifolds of the period-4 saddle, fold at the critical curves (L\ or L3), and cross into A'. We conclude that as long as a mixed absorbing area exists inside the basin of attraction (and no other attractor exists within the mixed absorbing area), the synchronized chaotic state will attract almost all points from its neighborhood. A is then a Milnor attractor, and its basin of attraction can be locally riddled only. In this way the formation in a supercritical transverse bifurcation of a mixed absorbing area that stretches along the synchronized chaotic state plays a major role for restraining the amplitude of the bursts away from the synchronized state. Surrounding the mixed absorbing area in Fig. 3.5 we find the absorbing area A (also referred to as the trapping region [29, 30]). Bounded by a finite number of segments Lk = Fk(Lo) of images of the critical curves LQ as given by (3.2), the
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Chaotic Synchronization: Applications to Living Systems
1.0
y
0.66 0.66
X
1.0
Fig 3.5. Magnification of the upper part of Fig. 3.4. Here the coupling parameter is s = —1.225. A denotes (part of) the synchronized chaotic set, and Pi and P4 are points of the period-2 repeller and the period-4 saddle cycle, respectively. Points Pg denoted by a small triangle belong to a period-8 repeller that is situated at the boundary of the basin of attraction.
absorbing area [13, 14, 15] has the property that trajectories that enter this area cannot leave it again, i.e., F(A) Q A. Moreover, trajectories that start in the neighborhood of the absorbing area will enter it in a finite number of iterations. The presence of an absorbing area is a characteristic feature of non-invertible maps. For our system of two coupled logistic maps, the absorbing area exists in an interval of the coupling parameter that includes part of the interval of asymptotic stability for the synchronized chaotic state and can stretch beyond this interval. When, as illustrated in Fig. 3.5, the mixed absorbing area falls fully within the absorbing area (and no attracting state other than A exists in A), almost all trajectories starting in the absorbing area will enter the mixed absorbing area, to finally be attracted by A. As we continue to increase the coupling parameter, the points P4 of the period-4 saddle cycle move further out along the transverse manifolds of the period-2 repeller, and the mixed absorbing area A' continues to grow until it fills out most of the absorbing area A- This is illustrated in Fig. 3.6, where e = —1.21. The critical curves Lfc will serve as an envelope to the unstable
Unfolding the Riddling Bifurcation
91
0.97
y
0.86 0.67
x
0.86
Fig 3.6. For e = —1.21 the period-4 saddle cycle has moved further out along the transverse manifolds of the period-2 repeller, and the mixed absorbing area A' covers most of the absorbing area .4. At the same time the period-8 repeller has moved closer to the common boundary of
A and A'. manifolds, and as long as the period-4 cycle falls within the boundary of the absorbing area, its unstable manifolds will be restrained to this area. Compared with Fig. 3.5, the period-8 repeller (and hence the basin boundary) has moved closer to the boundary of the absorbing area. At e = —1.205 a crisis takes place as the boundary of the absorbing area comes in contact with the basin boundary, and the period-8 repeller crosses into the region delineated by the critical curves. We can see that this happens in points where the boundaries of A and A' coincide. This marks the transition from local to global riddling of the basin of attraction. Figure 3.7 shows the situation immediately after this contact bifurcation has occurred. Here, e = —1.18. The points of the period-8 repeller now fall inside the region delineated by the images of the critical curves, the absorbing area has ceased to exist, and direct access has been opened for points starting near the transversely unstable period-2 cycle (and its dense set of preimages) to diverge to infinity. This is a typical example of global riddling. One can examine this transition in more detail by constructing the preimages of those tongues of the
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1.0
y
0.55 0.55
X
1.0
Fig 3.7. Crossing of the period-8 repeller through the segments of critical curves and unstable manifolds that define .4, and A' marks the transition from locally to globally riddled basins of attraction. There is now direct access from the neighborhood of the transversely unstable period-2 cycle (and its dense set of preimages) to infinity.
basin of infinity that have penetrated into the region A fl A'. Immediately after the boundary crisis these preimages can be followed backwards along the unstable manifolds that define the boundary of A' towards the period-4 saddle cycle P4 and from here along the transverse manifolds back towards the period-2 repeller P
Unfolding the Riddling Bifurcation
93
a dense set of systems having a large or even an infinite number of attracting cycles [31, 32]. The presence of such regions with unremovable homoclinic tangencies causes enormous problems, mathematically as well as numerically, and we shall not discuss their consequences here. We consider the above scenario describes the generic transition from locally to globally riddled basins of attraction following a supercritical riddling bifurcation in coupled map systems when the asynchronous cycle born in the bifurcation does not stabilize before the contact bifurcation between the absorbing area and the basin of attraction has occurred. Note that when the contact bifurcation occurs, the boundaries of the absorbing and the mixed absorbing areas coincide. Hence, the bifurcation involves a minimal, invariant absorbing area. In Sec. 3.5 we shall discuss a very different and much more complicated riddling scenario that arises in our system for a = a\. Here, the asynchronous saddle cycle born in the riddling bifurcation stabilizes in an inverse perioddoubling bifurcation to produce an attracting state inside the absorbing area for the synchronized chaotic set. First, however, let us consider a generic scenario for the appearance of globally riddled basins of attraction following a hard riddling bifurcation that can also be observed for a = ao-
3.4
Example of a Hard Riddling Bifurcation
At the other end of the interval of asymptotic stability for a = ao, a transverse destabilization of the symmetric period-6 cycle takes place . This again occurs via a period-doubling bifurcation (I/_L = — 1). However, evaluation of the parameters Ci shows that the bifurcation is subcritical (d < 0). Hence, before the transition occurs, the points of a period-12 repeller are situated on either side of the symmetric period-6 cycle. This situation is illustrated in Fig. 3.8(a) where we also can see the fractal boundary of the basin of attraction for the synchronized chaotic state. Closer examination reveals that the points of the period-12 repeller fall at cusp points of the basin boundary, and any neighborhood of these points contains a positive measure set of initial conditions from which the trajectories diverge to infinity. Figure 3.8(b) is a magnification of part of the structure in Fig. 3.8(a). Here, the coupling parameter e = —1.3. The figure shows (the upper part of) the synchronous chaotic state with three points of the symmetric period-6 cycle
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Chaotic Synchronization: Applications to Living Systems
-0.05
X
1.25
0.7
X
0.94
Fig 3.8. (a) Overview of the situation in phase plane before the transverse destabilization of the symmetric period-6 cycle. The figure shows the asymmetric period-12 repeller with its
longitudinal manifolds, (b) Magnification of part of (a). Note how the absorbing area falls inside the mixed absorbing area, which now exists before the riddling bifurcation. situated along the main diagonal. Also shown is the absorbing area as bounded by segments of the critical curves L\ — L\± (heavy lines). The dots indicate where these segments connect. Outside the absorbing area we find six points of the period-12 repeller (shown as circles with crosses through). As noted above, these points fall at cusp points of the basin boundary. In the longitudinal direction the unstable manifolds of the period-12 repeller again seem to wind around the synchronized chaotic state. However, since these manifolds now fall outside the absorbing area, we can not be sure that they will always remain bounded in the vicinity of the synchronous set. In the present situation this appears to be the case, since the unstable longitudinal manifolds all connect to the critical curves and, hence, define a mixed absorbing area (which now exists before the riddling bifurcation). As we approach the bifurcation, the points of the period-12 repeller move through the critical curves Lk, and the absorbing area ceases to exist. Unstable manifolds now protrude out through the images of the critical curves, and trajectories starting outside of the longitudinal manifolds of the period-12 repeller will diverge. This situation is illustrated in Fig. 3.9 for e = —1.307. Finally, at the point of bifurcation where the period-12 repeller disappears, direct access is opened from the symmetric period-6 cycle (as well as from its dense set of preimages) to infinity. The emergence of global riddling in this way
Unfolding the Riddling Bifurcation
95
0.94
y
0.7 0.7
x
0.94
Fig 3.9. For e — —1.307 the period-12 repeller has moved across the critical curves, and the absorbing area has ceased to exist.
occurs simultaneously with the destabilization of the period-6 cycle in a local bifurcation. Based on the above results we conclude that the riddling bifurcation will be hard if it is associated with a subcritical destabilization of some orbit. The subcritical bifurcation does not produce asynchronous saddle cycles whose unstable manifolds can restrain the bursts of trajectories. It is possible, of course, that saddle cycle(s) of appropriate periodicity could exist outside the repelling cycle(s) involved in the transverse destabilization (an example is provided by the model considered by Venkataramani et al. [11, 12]). In this case the subcritical riddling bifurcation might not directly produce a globally riddled basin of attraction.
3.5
Destabilization Scenario for a = oi
Our last scenario concerns the sequence of events that take place in connection with the destabilization of the synchronized chaotic state for a = a\ and e = — 1.1. This scenario starts with the transverse destabilization of the symmetric period-2 cycle in a supercritical period-doubling bifurcation for e = —1.1560. The blowout bifurcation at which the transverse Lyapunov exponent becomes
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Chaotic Synchronization: Applications to Living Systems
positive and the chaotic state loses its average attraction occurs at e = —1.0385. Hence, we shall follow the bifurcations that take place in the interval between e = -1.1560 and e = -1.0385. For a — ai, the synchronous chaotic state consists of two separate bands. The absorbing and mixed absorbing areas each therefore also consists of two regions that are mapped one into the other under the action of F. Figure 3.10(a) shows the upper part of the phase plane after the transverse period-doubling has occurred. Here, e — —1.155. We note the two points of the asynchronous period-4 saddle cycle situated along the transverse manifolds of the symmetric period-2 repeller. The points of the period-4 saddle are indicated as circles with crosses through. Together with segments of the critical curves L\ and L3, the unstable manifolds of the period-4 saddle define a mixed absorbing area A'. Around this area we find a large absorbing area A as bounded by segments of the iterates L\, L2, £3 and L4 of LQ.
0.75
X
0.95
0.85
x
0.89
Fig 3.10. (a) Upper part of phase plane for a = a^ and e = —1.155. We observe a larger and a smaller absorbing area together with a mixed absorbing area delineated by the unstable manifolds of the asymmetric period-4 saddle, (b) Magnification of part of the phase plane for e = —1.15597. Note how the transverse manifolds of the period-2 repeller pass right between the critical curves.
Trajectories starting inside the mixed absorbing area cannot escape from it. Except for a measure zero set of trajectories starting from points on the external branches of the stable manifolds of the asynchronous period-4 cycle (and possible preimages of these manifolds), trajectories starting from a neighborhood of the mixed absorbing area will move along the unstable manifolds
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of the period-4 saddle points towards the points where they meet the critical curve L3 (or L{). After a folding here they will be mapped into the mixed absorbing area, soon to be trapped in the smaller absorbing area denned by the critical curves L\, L3, L5, Lj, L$, L§, L\Q, and Lu- The transverse manifolds of the synchronous period-2 cycle pass right between the critical curves L5 and L7. Hence, in spite of the fact that the period-2 cycle has a dense set of preimages along the main diagonal, not a single trajectory will be able to leave the smaller absorbing area. Comment: The smaller absorbing area represents the absorbing area that existed before the homoclinic bifurcation in which the four-band chaotic attractor for the individual map has merged into a two-band attractor. The smaller absorbing area is destroyed in the homoclinic bifurcation, and as soon as the nonlinearity parameter a exceeds a\ by as little as a part in 1011 one can start to observe how trajectories escape along the transverse manifold of the synchronous period-2 cycle. Immediately after the transverse destabilization of the period-2 cycle (e.g., for £ = —1.15597 as shown in Fig. 3.10(b)) the period-4 saddle points will be situated very close to the main diagonal, and its unstable manifolds stretch along the synchronized chaotic state as a narrow band from which trajectories cannot escape. Hence, we again observe that a supercritical transverse bifurcation leads to a soft riddling transition with a locally riddled basin of attraction and with small and smoothly growing bursts of trajectories away from the synchronized state. As we move further away from the bifurcation point, however, a completely different sequence of events takes place. At e = —1.09571 the period-4 saddle cycle undergoes an inverse, subcritical period-doubling in the direction of its unstable manifold. This produces a stable asynchronous period-4 cycle surrounded on both sides by the points of a period-8 saddle cycle. This situation is illustrated in Fig. 3.11, where a point P4 of the stable period-4 cycle is marked by a small square and the neighboring points Ps of the period-8 saddle by circles with small crosses. Situated along the main diagonal and embedded in the synchronized chaotic attractor A, the upper point of the period-2 repeller is indicated by a small triangle. Also shown in this figure are the stable and unstable manifolds of the period-8 saddle. The white region B(oo) in the top right corner belongs to the basin of infinity. With a gray shading, the area B(A) is the basin of attraction for the synchronized chaotic state. As in the previous figures, the boundary between these two basins is
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Pig 3.11. The asymmetric period-4 cycle has undergone an inverse, subcritical period-doubling transition producing a stable period-4 cycle F 4 and a period-8 saddle P g . The stable manifolds of this saddle delineate the immediate basin for the period-4 node. The figure was obtained for e = -1.08.
fractal. The inverse period-doubling bifurcation at e = -1.09571 has produced a new attracting state inside the former basin of attraction for the synchronized chaotic state. The immediate basin for the period-4 cycle (shown hatched in Fig. 3.11) is defined by the stable manifolds of the period-8 saddle. Emanating from the symmetric period-2 repeller, these manifolds delineate a basin that stretches as a set of tongues all the way down to the main diagonal. Immediately after the stabilization of the period-4 cycle these tongues will be very narrow and they will not intersect the smaller absorbing area. As a consequence, trajectories that start within this area will not be able to reach the stable period-4 cycle, and the basin of attraction for the synchronized chaotic state will remain locally riddled only. Besides the immediate basin, the basin of attraction for the asynchronous period-4 cycle also consists of a set of secondary tongues. Most prominent among these are the tongues that emanate from the points where the critical curves L\ and L3 intersect the main diagonal. These secondary tongues are
Unfolding the Riddling Bifurcation
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also clearly visible (hatched) in Fig. 3.11. Like the primary tongues, they are delineated by sharp and well-defined edges. If the value of the nonlinearity parameter a exceeds a\ by the smallest amount, the smaller absorbing area ceases to exist. The basin of attraction for the asynchronous period-4 cycle will then include tongues that emanate from the dense set of preimages of the period-2 repeller in the synchronized chaotic set A, and the basin of attraction for A will be globally riddled. In this case the transition to global riddling is accomplished via two local bifurcations: first the transverse supercritical period-doubling of the symmetric period-2 cycle and thereafter the stabilization of the asynchronous period-4 cycle in an inverse subcritical period-doubling. However, the new tongues tend to be extremely narrow, and in the numerical calculations they show up only as randomly scattered points within the area that has otherwise been assigned to B{A). A similar transition to global riddling has recently been described by Astakhov et al. [33] for a pair of nonlinearly coupled logistic maps. In their main scenario it is the fixed point that first undergoes a transverse destabilization in the form of a supercritical period-doubling. After similar period-doubling bifurcations of the symmetric period-2, period-4, period-8, and period-16 cycles, the asynchronous period-2 saddle cycle produced in the first transverse bifurcation stabilizes in an inverse, subcritical pitchfork bifurcation, giving way to a globally riddled basin of attraction for the synchronous chaotic state. It is worth noting, however, that the globally riddled basins of attraction created through these processes have a fairly unusual structure. Firstly, the main tongues are delineated by the stable manifolds of a saddle cycle (or a pair of saddle cycles). In contrast, in the commonly described form of global riddling [1, 2, 29, 30], the repelling tongues are defined only in terms of bundles of trajectories that follow an unstable manifold away from the synchronized chaotic state. In addition, there is a prominent set of secondary tongues that also have sharp and well defined edges. Finally, the remaining tongues form an extremely thin structure that shows up in the numerical calculations only as randomly scattered points from which the trajectories eventually reach the asymmetric point attractor. As the coupling parameter is further increased, the phase portrait starts to become complicated. At e = —1.085, the asynchronous period-4 cycle is transformed from a stable node into a stable focus (i.e., the eigenvalues become
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complex conjugated). At e = -1.0625 a subharmonic saddle-node bifurcation [34] takes place in which a stable period-12 cycle is born together with a period12 saddle. This situation is illustrated in Fig. 3.12 where e = —1.0615.
Fig 3.12. A stable period-12 cycle has been born together with a period-12 saddle in a subharmonic saddle-node bifurcation. The stable manifold of the period-12 saddle defines disc-shaped immediate basins for the period-12 node. Here, e = —1.0615.
We now have two coexisting stable solutions in the repelling tongues emanating from the symmetric period-2 repeller. In Fig. 3.12, a point P4 of the period-4 focus is indicated by a square drawn with relatively heavy lines and the points Pn of the period-12 node by finer squares. The immediate basin of attraction for the period-12 cycle takes an unusual form namely as a set of discs surrounded by topological circles which are formed by the stable manifold of the period-12 saddle (points of which are shown as small circles). This peculiar structure, in which the same stable manifold as a closed curve approaches the saddle point from both sides, is made possible by virtue of the non-invertibility of the map F. This non-inverfcibility allows the preimages of the period-12 saddle cycle to serve as points of separation between the two directions of the manifolds. The complete basin of attraction for the period-12 node consists of these immediate disc-formed domains together with all their preimages in
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the main tongues as well as in their preimages. A similar basin structure was observed by Maistrenko et al. [35] for a two-dimensional, non-invertible map derived from a problem in radiophysics. The next transformations of the phase portrait to occur involve a Hopf bifurcation of the period-4 focus at e = —1.0581 and a Hopf bifurcation of the stable period-12 solution at e = —1.0562. The second Hopf bifurcation produces a period-12 invariant circle (referred to as a torus) Tn corresponding initially (with high probability) to quasiperiodic motion. With further increase of the coupling parameter, the dynamics becomes non-invertible and the torus breaks down [36], producing chaos filled by windows of periodicity. In both cases we can have either a single or two coexisting and mutually symmetric attractors. For e € [—1.054, —1.053], for instance, the dynamics is reduced to two symmetrically located stable period-12 solutions and two saddle cycles of the same period. 1.03
y
0.94 0.6
X
0.78
Fig 3.13. For e = —1.0513, a single 12-piece chaotic attractor exists in the repelling tongues emanating from the symmetric period-2 repeller. As the coupling parameter is further increased, the 12-piece chaotic attractor merges into a 4-piece attractor finally to disappear together with its basin of attraction in a boundary crisis.
For other values of the coupling parameter, the dynamics produced through
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the breakdown of Tn involves two coexisting chaotic attractors. For e = — 1.0517, for instance, two 5-12-piece chaotic attractors exist. At e = —1.05134, a single 12-piece chaotic attractor is born and for slightly higher values of the coupling parameter, the 12-piece attractor merges into a four-piece attractor. This last process gives rise to the appearance of so-called rare points [37], indicating that merging takes place across a fractal basin boundary. Finally, at e = —1.0458 the chaotic attractor and its basin of attraction disappear in a boundary crises, leaving only a chaotic saddle of period 4 in the region around the original asynchronous period-4 cycle. Figure 3.13 shows a phase portrait for £ = —1.0513. For this value of the control parameter, our coupled map system F displays a 12-piece chaotic attractor situated in the repelling tongues issued from the points of the symmetric period-2 cycles. As described above, this attractor has been produced via breakdown (loss of differentiability) of the 12-piece torus T\2- Also shown in the figure are the locations of the period-4 and period-12 cycles, both of which are now unstable focuses. This completes our discussion of the complex scenario that unfolds after the supercritical transverse destabilization of the symmetric period-2 cycle. After the asynchronous chaotic attractor has disappeared in a boundary crisis, no other attracting set is observed inside the larger absorbing area, and the basin of attraction for the synchronized chaotic state remains locally riddled until, at e = —1.0385 a blowout bifurcation takes place. In this bifurcation the typical transverse Lyapunov exponent for trajectories on A becomes positive, and the synchronized chaotic state loses its average attraction. However, the larger absorbing area A still exists inside the basin of attraction, and the blowout bifurcation leads to on-off intermittency. The chaotic attractor spreads over the whole area of the absorbing set. However, trajectories starting near the synchronization manifold cannot diverge to infinity. For slightly higher values of the coupling parameter, the two-dimensional chaotic attractor undergoes an inverse boundary crisis by which it decomposes into two mutually symmetric chaotic attractors. For e = — 1, these attractors are both restricted to one-dimensional manifolds, and as discussed in Sec. 2.6 one can observe the phenomenon of intermingled basins of attraction. Each attractor here has a basin that is riddled with initial conditions that lead to the other attractor. The mutually symmetric chaotic attractors finally disappear in a boundary crisis at e = —0.935. When this occurs, the larger absorbing area in Fig. 3.10 has made contact with the basin boundary and, hence, has ceased to exist.
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Fig 3.14. (a) Phase diagram illustrating the region (vertically hatched) in which attracting states exist inside the repelling tongues emanating from the symmetric period-2 cycle. The triangular region to the left is the region where the symmetric period-2 cycle is transversely stable (compare with Fig. 3.2). (b) Magnification illustrating the region of stability for the period-12 cycle (horizontally hatched). The curve marked CB represents the contact bifurcation.
The phase diagram in Fig. 3.14(a) gives an overview of the main bifurcations involved in the above scenarios and clarifies the difference between the scenarios observed for a — ao (Sec. 3.3) and for a = a\ (Sec. 3.5). In this diagram the hatched region to the left represents parameter values for which the symmetric period-2 cycle is transversely stable (compare with Fig. 3.2). At the right hand edge of this region, the period-2 cycle undergoes a supercritical transverse period-doubling producing a period-4 saddle cycle outside the main diagonal. The vertically hatched finger to the right of this bifurcation curve represents the regions of parameter space in which attracting states exist in the repelling tongues emanating from the points of the period-2 repeller. The left hand edge of this finger is the bifurcation curve in which the asymmetric period-4 saddle stabilizes in an inverse, subcritical period-doubling. The dashed curve running through the middle of the finger is the Hopf-bifurcation curve for the period-4 cycle, and the right hand edge of the finger represents the boundary crisis in which a 4-piece chaotic attractor disappears through collision with its basin boundary. The curve marked CB represents the contact bifurcation between the absorbing area and the basin boundary for the synchronized chaotic state A. For a = ao, as the coupling parameter is increased, the contact bifurcation occurs
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before the stabilization of the asynchronous period-4 cycle. This leads to the scenario that we have described in Sec. 3.5. Here, the transition from local to global riddling of the basin of attraction for A takes place via the contact bifurcation. For a = a\, on the other hand, the asymmetric period-4 cycle stabilizes before the contact bifurcation, and the scenario described in the present section takes place. If the nonlinearity parameter is precisely equal to a\, we do not observe a transition to global riddling. For a value of a slightly larger than a\, however, we observe a transition to the peculiar form of global riddling that manifests itself in the form of randomly scattered dots within the region otherwise assigned as the basin for A [33]. Figure 3.14(b) provides a little more detail to the bifurcation structure. Again, the vertically hatched region represents parameter values for which stable solutions exist inside the repelling tongues issued from the symmetric period-2 cycle. The left edge is the bifurcation curve in which the asynchronous period-4 cycle stabilizes, and the right edge represents the boundary crisis where the repelling tongues disappear. The narrower, horizontally hatched area is the region where the stable period-12 cycle exists. This cycle arises in a subharmonic saddle-node bifurcation to be destabilized at higher values of the coupling parameter in a Hopf bifurcation leading to the above mentioned 12-piece torus T\2. The dashed curve running through the middle of the finger again represents the bifurcations in which the asynchronous period-4 cycle loses its stability via a Hopf bifurcation. At the other end of the interval of asymptotic stability for a — a\, the transverse destabilization takes place in the form of a subcritical pitchfork bifurcation of the symmetric period-2 cycle at e = —1.46. This leads directly to a globally riddled basin of attraction. Let us finally note that for a = a\, there is an additional interval of weak stability for the synchronized chaotic state around e = 0.3. However, this interval does not contain regions of asymptotic stability for the chaotic set. Nor is there an absorbing area. Hence, the basin of attraction is globally riddled in the whole interval.
3.6
Coupled Intermittency-III Maps
The purpose of the following three sections is to discuss the same type of phenomena that we have examined for coupled logistic maps for a very different pair of maps in which the transition to chaos takes place via a subcritical
Unfolding the Riddling Bifurcation
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period-doubling rather than via the usual Feigenbaum cascade of supercritical period-doubling bifurcations. This also allows us to present some of the phenomena from a different point of view. Let us consider the two-dimensional map F = F^£ : R2 —> R2 defined by
P. J x \ _ > J Ux)+£(y-x) \ F -\yi^\U(y) + e(z-y)f
where /M = / : R1 -> R1 is the one-dimensional map
/M : x -»• - ((1 + n + 7) x + x3) e~bx2 + jx 1
1
/o1Q\ (3 19) ' (3.20)
and e 6 R is the coupling parameter, fi E R is used as the bifurcation parameter for the individual map, and in most of the analysis the remaining parameters are kept constant at b = 0.14 and 7 = 0.02, respectively. The one-dimensional map /M has previously been investigated by Laugesen et al. [38] with the aim of explaining the anomalous statistics for the length of the laminar phases associated with the strongly nonuniform reinjection mechanism for 7 = 0. These authors have also examined part of the bifurcation structure for a system of two symmetrically coupled maps corresponding to F^. An antisymmetric version of Eq. (3.19) has been considered by Laugesen et al. [39] as an example of a map that exhibits type-II intermittency with a nearly homogeneous reinjection mechanism. For 7 < 1, the map f^x) has a unique fixed point x* = 0. Here, the slope is / (0) — —(1 + A*), implying that the fixed point is stable for — 2 < n < 0. At fi = 0, a period-doubling bifurcation takes place, and since / (0) = 0 and / (0) = — 6 (1 — b (1 + 11 + 7)) this bifurcation is subcritical for 6(1 + 7) < 1. The function f^ix) also provides an efficient reinjection mechanism: When x becomes large, the factor e~bx becomes very small, and the next iterate may fall close to the unstable fixed point. The parameter 7 has been introduced to control the reinjection process. For b = 0.14 and 7 = 0.02, the iterate of the critical point f(xc) (/ (xc) = 0) falls outside the point XQ (/ (zo) = 0) where the map changes sign. Hence, the critical point is not mapped directly into the neighborhood of the unstable fixed point, and the reinjection is reasonably uniform. Figure 3.15 shows the map /M(a;) with a typical itinerary. The bifurcation diagram in Fig. 3.16 illustrates how the stable fixed point x* = 0 for (j, = 0 undergoes a subcritical period-doubling. Hereafter, the map fn{x) displays type-Ill intermittency, interrupted by periodic windows. It
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Chaotic Synchronization: Applications to Living Systems 10
'
8
'
'
'
LI
S
/
/' \
6
•
4 2 y
W"
o
/"Xo
V
-2
\
-4
-
-6 -8
':/
-10 -10 -8 -6 -4 -2 0 x
2
4
6
8 10
Fig 3.15. Iteration of the map f^x) = - ((1 + n + 7) x + x3) e~bx2 + •yx for b = 0.14, 7 = 0.02, and n = 0.1. The iterate of the critical point xc falls outside the point xo where the map crosses the horizontal axis. This provides a reasonably homogeneous reinjection near the unstable fixed point x* = 0.
appears, however, that the prevalence of such windows is relatively low, and we shall assume that the individual map generates chaotic dynamics for the parameter values considered in the following analyses. The global dynamics of fn{x) is such that it allows for the existence of a dense set of unstable periodic orbits as required for deterministic chaos to occur. In the considered parameter range, the number of cycles of period 2, 4, and 6 are 3, 12, and 16, respectively. Along the main diagonal {x = y}, the coupling vanishes, and the dynamics coincides with that of the one-dimensional map f^(x). If / C M1 denotes a chaotic attractor for / M (z), then A = {x = y G / } will be a one-dimensional invariant chaotic set for the coupled map system (3.19). This implies that the main diagonal is an invariant manifold for F^. For any point on this manifold, F^ has an eigenvector along the diagonal and an eigenvector perpendicular to it. In accordance with our results in Chapter 2, the corresponding eigenvalues are
"II =
flip)
(3.21)
Unfolding the Riddling Bifurcation
X
107
0
Fig 3.16. One-parameter bifurcation diagram for f^(x) with b = 0.14 and 7 = 0.02. For H = 0, the stable fixed point at x* = 0 undergoes a subcritical period-doubling and the system hereafter exhibits type-Ill intermittency. Considered from the right, the observable windows have periodicities 16, 9, 20, 11, 24, 13, etc. and
vx = / ; ( * ) - 2e,
(3.22)
respectively. As previously noted, Alexander et al. [29, 30] have shown that if (i) A is a finite union of intervals, and (ii) the invariant measure of fp{x) on A is absolutely continuous, then a negative value of the transverse Lyapunov exponent Ax guarantees that A attracts a positive Lebesgue measure set of points from any two-dimensional neighborhood Ug(A). This implies that the synchronized chaotic state is at least weakly attracting. For our system of two symmetrically coupled intermittency-III maps, the condition (ii) of an absolutely continuous invariant measure for /M(a?) is not necessarily fulfilled. Nonetheless, we shall use the transverse Lyapunov exponent as given by 1
N
I <
(3.23)
as indicator of the average transverse stability for the synchronized state. As before, [x„ = f™{x)}™ denotes a typical itinerary of A. Figure 3.17 shows the region of /ie-parameter plane in which Ax < 0 (hatched with downwards sloping lines). For /z = 0, the interval of weak stability extends from e = — 1 to e — 0. As /x increases, the interval gradually shifts towards more negative
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2.0 1.5 1.0 0.5 £
0.0 -0.5 -1.0 -1.5 -2.0 -0.1
0.0
0.1
0.2
0.3
0.4
0.5
Fig 3.17. Region in parameter space where the synchronized chaotic state is weakly stable (downwards sloping hatch). Also shown are the regions in which the fixed point and the symmetric period-2 cycle (upwards sloping hatch) are transversely stable. Since there is no overlap between these regions, the synchronized chaotic state is never asymptotically stable.
values of the coupling parameter while at the same time decreasing slightly in width. Note the characteristic fluctuations of the borderlines for the region of negative transverse Lyapunov exponent. Additional stability regions arise whenever /M(a;) displays a periodic window. For the synchronized chaotic state to be asymptotically (or strongly) stable all orbits embedded in this state must be transversely stable. For the fixed point, for instance, the transverse eigenvalue is u± — f (0) — 2e, and the condition of transverse stability for \i > 0 gives
-l-f<e<-f-
(3-24)
The interval in which the fixed point is transversely stable is also shown in Fig. 3.17. To a large extent this interval coincides with the range of weak stability for the chaotic set A. For comparison with the region of weak stability, Fig. 3.17 also shows the region in which the symmetric period-2 orbit is transversely stable (hatched with upwards sloping lines). This region falls entirely in the realm where e > 1. Since there is no overlap with the region in which Aj_ < 0, the synchronized chaotic state cannot be asymptotically stable. Neither can we observe the actual riddling bifurcation.
Unfolding the Riddling Bifurcation
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Fig 3.18. Regions of transverse stability for the symmetric period-4 cycles (a), and the symmetric period-6 cycles (b). Destabilization occurs either through a pitchfork or through a period-doubling bifurcation, and both of these bifurcations can be either sub- or supercritical.
To complete the investigation of the transverse stability of low-periodic orbits embedded in the synchronized chaotic state, Figs. 3.18(a) and (b) show the bifurcation curves in which the symmetric period-4, respectively the period-6 orbits lose their transverse stability. Again, the regions of transverse stability are hatched. We note that there are regions of parameter space where all of these cycles are transversely stable while at the same time Ax < 0.
3.7
The Contact Bifurcation
As previously noted, the emergence of an observed riddling after the transverse destabilization of the first low periodic orbit depends on the global dynamics of the system. Here, the existence of an absorbing area inside the basin of attraction for the synchronized chaotic state plays an important role. Figure 3.19 shows the construction of the boundary dA of the absorbing area for the synchronized chaotic state of FMi£ with /x = 0.07 and e = —0.585. The critical curve LQ = {(x,y) : [DF^ = 0}, at which the Jacobian of F^ vanishes, does not directly play a role. Its first iterate L\ — F^E(Lo) produces the smooth curve that outlines the form of the absorbing area along the upper right periphery. L\ is also involved in defining dA near the lower left corner. The second iterate L2 demarcates part of the boundary in the upper left corner. It is clear that the absorbing area can have a very complicated boundary, and that
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15 10 5
y o
-5 -10 -15 -15
-10
-5
0 x
5
10
15
Fig 3.19. Construction of the boundary of the absorbing area through iterates of the critical curves, /i = 0.07 and s = -0.585. Trajectories started within the absorbing area are restrained by the nonlinearities of the map F^e to remain in this area. a large number of images of the critical curves may be involved in determining the precise border. In the present case, the construction of the absorbing area may be simplified considerably, however, by noting that dA is symmetric both with respect to the line {x — y} and with respect to {x = —y}. This symmetry allows us to determine the most essential aspects of dA from L\ and L%. As the coupling parameter e varies, the absorbing area A(A) may grow or decrease. At the same time, the basin of attraction B{A) will also change its size, and at a certain parameter value a global bifurcation (a crisis) may occur in which dA makes contact with the basin boundary. When this occurs, the absorbing area ceases to exist, and trajectories repelled from the synchronized chaotic set are no longer restrained by nonlinear mechanisms from approaching other attractors (or to go to infinity). As discussed for two coupled logistic maps in Sec. 3.3, this contact bifurcation typically marks the transition from locally to globally riddled basins of attraction. For \i = 0.07, the contact bifurcation occurs approximately at e = —0.586. Figure 3.20 shows the variation of dA with e for /J, = 0.07. This variation is nonuniform, such that increasing the numerical value of e causes the corners farthest away from the main diagonal to move outwards while at the same time
Unfolding the Riddling Bifurcation
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20 15 10 5 y o -5 -10 -15 -20 -20 -15 -10
-5
0 x
5
10
15
20
Fig 3.20. Variation of the boundaries of the absorbing area (curves 1, 2, and 3) and the basin of attraction (curves 4 and 5) with the coupling parameter for fi = 0.07. The contact bifurcation occurs for e ~ —0.595. This bifurcation marks the transition from local to global riddling of the basin of attraction.
points near the line {x — —y} move in a nonmonotonous manner. Curve 1 applies for e = —0.4, and curves 2 and 3 for e = —0.5 and —0.585, respectively. Also plotted in Fig. 3.20 is the boundary dB of the basin of attraction for e = —0.585 (curve 4) and e = —0.5 (curve 5). Since fp(x) is attracting for all x, the basin of attraction for the coupled map system stretches to infinity along the main diagonal. For e = —0.585, the system is very close to the contact bifurcation, and as the coupling constant is reduced to —0.595, the border of the absorbing area touches the basin boundary. For b = 0.14, 7 = 0.02, and /x = 0.07, the longitudinal Lyapunov exponent is Ay = 0.2542. The interval of negative transverse Lyapunov exponent stretches from a blowout bifurcation at s = —1.1 to a blowout bifurcation at e = —0.2. In this interval the synchronized chaotic state is either a Milnor attractor (that attracts almost all points from its neighborhood) or it is a weak Milnor attractor that attracts a positive (but not the full) Lebesgue measure set of points from its neighborhood. As shown above, the global bifurcation at which the absorbing area makes contact with the basin boundary occurs approximately at £ = —0.595, and for —1.1 < e < —0.595 the coupled map system displays global riddling.
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Fig 3.21. Global riddling of the basin of attraction for the synchronized chaotic state, fj, = 0.07 and e = —0.7. Points that are attracted to the synchronized state are indicated by black dots, and points that are repelled from this state are left blank. Figure 3.21 shows a typical example of the basin of attraction for the synchronized chaotic state in this coupling interval. Here, e = —0.7. Points that are attracted by the synchronized state are indicated by black dots, and points that are repelled from this state are left blank. The outer borders of the basin of attraction have remained nearly unaffected by the contact bifurcation (compare, e.g., with the basin of attraction in Fig. 3.20). However, we note the characteristic holes that have developed in the basin, some of which emanate directly from the main diagonal. Arbitrarily close to any point that is attracted by the synchronized chaotic state one can now find a positive measure set of points that are repelled from it, and the basin of attraction is a fat fractal. As e is reduced to —1.1 (i.e., relatively close to the blowout bifurcation), the basin of attraction for the synchronized chaotic state takes the form depicted in Fig. 3.22(a). The transversely unstable period-2 orbit iterates between x = —5.67 and x = 5.67. In the magnification of part of the basin of attraction displayed in Fig. 3.22(b) we have indicated the position of some of the preimages (s_2 and X-i) to the unstable period-2 orbit. One can clearly observe how the basin of attraction is riddled with tongues emanating from the period-2 cycle and its preimages. As discussed in Sec. 2.3, such repelling tongues can take two
Unfolding the Riddling Bifurcation
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Fig 3.22. Basin of attraction for the synchronized chaotic state for /i = 0.07 and e = —1.1 (a). Note how the repelling tongues emanate from the preimages of the transversely unstable period-2 cycle (b).
different forms, depending on the relative sizes of the transverse and longitudinal eigenvalues [7, 8]. In the present case, the transverse eigenvalue is numerically the larger, and the tongues are tangential to the invariant manifold. For comparison with the basin of attraction, Fig. 3.23 shows the first three preimages of the main diagonal. Since the map F is noninvertible, in addition to itself the diagonal has other sets of preimages. Points that map onto the invariant manifold in one iteration of F^E fall on the curves drawn by a heavy line. Points that map onto {a; = y} in two iterations of F^£ are indicated by a weaker line, and points that map onto the diagonal in three iterations are indicated by a thin line. We note how the structure outlined by these curves forms the backbone of the basin of attraction. Each of these curves contains a dense set of preimages of the transversely unstable period-2 cycle, and from each of these points repelling tongues emanate to both sides, contributing all together to the complexity of the attracting domain. One may wonder what happens to trajectories that are not attracted to the synchronized state. Some of these go to infinity. However, as described for our system of coupled logistic maps in Sec. 3.5, others can go to other attractors in phase space provided that such attractors exist. To illustrate this point, Fig. 3.24 shows the basins of attraction for two coexisting chaotic attractors. Here, /J, = 0.2 and e = - 0 . 9 . The heavy line along the main diagonal represents the synchronized chaotic state AQ. In the lower left corner the black
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10
5
y
o
-5
-10 -10
-5
0 x
5
10
Fig 3.23. First three preimages of the invariant set {x = y} drawn with decreasing line width. These preimages form the backbone of the basin of attraction in Fig. 3.22.
points above the diagonal show part of its basin of attraction, the full basin being symmetric both with respect to {x = y} and with respect to {x = —y}. The black points below the diagonal represent the basin of attraction for a coexisting antisymmetric, two-dimensional chaotic attractor A\. This basin is also symmetric with respect to {x — y} and {x = —J/}. The antisymmetric chaotic attractor is shown in the middle of the figure around the line {x = —y}If fi is increased to /i = 0.22, the two-dimensional antisymmetric attractors are replaced by a one-dimensional chaotic attractor on the line {x = —y}, and for a certain interval of the coupling constant, this attractor is weakly stable. In the simultaneous presence of two chaotic attractors both restricted to onedimensional manifolds in phase plane, the attracting domains take the form illustrated in Figs. 3.24(a) and (b). Here, e = -0.89 and fx = 0.219494. The basin of attraction for each of the two co-existing chaotic states is riddled with initial conditions that belong to the basin of attraction of the other state. This is a new example of intermingled basins of attraction [7, 8, 29, 30]. At the same time, the basins are riddled with holes that belong to the basin of infinity.
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Fig 3.24. Basins of attraction for two coexisting chaotic attractors: The synchronized chaotic state A0 on the main diagonal, and the two-dimensional, antisymmetric chaotic state Ai around {x = —J/}. Both basins are symmetric with respect to {x = y} as well as {x = —y}.
Fig 3.25. Intermingled basins of attraction in the presence of two one-dimensional chaotic attractors. Black points are attracted to chaotic sets along {x — y} (a) and along {x = —y} (b), respectively . Here, e = - 0 . 8 9 and fj, = 0.219494.
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Conclusions
As the parameters of the coupled map system are changed we may pass out through the blowout bifurcation where the synchronized state loses its average transverse stability and becomes a chaotic saddle. Almost all trajectories may then be repelled from the synchronized state. However, if the parameters are such that an absorbing area still exists, the dynamics will be restrained by nonlinear mechanisms to occur within this area. To investigate the phenomenon of on-off intermittency in our system of coupled intermittency maps we have chosen a set of parameter values corresponding to a relatively weak coupling and to a situation close to the subcritical perioddoubling bifurcation in the individual map. The weak interaction allows the two subsystems to operate nearly independent so that turbulent bursts can occur in one map while the other operates in a laminar phase. The small value of /i causes the laminar phases of the type-Ill intermittency of the individual maps to be relatively long. Figure 3.26 displays the global dynamics of the on-off intermittency observed for fi = 0.1, 7 = 0.13, b — 0.18697, and e = —0.1. In its temporal behavior the system exhibits long laminar phases in which the excursions in phase plane are relatively small, interrupted by occasional bursts in which the system moves out in phase space, preferably along one of the main axes. As illustrated in the figure, the global dynamics is restricted to taking place within the absorbing area, the boundaries of which are defined by the heavy lines surrounding the attractor. Figure 3.27(a) provides a close up of the dynamics near the transversely unstable fixed point. Here y, = 0.001, 7 = 0.13, b = 0.18697, and e = -0.0002. For these parameters the longitudinal and transverse eigenvalues for the fixed point are z/y = — 1 — \± = —1.001 and i/± = — 1 — fi — 2e — —1.0006, respectively. Inspection of the figure shows how trajectories reinjected into the vicinity of the fixed point are repelled in all directions, producing the typical structure of a repelling tongue. If the coupling parameter is changed to e = —0.002, the eigenvalues of the fixed point become v\\ = —1.001 and vL = —0.997. Hence, the fixed point is now a saddle. The bifurcation that has occurred is a subcritical perioddoubling, and close to the fixed point we now have a repelling, antisymmetric period-2 cycle. The local dynamics generated in this situation is illustrated in
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8 6 4 2 y
o -2 -4 -6 -8 - 8 - 6 - 4 - 2
0 x
2
4
6
8
Fig 3.26. Global picture of the on-off intermittency observed for /J, = 0.1, 7 = 0.13, b = 0.18697, and e = —0.1. The solid lines represent the borders of the absorbing area.
Fig 3.27. Close up of the dynamics near the unstable fixed point (a). After a subcritical period-doubling the local dynamics is controlled by the repelling antisymmetric period-2 cycle (b). The position of the repelling period-2 cycle is marked by the two crosses.
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—I ^
(b) — |
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(b)
1—|
• \i
^i ( b )
^(c)
1
1
• \i
Fig 3.28. Bifurcation sequences in the two main scenarios describing the breakdown of chaotic synchronization.
Fig. 3.27(b). The global dynamics has remained largely unchanged and is still an example of on-off intermittency. In the local dynamics, however, we observe how the repulsion is controlled by the unstable period-2 cycle. Based on the above results we may distinguish between two main scenarios as sketched in Fig. 3.28. Here, [i^ and (j,^ denote the points for some parameter fi at which the riddling, respectively, the blowout bifurcation occurs. // c ' represents the point at which the contact bifurcation between the absorbing area and the basin of attraction takes place. The riddling bifurcation is a local bifurcation, and the blowout is quasi-local in the sense that it is controlled by the average conditions along the low-dimensional chaotic attractor. The contact bifurcation, on the other hand, is a global bifurcation that depends on the nonlinear character of the system. In the first scenario (a), the contact bifurcation occurs between /j,^ and /z^. In this case we can observe a local riddling in the interval between //"' and /z^, global riddling from /z^ to /z^, and an abrupt loss of synchronization in the blowout bifurcation. In the second scenario (b), the contact bifurcation happens after the blowout. In this case we have local riddling in the entire interval from /i(°) to n®, and on-off intermittency between /z^ and fi^. Another distinction is associated with the question whether or not stable synchronous states arise within the absorbing area. Finally, it should be noted that in the case that the riddling bifurcation is soft, there will initially be a mixed absorbing area stretching along the synchronized chaotic state, and additional transformations
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must take place before global riddling can be observed.
Bibliography [1] J. C. Sommerer and E. Ott, A Physical System with Qualitatively Uncertain Dynamics, Nature (London) 365, 138-140 (1993). [2] E. Ott and J. C. Sommerer, Blowout Bifurcations: The Occurrence of Riddled Basins and On-Off Intermittency, Phys. Lett. A 188, 39-47 (1994). [3] P. Ashwin, J. Buescu, and I. Stewart, Bubbling of Attractors and Synchronization of Chaotic Oscillators, Phys. Lett. A 193, 126-139 (1994). [4] P. Ashwin, J. Buescu, and I. Stewart, From Attractor to Chaotic Saddle: A Tale of Transverse Instability, Nonlinearity 9, 703-737 (1996). [5] J. F. Heagy, T. L. Carroll, and L. M. Pecora, Desynchronization Orbits, Phys. Rev. E 52, R1253-R1256 (1995).
by Periodic
[6] Y.-C. Lai, C. Grebogi, J. A. Yorke, and S. C. Venkataramani, Riddling Bifurcation in Chaotic Dynamical Systems, Phys. Rev. Lett. 77, 55-58 (1996). [7] Yu.L. Maistrenko, V.L. Maistrenko, A. Popovich, and E. Mosekilde, Transverse Instability and Riddled Basins in a System of Two Coupled Logistic Maps, Phys. Rev. E 57, 2713-2724 (1998). [8] Yu.L. Maistrenko, V.L. Maistrenko, A. Popovich, and E. Mosekilde, Role of the Absorbing Area in Chaotic Synchronization, Phys. Rev. Lett. 80, 1638-1641 (1998). [9] Yu. Maistrenko, T. Kapitaniak, and P. Szuminski, Locally and Globally Riddled Basins in Two Coupled Piecewise-Linear Maps, Phys. Rev. E 56, 6393-6399 (1997). [10] T. Kapitaniak, Yu. Maistrenko, A. Stefanski, and J. Brindley, Bifurcations from Locally to Globally Riddled Basins, Phys. Rev. E 57, R6253-R6256 (1998).
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[11] S.C. Venkataramani, B.R. Hunt, E. Ott, D.J. Gauthier and J.C. Bienfang, Transitions to Bubbling of Chaotic System, Phys. Rev. Lett. 77, 5361-5364 (1996). [12] S.C. Venkataramani, B.R. Hunt, and E. Ott, Bubbling Transition, Phys. Rev. E 54, 1346-1360 (1996). [13] C. Mira, L. Gardini, A. Barugola, and J.-C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps (World Scientific, Singapore, 1996). [14] R.H. Abraham, L. Gardini, and C. Mira, Chaos in Discrete Dynamical Systems (Springer-Verlag, Berlin, 1997). [15] G.-I. Bischi, L. Stefanini, and L. Gardini, Synchronization, Intermittency and Critical Curves in a Duopoly Game, Mat. Comp. Sim. 44, 559-585 (1998). [16] P. Ashwin and J.R. Terry, On Riddling and Weak Attractors, Physica D 142, 87-100 (2000). [17] E. Ott, J.C. Sommerer, J.C. Alexander, I. Kan, and J.A. Yorke, Scaling Behavior of Chaotic Systems with Riddled Basins, Phys. Rev. Lett. 7 1 , 4134-4137 (1993). [18] Yu.L. Maistrenko, V.L. Maistrenko, O. Popovych, and E. Mosekilde, Unfolding of the Riddling Bifurcation, Phys. Lett. A 262, 355-360 (1999). [19] Yu.L. Maistrenko, V.L. Maistrenko, O. Popovych, and E. Mosekilde, Desynchronization of Chaos in Coupled Logistic Maps, Phys. Rev. E 60, 2817-2830 (1999). [20] J. Laugesen, E. Mosekilde, Yu.L. Maistrenko, and V.L. Maistrenko, On-Off Intermittency and Riddled Basins of Attraction in a Coupled System, Adv. Complex Syst. 1, 161-180 (1998). [21] M. Manscher, M. Nordahn, E. Mosekilde and Yu.L. Maistrenko, Riddled Basins of Attraction for Synchronized Type-1 Intermittency, Phys. Lett. A 238, 358-364 (1998). [22] I. Gumowski and C. Mira, Recurrences and Discrete Dynamical Systems (Springer Verlag, New York, 1980).
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[23] C. Mira and T. Narayaninsamy, On Two Behaviors of Two-Dimensional Endomorphisms. Role of the Critical Curves, Int. J. Bifurcation Chaos Appl. Sci. Eng. 3, 187-194 (1993). [24] L. Gardini, R. Abraham, R.J. Record, and D. Fournier-Prunaret, A Double Logistic Map , Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 145-176 (1994). [25] P. Celka, A Simple Way to Compute the Existence Region of ID Chaotic Attractors in 2D-Maps, Physica D 90 235-241 (1996). [26] I.U. Bronstein and A.Ya. Kopanskii, Smooth Invariant Manifolds and Normal Forms (World Scientific, Singapore, 1996). [27] O. Popovych, Asymptotic Behaviour and Stability of Solutions of Coupled Oscillators, PhD Thesis (Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, 1999). [28] C. Mira, C. Rauzy, Yu. Maistrenko, and I. Sushko, Some Properties of a Two-Dimensional Piecewise-Linear Noninvertible Map, Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 2299-2319 (1996). [29] J.C. Alexander, J.A. Yorke, Z. You, and I. Kan, Riddled Basins, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 795-813 (1992). [30] J.C. Alexander, B.R. Hunt, I. Kan, and J.A. Yorke, Intermingled Basins for the Triangle Map , Erg. Theor. Dyn. Syst. 16, 651-662 (1996). [31] S.E. Newhouse, Diffeomorphisms with Infinitely Many Sinks, Topology 13, 9-18 (1974). [32] S.V. Gonchenko, L.P. Shil'nikov, and D.V. Turaev, Dynamical Phenomena in Systems with Structurally Unstable Poincare Homoclinic Orbits, Chaos 6, 15-31 (1996). [33] V. Astakhov, A. Shabunin, T. Kapitaniak, and V. Anishchenko, Loss of Chaos Synchronization through the Sequence of Bifurcations of Saddle Periodic Orbits, Phys. Rev. Lett. 79, 1014-1017 (1997). [34] G. Iooss and D.D. Joseph, Elementary Stability and Bifurcation Theory, 2nd Ed. (Springer-Verlag, Berlin, 1980).
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[35] V. Maistrenko, Yu. Maistrenko, and I. Sushko, Noninvertible TwoDimensional Maps Arising in Radiophysics, Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 383-400 (1994). [36] V.I. Arnol'd (Ed.), Dynamical Systems V, Encyclopedia of Mathematical Sciences (Springer-Verlag, Berlin, 1994). [37] Yu. Maistrenko, I. Sushko and L. Gardini, About Two Mechanisms of Reunion of Chaotic Attractors, Chaos, Solitons and Fractals 9, 1373-1390 (1998). [38] J. Laugesen, N. Carlsson, E. Mosekilde, and T. Bountis, Anomalous Statistics for Type-Ill Intermittency, Open Sys. Information Dyn. 4, 393-405 (1997). [39] J. Laugesen, E. Mosekilde, T. Bountis, and S.P. Kuznetsov, Type-II Intermittency in a Class of Two Coupled One-Dimensional Maps, Discr. Dyn. in Nature and Society 5, 233-245 (2000).
Chapter 4 TIME-CONTINUOUS SYSTEMS 4,1
Two Coupled Rossler Oscillators
As discussed in the preceding chapters, recent studies of chaotic dynamics restricted to low-dimensional manifolds of total phase space have led to the discovery of a variety of new phenomena, including riddled basins of attraction [1], blowout bifurcations and on-off intermittency [2], and attractor bubbling [3]. Full synchronization, in which both the phases and the amplitudes develop precisely in the same manner, can be achieved through the coupling of two (or more) identical chaotic oscillators [4]. Experimental evidence for this type of synchronization in different electronic systems has been obtained by Volkovskii and Rul'kov [5], Pecora and Carroll [6, 7], and Rul'kov et al. [8]. Chaotic synchronization has also been observed in coupled laser systems [9], and del Rio et al. [10] have shown that synchronization of chaotic oscillators can be achieved with unidirectional as well as with bidirectional coupling. Surveys of many of these results may be found in the publications by Rul'kov [11] and by Pecora et al. [12]. When the trajectories of a system of two coupled JV-dimensional oscillators are restricted to an iV-dimensional subset of total phase space (the synchronization manifold), the spectrum of Lyapunov exponents can be divided into N longitudinal exponents that characterize the motion in the synchronization manifold, and N transverse exponents that describe the response to perturbations perpendicular to this manifold. A necessary condition for synchronization 123
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is that the largest transverse Lyapunov exponent is negative. In this case, the synchronized state is at least weakly stable [13, 14]. As described in Chapter 2, this implies that the state attracts a finite Lebesgue measure set of points from its neighborhood, but not necessarily the full measure. Related to the existence of particular orbits for which the transverse Lyapunov exponent is positive, there may still exist a positive measure set of points within the neighborhood of the synchronized chaotic state from which the trajectories diverge to infinity or approach some other attracting set. This produces the characteristic riddled basins of attraction with repelling tongues issuing from a dense set of points in the synchronization manifold. A major question of concern, particularly in connection with the electronic applications, relates to the sensitivity of the synchronized state to a small mismatch in the parameter values between the interacting subsystems. This problem has been considered, for instance, by Rul'kov et al. [15], by Abarbanel et al. [16], and by Johnson et al. [17]. Rosenblum et al. [18] have demonstrated how one can identify and characterize the resulting state of phase synchronization, and Astakhov et al. [19] have examined how the bifurcations that control the loss of synchronization are affected by a parameter mismatch. Because of inevitable drift of the parameters during operation, passive synchronization methods may not suffice in practice, and a variety of active controls are presently being investigated. In particular, Volkovskii [20] has shown how one can obtain in- and out-of-phase synchronization as well as frequency, phase and phaseshift key-in modulation and demodulation of coupled electronic circuits. Other types of active synchronization methods have been studied by Fradkov and Markov [21] and by Suykens et al. [22]. In-phase and anti-phase synchronization have also been reported by Bohr et al. [23] in a parallel computer simulation of 256 chaotically oscillating kidney nephrons. In Chapters 2 and 3 we established a detailed description of the processes by which synchronization is lost in a system of two linearly coupled logistic maps [24, 25, 26], emphasizing the distinction between sub- and supercritical riddling bifurcations and the role of the so-called absorbing and mixed absorbing areas [27]. The purpose of the present chapter is to extend this analysis to systems of coupled time-continuous oscillators. The presentation is based on our recent investigations of chaotic synchronization and riddled basins of attraction in a system of two coupled Rossler oscillators [28]. This investigation applied a Lyapunov function approach to obtain a sufficient condition for chaotic synchro-
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nization and a calculation of the largest transverse Lyapunov exponent to establish the necessary condition. We also studied the transverse period-doubling, pitchfork and Hopf bifurcations, by which low-periodic orbits embedded in the synchronized chaotic state become transversely unstable. Finally, we showed how the characteristic riddled basin structure for time-continuous systems can be explained in terms of trajectories that approach an unstable low-periodic orbit. The above analysis was subsequently extended to the case where there is a mismatch between the two subsystems [29]. With this aim we generalized the approach developed by Johnson et al. [17] for the case of unidirectional coupling in order to show that synchronization in the neighborhood of the synchronization manifold for the identical subsystems can still occur, although the trajectories will exhibit characteristic fluctuations in the directions transverse to this manifold. We determined both the shift of the synchronization manifold and the amplitude of the transverse fluctuations in terms of the coupling strength and the mismatch parameter. We have also examined the problem of partial synchronization (or clustering) in diffusively coupled arrays of identical Rossler oscillators [30]. As discussed in Chapter 2, the term partial synchronization denotes a dynamic state in which groups of oscillators synchronize with one another, but there is no synchronization among the groups. By combining numerical and analytical methods we proved the existence of partially synchronized states for systems of three and four oscillators and described the dynamics within the clusters. Finally, we analysed the emergence of synchronized states in larger arrays of chaotic oscillators. In Chapter 5 we shall demonstrate how similar phenomena can arise in a system of coupled Sherman oscillators [31] describing the characteristic spiking and bursting behavior of pancreatic /3-cells. There is ample evidence to show both that the behavior of the individual cell model can become chaotic [32, 33] and that neighboring cells synchronize [34, 35].
4.2
Transverse Destabilization of Low-Periodic Orbits
Let us start by establishing the conditions for synchronization of two coupled Rossler systems and examine the mechanisms by which this synchronization is lost. A system of two linearly coupled, identical oscillators may be cast into
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the form x= f(x) + C(y - x), V= f(y) + C(x - y),
(4.1)
3
where x, y £ R represent the state variables of the individual oscillator and C is the coupling matrix. For simplicity, this matrix will be chosen as C =diag{di, d2, c/3}. This implies that there is no cross coupling between different variables in the two oscillators. In the case of coupled Rossler systems x = (2:1, £2, £3) G R3, and the function /(•) takes the form [36]
(
-x2 -x3
\
+ ax2 • (4.2) b + x3(x! -c) ) Here, time has been scaled so that the natural (angular) frequency is 1. If the parameters are chosen to be a = 0.42, b — 2.0, and c = 4.0, the uncoupled system x = f(x) can be assumed to generate a chaotic attractor [37]. The synchronization manifold x = y is invariant in the six-dimensional phase space (x, y) of the coupled system (4.1). Complete (or full) synchronization occurs when the interacting oscillators asymptotically exhibit identical chaotic behaviors, i.e., || x(t) — y(t) |—>• 0 as t —> 00 for any initial condition (a;(0), j/(0)) within some neighborhood of the synchronous state A [12]. Introducing coordinates along and transverse to the synchronization manifold 77 = y + x, and £ = y — x, the linearized equation for the transverse perturbations takes the form ft=[Df(s(t))-2C}5Z. (4.3) Here, Df denotes the Jacobian matrix evaluated along some solution s(t) C A. Hence, the local stability properties of the synchronized motion can be derived by analyzing the stability of the equilibrium point 5£ = 0 of Eq. (4.3). Applying the Lyapunov function U =\\ 5£ ||2 to Eq. (4.3) we obtain [29] the domain D\ of transverse stability in the space of coupling parameters as shown in Fig. 4.1. Namely, if (^1,^3) G D\ and d2 > a/2 then the equilibrium point 5£ = 0 is asymptotically stable, which implies synchronization. The Lyapunov function method provides the sufficient conditions for synchronization and, therefore, underestimates the real synchronization region. The blowout bifurcation curve (curve 2 in Fig. 4.1) determines the moment where the largest transverse Lyapunov exponent of the synchronized chaotic attractor changes sign [2]. This curve represents the necessary condition for synchronization without intermittent bursting. Xl
Time-Continuous Systems
I I
synchronization region
Fig 4.1. Synchronization of two identical Rossler systems. The region of complete synchronization Dx as determined by the Lyapunov function method. Curve 2 is the blowout curve (calculated for rf2 = 1-0). This curve delineates the region of weak transverse stability of the synchronized state.
-
4. 2
\ '
127
'
J, The exact domain of synchronization in the space of coupling parameters has to be obtained by means of numerical experiments. As in the case of coupled logistic maps, asymptotic stability requires that not only a typical trajectory on the synchronized chaotic set is transversely stable (as used for calculation of the Lyapunov exponent), but that all orbits on the chaotic set are stable in the direction transverse to the synchronization manifold. The normal approach to this problem is to investigate the transverse stability of different low-periodic orbits embedded in the synchronized chaotic state A [38, 39]. As demonstrated by several authors [1, 2, 28, 38], the transverse stability properties of individual periodic orbits do not necessarily coincide with those averaged over the chaotic attractor. Figure 4.2 shows the bifurcation diagram for the synchronized period-1 saddle cycle in the plane (d\,d$), and Fig. 4.3 displays a projection of this cycle and of the synchronized chaotic state for our coupled Rossler oscillators. In Fig. 4.2, region I is the domain of transverse stability for the period-1 cycle. This cycle is transversely unstable in region II. When crossing part P of the bifurcation curve, the period-1 cycle undergoes a pitchfork bifurcation. Similarly, when crossing the parts denoted PD and H, destabilization occurs via a transverse period-doubling or Hopf bifurcation. As discussed in Chapter 2 and 3, each of these transverse bifurcations may be either sub- or supercritical [26]. Figures 4.4 and 4.5 illustrate how the transverse destabilization of the period-1 saddle cycle occurs in a supercritical period-doubling and in a subcritical pitchfork bifurcation, respectively. In the first case, a period-2 saddle cycle situated symmetrically with respect to the synchronization plane is created in the destabilization process. In the second
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P
I
C PD
.«.—
Fig 4.2. Bifurcation diagram for the transverse destabilization of the period-1 saddle cycle embedded in the synchronized attractor. Parts P correspond to the loss of transverse stability via a pitchfork bifurcation, PD via period doubling, and H via a Hopf bifurcation.
stability region for period—1 cycle
n H
P
*"
i
1 .o
o.o
I
T
iS
0.0
4.0
;
" •
Fig 4.3. Period-1 saddle cycle embedded in the synchronized chaotic attractor. Its period is T « 2K. S illustrates the Poincare section of the attractor.
J" *>K
k
I 0.0
1 4.0
I
case, two mutually symmetric, doubly unstable period-1 cycles existing before the transverse destabilization, approach the symmetric period-1 saddle cycle and destroy its stability as they are annihilated [28]. In Fig. 4.6 we have combined the bifurcation diagrams obtained for period-1, -2, -3, and -4 saddle cycles to determine which is the first to lose its transverse stability. (Here and in the following text we use the term period-i¥ saddle cycle to denote an iV-loop orbit with a period of approximately NT, with T being the period of the period-1 cycle.) The moment of blowout bifurcation is displayed as a bold curve. The rightmost envelope of the destabilization curves for low-periodic orbits gives the moment when the first of the considered saddle
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-0.3
Fig 4.4. Antisymmetric cycle of period ss 2T that arises after the supercritical transverse period doubling bifurcation. The synchronization manifold is shown as a projection on the
a)
Fig 4.5. Different projections of the two doubly unstable period-1 cycles (dashed curves) and the transversely stable period-1 saddle cycle before the subcritical pitchfork bifurcation at d, = 1.0, d2 = 1.0, and d3 « -1.75.
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cycles becomes transversely unstable. Part I of this envelope corresponds to a period-doubling bifurcation of the period-3 cycle, part II to a period-doubling bifurcation of the period-1 cycle, part III to a period-doubling bifurcation of the period-2 cycle, and part IV to a pitchfork bifurcation of the period-1 cycle.
d*
Fig 4.6. Bifurcation diagrams for the transverse destabilization of low-periodic orbits and the blowout bifurcation curve (solid curve). We consider the rightmost envelope to determine the occurrence of the first transversely unstable (periodic) orbit embedded in the chaotic attractor.
blowout period-1
period-3 _.._.._.._... period-4
period—2
4.3
Riddled Basins
In the previous section we showed how the fully synchronized chaotic state loses its transverse stability via desynchronization of particular orbits in a socalled riddling (or bubbling) bifurcation. Riddled basins of attraction [1, 2] may be observed immediately after the riddling bifurcation where the maximal transverse Lyapunov exponent is still negative for typical trajectories. Under the assumption that the first orbits to become transversely unstable in the coupled Rossler systems are low-periodic cycles [39], the riddling bifurcation curve can be represented by the rightmost envelope in Fig. 4.6. We conclude that riddled basins of attraction may occur in the domain that is delineated by the riddling bifurcation curve on one side and by the blowout bifurcation curve on the other. This domain is illustrated in Fig. 4.7. Fixing the parameters d\ = 0.1, d2 = 1.0, and d% — —1.86 (point A in Fig. 4.7) we
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have calculated the two-dimensional cross-section of the basin of attraction with £2 = £3 = 0,772 = 773 = 1 displayed in Fig. 4.8. The basin shows clear evidence of riddling in the form of a dense set of tongues of initial conditions from which the trajectories are repelled from the synchronized chaotic state. Another cross-section of the basin is presented in Fig. 4.9. Here, the values £3 + 2/3 = % = 0.1, £2 = £3 = 0, and £1 = 1.0 are fixed. Black points are points from which the trajectory diverges to infinity. In addition, in order to clarify the structure of the repelling tongues throughout the attractor, we have investigated the long term behavior of trajectories that start from points 77 6 •A, £ = (0.01,0,0), i.e., with longitudinal coordinates belonging to the attractor and a small transverse perturbation of the form (£1,0,0,0). Figure 4.10 shows the projections of this set onto the (2:1,2:2) and (2:1,2:3) planes, respectively. Black points again correspond to initial values of trajectories that diverge to infinity, and white points to those that converge to the synchronized state.
Fig 4.7. Domain in the (di,d3) plane where the synchronized state is transversely stable on average but there is a transversely unstable periodic orbit embedded in the attractor (da = 1.0).
-0.4
0.0
0.4
0.8
d
x
In order to explain the structure of the basins observed in Figs. 4.8, 4.9, and 4.10 we have determined the Poincare section S of the individual Rossler oscillator for fixed 2:1 = 0.44 as illustrated in Fig. 4.3 [28]. The invariant attracting set of the corresponding Poincare map is shown in Fig. 4.11(a). Considering this set to be one-dimensional, we have constructed the one-dimensional map which acts on the set. This is shown in Fig. 4.11(b). As one can see from Fig. 4.6, the period-1 saddle cycle is transversely unstable for the considered parameter values. The fixed point P of the one-dimensional map corresponds
132 Chaotic Synchronization: Applications to Living Systems
Fig 4.8. Cross-section % = % = 1.0, £2 = £3 = 0 of the basin of attraction for the synchronized state. The coupling parameters are d\ = 0.1, efe = 1.0, and d% = —1.86. This corresponds to point A in Fig. 4.7.
Fig 4.9. The cross-section £1 = 0.1, £2 = £3 = 0, and % = 1.0 of the basin of attraction for the synchronized state. Black points correspond to initial conditions from which the trajectory diverges to infinity. The four orbits starting from the preimages P ^ 4 are shown in gray.
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133
Fig 4.10 Here two projections of the synchronized attractor are presented. Adding a transverse perturbation & = 0 01 to the synchronized state on the attractor, the black points correspond to initial conditions of orbits that diverge to infinity. The four orbits starting from the preimages P^ 4 are shown in gray
to the period-1 saddle cycle, and its preimages P"1, P~ 2 , Pf'f', and P{2 3,4 correspond to orbits that approach the cycle in one, two, three, and four loops, respectively. Considering the various preimages of the 1-D map, we can identify different orbits of the Rossler system that approach the period-1 cycle, namely the orbits which contain the points P, P - 1 , P~ 2 , P - 3 , P - 4 , Plotting further preimages of the fixed point P we can find the "inset" of the period-1 cycle. This set includes the trajectories that share the property of transverse instability with the period-1 cycle, and these orbits form the skeleton of the riddling structures in Figs. 4.8 and 4.9. The orbits that start from P ^ 4 are shown in Figs. 4.8 and 4.9. We observe that these orbits correspond to the most transversely unstable parts of the synchronized attractor. That the preimages do not coincide with the repelling points in the right part of Fig. 4.9 can be explained by means of Fig. 4.12. Here the projection of the synchronized attractor onto the (xz, x\) plane is presented. The dashed line shows the corresponding projection of the plane from Fig. 4.9. The left part of the attractor is nearly flat, and this allows us to reveal the location of transverse instabilities by investigating the two-dimensional crosssection in Fig. 4.9 "along" the attractor. The same is not the case for the right hand part of Fig. 4.9.
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(a)
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P4
1.4 *3
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(b)
1.4
p^
P
to Living
/r3 /
^ \ P
P
1
*2
- * \ ~ / \
./
- fM
1.1 0.8
_t p -4 1
-1
I
P"1
'*1
/-4
0.6 0.6
/ ^
0.8
\ P
1.2
1.4
Fig 4.11. Invariant set of the Poincare map 5 (cf. Fig. 4.3) for the individual Rossler attractor (a) and 1-D map acting on this invariant set (b). The fixed point P corresponds to the period-1 saddle cycle for the Rossler system. P^' are the (m different) preimages of P of order i.
4.4
Bifurcation Scenarios for Asynchronous Cycles
The next problem to consider is the bifurcation scenarios for the appearance of stable cycles and various types of more complex dynamics outside the synchronization manifold. With this purpose we shall construct a one-dimensional bifurcation diagram along the diagonal line in coupling parameter space passing through the riddling bifurcation points R\ and R% as depicted in Fig. 4.6 (dashed line). This scan was chosen in such a way that both riddling bifurcation points correspond to destabilization of the synchronous period-1 cycle. Accordingly, we shall be mainly interested in solutions arising from this cycle. Let us introduce rfasa new coupling parameter such that its variation corresponds to the motion of a point along the scan. Expressed in terms of this new parameter, our system has the form (4.1) with the coupling matrix C = diag {d - 0.6,1.0, -3.1d + 0.7} . The leftmost point of the scan corresponds to d = 0 while the rightmost point is reached for d = 1.0. Figure 4.13 shows the behavior of the numerically largest multipliers /xy and fi± of the synchronous period-1 cycle. The largest longitudinal multiplier is obviously independent of d and numerically larger than 1. The region of transverse stability is located between the points R\ and R2 where the largest transverse multiplier has a modulus less than one. We recall that R\ (d = 0.2408) corresponds to a supercritical period-doubling
Time-Continuous Systems
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Fig 4.12. Projection of the Rossler attractor on the (xi,x3)plane. The dashed line represents a projection of the cross-section from Fig. 4.9.
bifurcation (fix = — 1) and i?2 (d = 0.7360) to subcritical pitchfork bifurcation The asynchronous period-2 cycle that arises at point R\ plays an important role in the global dynamics of the coupled system. Figure 4.14 shows the evolution of this cycle in phase space with decreasing d. For d = 0.240, the asynchronous period-2 cycle is still a saddle cycle (shown as a dotted curve in Fig. 4.14). At d = 0.220 the cycle stabilizes via a subcritical pitchfork bifurcation. (Note that although the period-2 cycle does not belong to the synchronization manifold, its symmetry properties allow the pitchfork to occur). The interval of existence for the stable asynchronous period-2 cycle is approximately 0.201 < d < 0.220. At the lower end of this interval the cycle is destabilized in a secondary Hopf (or Neimark-Sacker) bifurcation, producing typically a quasiperiodic motion. Numerically we can observe a stable torus that arises at this bifurcation point. Two different projections of this torus are shown in Fig. 4.15, and the corresponding bifurcation diagram is presented in Fig. 4.16. Finally, this bifurcation branch leads to a stable asynchronous chaotic set via torus destruction at d = 0.197. This is illustrated by the sequence of Poincare sections in Fig. 4.17. This whole development is similar in many respects to the scenario described for two coupled logistic maps in Chapter 3. Another chaotic set, a chaotic saddle, is born as a result of a period-doubling cascade of the unstable asynchronous orbits associated with the subcritical pitchfork bifurcation at d = 0.220. This is illustrated in Fig. 4.18, where we
136
Chaotic Synchronization: Applications to Living Systems
1
-
longitudinal multiplier
_z
1
'o.22
' 0.240/
,1/ ££''
\ ""~V''/<
transverse multiplier
transverse stability region
0.ioi(NS)
-
-
-
-
-
—*•
1
Fig 4.13. Variation of the modulus of the largest multipliers of the period-1 cycle, p,\\ (longitudinal) and /ij_ (transverse) along the dotted line of Fig. 4.6. The horizontal line in the middle of the graph (around d = 0.4) where two multipliers coincide with one another corresponds to the existence of two complex conjugated multipliers.
1
i
Fig 4.14. Phase space representations of the asynchronous period-2 cycle for different values of the coupling parameter d. An unstable cycle is shown by a dashed line while a solid line denotes a stable cycle. (P) and (NS) correspond to the pitchfork and Neimark-Sacker (or secondary Hopf) bifurcation points that occur, respectively, at d ~ 0.220 and d ^ 0.201.
Fig 4.15. Two different projections of the stable invariant torus that exists for d = 0.2. (a) projection onto the (x\,yi) plane, and (b) onto the (xi,Xz) plane.
Time-Continuous Systems
1
3
1
,
137
1
Fig 4.16. Bifurcation diagram - in the parameter region where - the stable asynchronous torus ap-
2
^R^^^MaJP'*1' 1
O -1
pears and is destructed. To the
- right in the figure we have the sta-
-2
j&^__. "' Q Q Q ^ ^ U ^ M "' 0.2
-
ble asynchronous period-2 cycle.
1
0.205
d display both the unstable period-4 cycle (a) and the unstable period-8 cycle (b) observed, respectively, for d = 0.2148 and d = 0.2142. Figure 4.19 provides a schematic overview of the whole bifurcation diagram where two parallel perioddoubling cascades can be seen in a parameter range to the left of d = 0.217. Although the chaotic saddle is unstable, it influences the transient behavior of the system. As an example, Fig. 4.20 shows a typical "disordered" transient in the region of the chaotic saddle (d = 0.209) for an orbit approaching the asynchronous period-2 cycle (shown bold). It is worth noting that all the above bifurcations occur while the synchronous chaotic state is weakly stable, cf. Fig. 4.19. This allows us to expect that before the blowout with decreasing d the coupled Rossler system will reveal riddled basins caused by different asynchronous attracting sets. Indeed, the basin may be riddled with initial conditions approaching a stable limit cycle, a torus, or a chaotic set. Figure 4.21 shows an example of this type of structure for d = 0.204 where the weakly stable synchronous chaotic attractor coexists with a stable asynchronous period-2 cycle. Black points represent initial conditions from which the trajectory approaches the stable period-2 cycle (indicated in the Poincare section by a white point P2). Along the horizontal axis, where the weakly stable synchronous chaotic set is located, we have marked the point Pi of the synchronous period-1 cycle. After the riddling bifurcation this cycle is doubly unstable. Let us now consider what happens as the coupling parameter d increases. We can then observe the appearance of an asynchronous chaotic saddle while
138
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x
3
Systems
3.8
1
1
P^r
h
•••/•..
i
'
i
i
'
i
-••*.
_U**r - • ; - ^ , " • - • ^ •
!
& -
&
V
v*.
••S^V
-H S i
.**>. -
0
i
.
2
i
i
i
2.2
2.4
2.6
X,
(A) Fig 4.17. Two-dimensional projections onto the (xi, x3) plane of the invariant set of the Poincare return map at different values of coupling: d = 0.201 (a), d = 0.198 (b), d = 0.1975 (c), and d = 0.197 (d). The figure illustrates the destruction of the torus and the formation of a stable, asynchronous chaotic set.
Time-Continuous Systems
139
Fig 4.18. Unstable asynchronous periodic period-4 and period-8 orbits arising through perioddoubling bifurcations at d = 0.2148 (left) and d = 0.2142 (right).
weak stability of the synchronous statecomplete synchronization"
stable asynchronous . chaotic set , stable torus
•«3 X -
—
''%%$' asynchronous
^z222& c^9}}?,
^Z asynchronous ,. ., ^y chaotic saddle J ;; '
w
-s,a 41?, J$
%$' asynchronous $, chaotic saddle fli
SN';
NS
f^
fN r ^
i
4+-
Cj Ci O
Fig 4.19. One-dimensional bifurcation diagram for solutions associated with the synchronous period-l cycle. A detailed description may be found in the text. In a region to the left of the pitchfork bifurcation points at d = 0.220 the weakly stable synchronous chaotic set coexists with stable asynchronous attractors.
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Fig 4.20. Typical behavior of orbits attracted by the stable asynchronous limit cycle in the presence of asynchronous chaotic saddle for d = 0.209. Note the long chaotic transient.
""-4-2
0
2
4
6
x
i
the synchronous chaotic state is still asymptotically stable. The corresponding bifurcation diagram is shown in the right-hand side of Fig. 4.19. It involves a subcritical pitchfork bifurcation of the synchronous period-1 cycle (P), and simultaneous reverse period-doubling bifurcations (PD), saddle-node bifurcations (SN) and Hopf bifurcations (H) of the symmetric pair of asynchronous period-1 cycles arising in the pitchfork bifurcation. The unstable period-2 cycles produced in the period-doubling bifurcations continue in parallel perioddoubling cascades leading finally to the emergence of two symmetric chaotic saddles. Note, however, that we do not observe asymptotically stable limit sets in the phase space for d > 0.736, at least not such which can be associated with the bifurcations described above. In fact, the globally riddled basin in Fig. 4.8 was calculated for parameter values in this region. Here, the basin of attraction for the synchronized chaotic state is riddled with initial conditions from which the trajectory diverges.
4.5
The Role of a Small Parameter Mismatch
When we consider synchronization phenomena in coupled chaotic systems, one of the first questions to arise relates to the influence of a parameter mismatch. An approach to this problem has been outlined by Johnson et al. [17]. In this section we shall extend that approach and describe the interesting effect of a shift of the synchronization manifold which can be observed numerically when the considered parameter mismatch can be expressed in a particular manner. The motion of the coupled system then takes place in the vicinity of some state £ = const 7^ 0 with occasional chaotic bursts of small amplitude.
Time-Continuous Systems
141
Fig 4.21. Globally riddled basin of attraction for d = 0.204. Black points represent initial conditions from which the trajectory is attracted to the stable asynchronous period-2 cycle P2. 71 and 72 are linear combinations of the longitudinal and the transverse coordinates, respectively, chosen such that the plane intersects the asynchronous period-2 cycle.
We start with the following model [29] x
= f(x) + C{y -x)
+ sg(x),
y=f(y) + c(x-y),
(4.4)
which accounts for the difference between the parameters of the two systems via the term eg(x). s is taken as a measure of the magnitude of the parameter difference. If the mismatch is restricted to a single parameter fi, we define e = fi - no and g(x) = ^ £ ^ In the case that a mismatch exists for more than one parameter, one can choose, for example, e = \fi\ = max; |/z, — ^oi| a n d g{x) = p | (/ {x, fi) — / (x, Ho))- We shall consider the case where the identical systems, i.e., with e = 0, are synchronized. The approximate solutions of (4.4) may be represented in the following form xe(t) = x0{t) + e
(t), ye(t) = y0(t) + ei/>(t)
(4.5)
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where we neglect higher order terms in e. xo(t) and yo(t) are the solutions of Eq. (4.4) with e = 0. By substituting (4.5) into (4.4) and comparing terms of order 1 and e we obtain the following equations for the unknown functions <j> and ip: 'cj>= Df (x0 (t)) 4> + C{^-)+g(xv
(*)),
(4 6)
=£/MW + co/.-v). Using Eq. (4.5) the transverse perturbation can be written as £(£) = x(t) — y(t) = £o(£) + e£iM- Here, £o(i) = xo(t) — yo{t) -> 0 with t -> co because of the assumed synchronization of the identical systems. Therefore, £(t)~e6(t) = eW*)-V(*)).
(4.7)
The equation for ^ ( i ) now follows from (4.6) ^
= [Df (x 0 (t)) - 2C] 6 + 9 (x0(t)) •
(4.8)
The non-homogeneous term g {xo(t)) arises due to the parameter mismatch. In the case of identical systems (g — 0), (4.8) has the trivial solution £i(t) = 0. Moreover, due to the assumption that identical systems are synchronized, this solution will be asymptotically stable. Hence, there exists a fundamental matrix solution $(i,io) of ^ ^ 1
=
[Df
(z o ( i )) _ 2C] $ ( i , i 0 ), *(*, to) = / := diag {1,1,1} ,
(4.9)
such that || $(Mo) ||-^ 0 with t ->• oo. In terms of $ the solution of Eq. (4.8) can be expressed as follows fc(i) = $ ( i , t„)£i,o + I *(*, r)g •> to
(I0(T))
~ f
$(t, r)g
(X0(T))
(4.10)
*> to
for initial values £i7o from some neighborhood of £i = 0. (Actually, this is the neighborhood that admits the asymptotic synchronization of the identical systems.) In the case when the identical systems are exponentially synchronized with a rate constant A, i.e., || xo(t) — yo(t) ||~ exp {—At} , A > 0, we have ||$(i,io)||<Mexp{-A(t-i0)}.
(4.H)
Time-Continuous
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143
where M > 0 is an appropriate constant. Equation (4.10) then implies that the transverse perturbations are of the order e: £ || < const-
9ix)
(4.12)
A
(see also [17]). It can be shown that condition (4.11) is fulfilled provided the criterion for synchronization of the identical systems is expressed in terms of the Lyapunov function method with the function || £ ||2 (see [29]). Indeed, in this case we can estimate —7 > sup^i/ max (a;), where fmax.(x) is a maximal eigenvalue of the matrix [Df(x) + DfT(x) - 2 (C + CT)] over the synchronized chaotic attractor A . In the paper by Johnson et al. [17], —7 was chosen as maximal instantaneous transverse Lyapunov exponent for the attractor. Now consider the case eg(x) = go = const, i.e., the mismatch parameter is included in an additive way. For example, in the Rossler system, the parameter b has this property. Problems involving an external constant force acting on one of the oscillators can also be described by Eq. (4.4) with eg(x) = go. Under this assumption, (4.10) turns into
m
/
$(i,r)dT
(4.13)
ffo-
For simplicity, we choose the coupling in the form C = al — diag {a, a, a}. Let us then examine the asymptotic properties of £(£) at large coupling magnitude a. Consider ty(t,to) — e~2Ct to be the fundamental matrix solution of ^ = - 2 C * . The substitution of $ = ( $ - * ) + * into (4.13) gives
««) = £ + A*(«,r)
V(t,T))dr
(4.14)
9a-
.J t0
From Eq. (4.9), the behavior of the difference $ — \l/ is governed by the equation d (* (t, r ) - * (t, r)) = -2C (* (t, r) - * (t, r)) + Df (x0 (t)) $ (*, r ) . (4.15) dt Integration of (4.15) yields $-*:
f
e-2aI(t-T')Df
(a,Q
^
$
(r,) r )
dT,dT
90
(4.16)
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and, hence
m
~ £ ~ [f f e~2aI{t~T')Df (*« (T')) $ (^i T) dT'dT 9o-
(4.17)
We refer to our original publication [29] for the proof of the estimation II ${T',T)
II< M e x p ( - 7 ( r ' - T ) ) ,
with M > 0, and 7 = 2a — m, m > 0. m and M do not depend on a. Assuming the function / to be smooth or to have a bounded derivative, ||D/ (xo ( T ' ) ) | | < I with some positive constant I. Therefore, by integrating Eq. (4.17) we obtain
«')-£
< -^U,P,.
2a(a-iy
(4.18)
Here |-| denotes a vector with modules taken of each component. Expression (4.18) shows that for large a (more precisely, for a » I, see [29]), the asymptotic behavior is given by
a*)
9o_
2a
ho a2
(4.19)
where the constant L does not depend on a or go. The inequality (4.19) is the main result of this section. For any pair of linearly coupled oscillators (4.4) with an additive parameter mismatch or a constant external force g0 acting on one of the oscillators, assuming that the corresponding identical oscillators are exponentially synchronized, one can observe the following asymptotic properties with increasing magnitude of the coupling • there is a parallel shift of the synchronous motion from the synchronization manifold by an amount go/2a; • the amplitude of chaotic bursts around the shifted state is of the order I go/a2, where I is a measure of the maximal numerical value of the Jacobian determinant in the points of the synchronized chaotic state; • for large coupling strength a, the bursts becomes small compared with the shift, and one can numerically observe the phenomenon of generalized synchronization where || x[t) — y(t) |—> const ^ 0. Let us hereafter illustrate the obtained results in the case of two coupled Rossler systems with C = al. Figure 4.22 shows the projection of the synchronized motion in the system of identical oscillators (go = 0) onto the (x\, y\)
Time-Continuous Systems
145
plane for coupling parameter a = 2. As it follows from the previous sections, the system is asymptotically synchronized for this coupling strength. Taking go = (0.1,0,0) and a = 2, numerical calculations demonstrate a behavior of x\{t) — Vi(t) as illustrated in Fig. 4.23. The presence of the shift away from xi — 2/1 = 0 is clearly observed. The dashed line shows the theoretically predicted asymptotic shift by Eq. (4.18). Figure 4.24 shows the same trajectory as a projection onto the (2:1,2/1) plane. The motion now takes place at some distance from the diagonal (which represents the projection of the synchronization manifold). The shifted set is broadened because of nonzero transverse perturbations, again in accordance with Eq. (4.18). With increasing coupling parameter a we can dampen the bursts around the shifted state. Figure 4.25 shows the dependence of x\{t) — y\{t) for three different parameter values. The shift decreases linearly with 1/a while the bursts are damped as 1/a 2 .
xry
Fig 4.22. Synchronous motion of identical coupled Rossler oscillators with a coupling parameter a = 2.
4.6
Fig 4.23. The behavior of &(*) = xi(t) - yi(t) for the parameter mismatch ga = (0.1,0,0) and the coupling a = 2. The dotted line represents our theoretical result (4.19) for the shift of the synchronization manifold.
Influence of Asymmetries in the Coupled System
Let us now consider t h e following a s y m m e t r i c coupling scheme
x=f{x)+pC(d)(y-x), y=f(y) + (2-p)C(d)(x~y),
(4.20)
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Chaotic Synchronization: Applications to Living Systems
'
1 '
'
1
«=2
xry.
ex=6 ^ I
Fig 4.24. (x1,2/1)-projection of the orbit (heavy line) for the parameter mismatch go = (0.1,0,0) and the coupling a = 2.
ct=12 ,
I
Fig 4.25. Behavior of £i(t) for the parameter mismatch g0 = (0.1,0,0) with different values of the coupling parameter a. As a is increased, the random bursts away from synchronization nearly disappear.
where parameter p introduces the asymmetry provided that p ^ 1. As discussed in Chapter 3, this type of asymmetric coupling is of interest when studying the transition from one- to two-cluster dynamics in ensembles of globally coupled, identical chaotic oscillators [51]. Under these conditions, our calculations show that the subcritical pitchfork bifurcation (occurring in the parameter point d = 0.736 of Fig. 4.19) transforms into a transcritical bifurcation. Figure 4.26 schematically displays the perturbed bifurcation vis a vis the original unperturbed one (thin lines). For p = 0.9, the transcritical riddling bifurcation occurs at d = 0.7360, and the associated saddle-node bifurcation at d = 0.7382. In the transcritical riddling bifurcation, a doubly unstable asynchronous period-1 solution passes through the synchronization manifold (from below) and exchanges its transverse stability with the synchronized period-1 saddle, leaving this as a transversely unstable solution. Yet another scenario arises if we perturb the coupled oscillators by introducing a mismatch between their parameters. To investigate this situation let us couple two nonidentical systems in the form
x=fl(x) + C(d)(y-x), y=My) + c(d)(x-y).
(4.21)
In this case, a symmetric period-1 saddle cycle no longer exists. The transverse pitchfork bifurcation scheme is transformed into two isolated cycle branches
Time-Continuous Systems
147
p=0.9 p=1.0
Fig 4.26. Perturbation of a subcritical transverse pitchfork bifurcation by a nonsymmetric coupling. The thin line shows the unperturbed case. The figure illustrates the appearance of a transcritical riddling bifurcation.
with one of them undergoing a saddle-node bifurcation [52]. This is illustrated in Fig. 4.27, where the thin curves represent the transverse pitchfork bifurcation in the fully symmetric case and the dotted curves represent the perturbed diagram in the case of a parameter mismatch. In the fully symmetric case (as shown in Fig. 4.19) we have a synchronous period-1 saddle and two symmetric, asynchronous and doubly unstable period-1 orbits to the left of the bifurcation point. To the right of the bifurcation point we have a transversely unstable synchronous period-1 cycle. To illustrate the effect of a parameter mismatch take fi(x) = f(x,b{) and J2{y) — f{y,b2) with a value of the parameter b
4.7
Transverse Stability of the Equilibrium Point
Application of numerical methods to synchronization problems, e.g., calculation of transverse Lyapunov exponents, investigation of the stability of periodic orbits embedded in the attractor, etc., naturally implies some limitations. In particular, this approach restricts the number of control parameters such that the results can be presented in terms of one- or two-dimensional scans in pa-
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d ^
Systems
Fig 4.27. The perturbed subcritical pitchfork (riddling) bifurcation in the case of nonidentical coupled oscillators. Fully drawn thin lines illustrate the unperturbed case.
PD*
rameter space. For example, using the results of Sec. 4.2, we are not able to predict what happens if instead of the coupling matrix C = diag {d\, 1, d^} we consider C = diag {di, 0,^3}. Similar problems arise with respect to the local behavior in a neighborhood of the bifurcation branches: Are the considered bifurcations subcritical or supercritical, for instance? This section is meant to investigate the extent to which one can replace detailed analysis of the synchronization of a pair of chaotic oscillators by determining the stability properties of a much simpler solution, such as the equilibrium point, which can be treated analytically. Let us fix the parameters of the Rossler system to be b = 2.0 and c = 4.0, with a to be varied. It is assumed that if the value of a is about 0.4, the Rossler system in general has a chaotic attractor. As a simple solution to system (4.1), we consider the synchronous equilibrium point O = (xi,x2,xz,x\,x2,x$) = (u,u) ,xi— 2 — 2^/1 — a/2, x2=xi /a, and 2)3= — x2. Figure 4.3 shows the position of this equilibrium point O in a particular projection. For the assumed parameter values we have x\= 0.22 and x2= — x3= —0.53. Although the point O does not belong to the attractor, one may hope that the transverse stability properties of orbits in the chaotic attractor are similar to those for the equilibrium point. In the following analysis we shall consider the coupling C = diag {di, d2, d^} with three free parameters. Conditions for transverse stability A stability analysis of the point O can be carried out analytically. For this purpose, we again introduce the transverse and longitudinal coordinates (x — y, x+y) and shift the origin to the equilibrium point. We denote the new variables as a vector u =((ui — u2), (u\— u) + (u2— u)) . f (u,/i) is defined as the right hand side of Eq. (4.1) with respect to the new variables and a new coupling
Time-Continuous
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149
parameter fi which we shall later define in terms of d;. The linear stability analysis of the equilibrium point 0 gives a characteristic equation of the form aiA 3 + a2A2 + a3A + a4 = 0,
(4.22)
where the coefficients depend on the parameters a,d\,d2, and d3. We note that, in general, a six-dimensional system gives a characteristic equation of the sixth order. However, for coupled Rbssler systems the linearization matrix is block-diagonal, with blocks corresponding to the transverse and longitudinal coordinates [37]. This implies that the characteristic polynomial becomes a product of two independent parts. We are interested in the characteristic roots that depend on the coupling parameters and correspond to the transverse directions. In this way, we arrive at the third order equation (4.22). Bifurcations occur when the roots of this equation have vanishing real parts, i.e., either A = 0 or A = ±ioj. This happens when one of the following conditions is fulfilled a 4 = 0, or aicii — a2C3 = 0.
(4-23)
In terms of the parameters ai,di,d 2 , and d3, the first condition of Eq. (4.23) takes the form a2di (8 + 7) + d 2 7 - 2ad 3 - 2ad1d2 (8 + 7) + 4a 2 did 3 - 8adid 2 d 3 - a (4 + 7) = 0 (4.24) with 7 = — 4 + 2\/4 — 2a. The second condition gives 2d\ (2a - 8 - 7) - 2d\ (7 + 8) + 4ad^ - 8(d?d3 + d\d2 + d\dz + d\di + d\dx+ d\d2) - 16did 2 d 3 + 2a (7 + 8) (dx + d2 + d3) + ( - - 2a 2 ) (di + d3) + 2 (2a - 2 7 - 17) (d : + d2) + 4 (2a - 7 - 8) (did 2 + d2d3 + did 3 ) + ( 2 a 7 - 2 - 6a2 + 17a - ^ M = 0.
(4.25)
Expressions (4.24) and (4.25) define surfaces in parameter space for the boundaries of the regions of transverse stability for the equilibrium point O. In fact, this equilibrium point is of saddle type. So, when speaking about the transverse stability of O we mean the existence of a one- or two-dimensional stable manifold in the transverse direction. To compare the transverse stability curves, defined by Eqs. (4.24) and (4.25), with the numerically computed
150
Chaotic Synchronization: Applications to Living Systems
—— . —
3
pitchfork Hopf
o
Fig 4.28. The critical values of coupling parameters where transverse bifurcations of the equilibrium point O occur. d,2 = 1.0. Solid lines correspond to pitchfork bifurcations, and dashed lines to Hopf bifurcations.
Supercr.
C ~~" -1
-0.5
O
0.5
1
_l 1.5
1
•s L 2
stability diagram for the chaotic attractor we insert d,2 = 1 and a = 0.42. Figure 4.28 shows the computed stability curves in the (c^efo) parameter plane. As determined by Eq. (4.25), the curve for the transverse pitchfork bifurcation is a hyperbola. The upper right envelope of the stability curves determines the region of transverse stability, like in the case of the chaotic attractor (Fig. 4.6). Its boundary consists of three parts: a Hopf bifurcation curve in the middle (dashed line) and two branches of a pitchfork bifurcation curve (solid lines). Now compare this threshold for the transverse destabilization of the equilibrium point with the destabilization curves for low-periodic orbits embedded in the chaotic attractor as illustrated in Fig. 4.29. Here, the dashed line corresponds to transverse destabilization of the equilibrium point O. As we can see, this analytically obtained curve reproduces the shape of the numerically obtained curves for destabilization of the low-periodic orbits quite well (even though, of course, the equilibrium point cannot undergo a period-doubling). It is also interesting to observe the shape of the transverse stability region in the case G?2 — 0 as shown in Fig. 4.30. This region is also bounded from the right hand side. Taking cfo = 0 (i.e., the coupling matrix becomes C — diag {d\, 0,0}) and increasing d\ from zero, stabilization for relatively small values of d\ is followed by a destabilization for larger values of the coupling parameter d\. This analytical result is in keeping with the numerical results obtained by Pecora et al. [12, 53]. Moreover, we can assert that with the cosidered coupling scheme considered (i.e., with d2 = 0), for any positive fixed ^3, the desynchronization threshold will persist with increasing d\. For cfo < —1.8, transverse stability cannot be achieved for any values of d\.
Time-Continuous Systems
1
1
M
1
'
I
•
151
1
1
d
o -1
Mi
/'
—
/
-
1
-2
S^-^ i
Siibcr. TSupercr. i
- 1
0.0
-0.4
0.4
i i i .
0.8
i
1.2
d, Fig 4.29. Comparison of the boundaries of transverse stability for the synchronous chaotic attractor (heavy line), for some of the main low-periodic orbits embedded in the attractor (thin lines), and for the equilibrium point O (dashed line) for the coupling matrix C = diag {di, 1, d$}.
1
_
d3
1
1
1
1
Transverse stability region
o
WM
9 -• I
-1.5
-1
.
I
i i
.
-0.5
/ i
v
^
0
pitchfork Hopf
Fig 4.30. Same as in Fig. 4.28 but for d2 = 0. Note the finite range for the coupling parameter d\ in which transverse stability can be achieved. This implies that for given values of the coupling parameters d2 and d3, increasing dx can lead to destabilization of the coupling.
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Chaotic Synchronization:
Applications
to Living
Systems
Criticality of the riddling bifurcations It was noticed in Sec. 4.2 that the desynchronization bifurcations (transverse period-doubling or pitchfork) for the low-periodic orbits embedded in the chaotic attractor tended to be supercritical in the upper left side of the bifurcation diagram in Fig. 4.6, while they are subcritical along the bottom. For the equilibrium point O, the type of criticality for the bifurcation can be determined analytically by evaluating the projection onto the one-dimensional subspace spanned by the eigenvector that corresponds to the critical eigenvalue [54]. Define the parameter \i = d%— cfo, where cfo is a critical value of d$ satisfying Eq. (4.25). Let v x be the eigenvector of the Jacobian matrix evaluated in the origin u = 0 with the coupling parameter fixed at the critical value (i = 0. Denote this Jacobian as f u (0). Hence, f u (0)vi = 0. Denote by Wi the corresponding adjoint eigenvector, i.e., f u T (0)wi = 0, chosen so that it satisfies the normalizing condition (vi, wi) = 1, where (•, •) denotes the scalar product. It is known [54] that locally the bifurcation branches of solutions can be represented by the series
f!=£Ulf2U2+;''
(4.26)
//(e) = e/ti + e /i 2 H , where the amplitude e is the projection on the subspace associated with the adjoint eigenvector wi is £=(u,wi). (4.27) By virtue of the symmetry of the system it can be shown that /ii = 0. Therefore, the sign of /*2 in Eq. (4.26) determines whether the bifurcation is subcritical or supercritical. Substituting (4.26) into the equation u = f (u, //) we find, by identification, a set of equations for the unknowns Ui, U2 and \i
= ~;^(fuu(0|vi|U2)'Wl)'
(428)
where U2 is to be determined from the linear system of equations fu(0)u 2 = - ( f u u ( 0 | v 1 | v i ) > w 1 ) ,
(4.29)
(u 2 ,wi) = 0. Here the following notations are used: A'(0) = (fu/x(0)vi, wi) determines the derivative of the critical eigenvalue with respect to /z at the point of bifurcation
Time-Continuous Systems
153
[54], [f uu (0|u|v)]j = dudu njvk, and fu/i(0) is the component-wise derivative of fu with respect to fi evaluated in the point (0,0).
Fig 4.31. Graph of the function (fuu(0|vi|u 2 ), wi) for fixed d2 = 1 and a = 0.42. This curve determines the sub- or supercritical character of the transverse destabilization of the equilibrium point O. The points T\, T2, and Q are defined in the text.
Our interest concerns the sign of H2- Clearly, A'(0) is positive when the stability region is left by increasing /i and negative in the opposite case. Therefore, to know the type of bifurcation the sign of {f uu (0|v 1 |u2), Wi) has to be determined. We omit here an analytical expression for this term due to the involved complexity. Figure 4.31 shows the graph of the function for fixed d^ = 1 and a = 0.42. The function changes sign at three points where the coupling parameter di = (a — 1) / [a (a — 2)] = 0.871 (point Q) or d,=
a ± \Ja(a2 + 3 - 4a) (2a - y/4 - 2a) I [2a (a - 2)] S {-0.035, -0.598} (4.30)
(points T2 and TO. Let us now calculate the sign of /X2- For fixed di = 1 we have sign(/x2) =
- , di >di= {a - 1) / [a {a - 2)] £ 0.871
(4.31)
+, d\
Hence, for d\ > d\ the bifurcated branch exists only for \i < 0, i.e., c^ < d$. This implies a supercritical stability loss in the region to the right of Q. On the other hand, for d\ < d\ a subcritical stability loss occurs along the pitchfork bifurcation curve. The transition between the supercritical and subcritical pitchfork bifurcation is indicated on the solid curve in the bottom of Fig. 4.28. Similar arguments can be applied to other values of aV Returning to the bifurcation diagrams for low-periodic orbits in Fig. 4.6, the above results suggest that as d\ becomes large enough, the riddling bifurcations
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Chaotic Synchronization:
Applications
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Systems
for low-periodic orbits tend to become supercritical, as observed for the equilibrium point O. Our numerical calculations confirm this conclusion. Indeed, a threshold d[ = 1.7 has been found such that for d\ > d[ the riddling pitchfork bifurcation of the synchronous period-1 cycle becomes supercritical.
4.8
Partial Synchronization of Coupled Oscillators
If the coupled system involves more than two coupled chaotic oscillators, a whole new range of additional phenomena can occur, including partial synchronization (or clustering) [40, 41, 42, 55] as well as various forms of wave-like dynamics. A state of partial synchronization is said to occur when the interacting oscillators synchronize with one another in different groups, but there is no synchronization among the groups. Interesting questions in this relation concern the types of partial synchronization that can exist with different coupling schemes. The analysis that we shall present in this section is mainly concerned with partial synchronization phenomena in an a