Topics in nonlinear dynamics with computer algebra

, k = 1,2

which gives: fll = 1 , f = 0, f = 12 13

( 1 . 7.30)

-

R cosO,

f14 = R cosO sin7/J , f15 = R sinO cos7/J f21 = 0, f22 = 1, f23 =

( 1 . 7.31)

f24 = R sinO sin7/J , f = 25

-

-

R sinO,

R cosO cos7/J

Then Lagrange ' s eq uations take the form ( 1 . 7.32)

d 8,Y 8q 1.

8,Y = A f l li 8q 1.

at - - .

+ A 2 f2i

, i = 1 ,2,3,4,5

METHOD OF GENERALIZED SPEEDS

where of course � = T-V. We will write th ese out in order to show how much algebra is involved in the computation. ( 1 . 7.33) for x: m:ie ' = A l ( 1 . 7.34) for y: m y ' = A 2 ( 1 . 7.35) for cp:

[

� m R2 0' sin'¢' + O¢ cos'¢' - Cp' ]

= R [ A I cosO + A 2 sinO]

= R sin'¢' [ A I cosO + A 2 sinO]

- m g R sin'¢' = R cos'¢' [ A I sinO - A 2 cosO] The five eq s . ( 1 . 7.33)-( 1 . 7.37) and the two constraint eq s . ( 1 .7.5) ,(1.7.6) represent seven eq uations in the seven unknowns x,y,CP ,O,,¢, ,A l and A 2 . A strategy for dealing with these is to solve the x and y eq s.(1. 7.33) ,(1.7.34) for A l and A 2 , and substitute these into into the other three eq s . ( 1 . 7.35)-(1. 7.37). Then differentiate the constraint eq s . ( 1 . 7.5) ,(1 .7.6 ) and substitute the resulting expressions for:ie ' and y ' into the three eq uations of motion, thereby eliminating x and y entirely. The resulting eq uations may be solved for Cp' ,0 and '¢' to give: ( 1 . 7.38)

( 1 . 7.39)

23

ME THOD OF GENERALIZED SPEEDS

24 ( 1 . 7.40)

't/J

=

-

i 0 y, cos't/J + iJ2 cos't/J sin't/J + � ft sin't/J

Substitution of eq s.(1. 7.8) into the results of the method of generalized speeds, eq s . ( 1 . 7.23)-( 1 . 7. 25), gives eq uations which are eq uivalent to ( 1 . 7.38)-( 1 . 7.40). 1.8 Computer Algebra Computer algebra is a welcome tool for handling the computations involved in deriving the eq uations of motion in a sufficiently complicated problem. E.g. , in the case of the rolling wheel presented in the previous section, both the method of generalized speeds and Lagrange's eq uations offer enough algebra to justify writing a small computer algebra routine to check the hand computation. Sample MACSYMA programs are presented in Appendix 2 . The process of generating the eq uations of motion b y computer has been automated in several commercially available packages, e.g. AUTOSIM and AUTOLEV (see Referenc es) . Regarding the q uestion of which approach is more efficient , the method of generalized speeds or Lagrange ' s eq uations, we see that once computer algebra is utilized, the dreary part of the derivation is thrust upon the computer. In this case both methods are easier on the user and the q uestion of efficiency has less significance. In order to be sure that the eq uations of motion have been derived cor rectly, a prudent strategy is to derive them by both methods and compare results. 1.9 Comments The method of generalized speeds is a relatively new approach to the problem of deriving the eq uations of motion for a mechanical system. Lagrange ' s eq uations, on the other hand, have been successfully used for more than two hundred years. I have presented them both in this chapter in order for the reader to better evaluate their relative merits.

METHOD OF GENERALIZED SPEEDS

The met hod of generalized speeds has it s ro ot s in a paper by T.R.Kane in 196 1 . Since t hen it has gained an ever-g rowing popularit y. The bible of t he met hod is t he t ext by Kane and Levinson, "Dynamics: Theory and Applicat ion" ( 1985 ) . An ext ensive set of not es has been made available during a one-week summer course t aught for many years at MIT by D .Rosent hal and A.H.von Flot ow. In spit e of t he many advocat es of t he met hod of generalized spee ds, t he met hod has met wit h considerable resist ance amongst dynamicist s. As an example, t ake t he crit icism by E.A.Desloge in t he J.Guidance,Cont rol and Dynamics, vo1. 10, pp. 120-122 ( 1987 ) . Desloge is concerned about t he relat ionship bet ween t he met hod of generalized speeds and t he Gibbs-Appell eq uat ions, concluding t hat "Kane ' s method is simply a part icular met hod of applying t he Gibbs-Appell eq uat ions . " In a lat er issue of t he same Journal, no less t han four lett ers t o t he edit or, respect ively by D.A.Levinson, J.E.Keat , D .E.Rosent hal and A.K.Banerjee, appeared, defending Kane's approach as being superior t o t hat of Gibbs and Appell. Perhaps part of t he resist ance t o welcoming t he met hod of generalized speeds is based on t he delicat e issue of what name t hey are known by. In t he hope of avoiding t he emot ional issues involved in t he cont roversy, I refer t o t he approach as t he met hod of generalized speeds in place of t he more common name of Kane's eq uat ions.

25

26

METHOD OF GENERALIZED SPEEDS

1 . 10 References " AUTOLEV a symbol manipulator for dynamics" , OnLine Dynamics, Inc., 1605 Honfleur Drive, Sunnyvale, CA 94087 " AUTOSIM for Multibody Dynamic Modelling" , Mitchell and Gauthier Associates, Inc. , 200 Baker Ave. , Concord, MA 01742 Banerjee,A.K . , " Comment on 'Relationship Between Kan e ' s Equations and the Gibbs-Appell Equations ''' , J . Guidance,Control and Dyna mics, 10:596-597 (1987) Desloge,E . A . , " Relationship Between Kane ' s Equations and the Gibbs-Appell Eq uations" , J. Guidance,Control and Dynamics, 10:120-122 ( 1987) Goldstein, H . , " Classical Mechanics" , second ed. , Addison-Wesley ( 1980) Kane,T.R. and Levinson,D. A., " Dynamics: Theory and Applications" , McGraw-Hill (1985) Keat ,J.E. , " Comment on ' Relationship Between Kane ' s Equations and the Gibbs-Appell Equations ''' , J . Guidance,Control and Dynamics, 10:594-595 (1987) Levinson,D . A . , " Comment on ' Relationship Between Kane ' s Equations and the Gibbs-Appell Equations ''' , J . Guidance,Control and Dynamics, 10:593 (1987) Rosenthal,D . E . , " Comment on ' Relationship Between Kane ' s Equations and the Gibbs-Appell Equations''' , J . Guidance,Control and Dynamics, 10:595-596 (1987) von Flotow,A.H and Rosenthal,D. , " Multi-Body Dynamics: An Algorithmic Approach Based on Kane ' s Equations" , notes from a short course 19-23 June 1989, Mass.Inst . Technology.

-

METHOD OF GENERALIZED SPEEDS 1.11 Exercises 1. A circular massless disk of radius r rotates wi th respect to an inertial frame E:e1,e2,e3 with constant angular speed w about the vertical axis e3. A pendulum consisting of a particle of mass m attached to a massless rod of length L is pinned to the disk at point P in such a way that the pendulum always remains in the plane

determined by the e3 axis and poi nt P .

_--t---� e

1

Fig.1.5. A pendulum attached to a rotating disk.

v

a. Find the velocity m of the mass m with respect to the inertial frame E. b. Find the acceleration a m of the mass m. c. Obt ain Lagrange's eq uations for this system using 0 as a generalized coordinate. d. Obtain the eq uations of motion by the method of generalized speeds for this system using 0 as a generalized coordinate, and using the following choices for the generalized speed u: a. u

=

0

b. u

=

{, +

w

[f+

sin 0

]

27

..

28

METHOD OF GENERALIZED SPEEDS

e. Determine approximate values for all equilibrium points of the system for w = 2, g = 1, r = 1

L

L

4"'

f. Determine the stability of the equilibria you found in part e. 2. A double plane pendulum consists of two massless rods and two particles arranged as in the Figure:

Fig. 1.6. A double plane pendulum. a. Derive the equations of motion by Lagrange' s equations. b. Derive the equations of motion by Kane' S equations, using

c. The 0 1-0 2 configuration space for this problem is a torus. Use conservation of energy to obtain a region in configuration space which is impossible to be reached by the motion emanating from the initia l cond ition:

For simplicity of computation, take m 1

=

m2

=

L1

=

L2

=

g

=

1.

3. A particle moves i n an inertial frame with axes x,y,z under the influence of no applied forces but subject to the single constraint equation:

METHOD OF GENERALIZED SPEEDS

x-zy+1 = O a. Obtain the equations of motion by the method of generalized speeds. b. Obtain the equations of motion by Lagrange' s equations. c. Comment on the following attempt to avoid using Lagrange multipliers, while remaining in the setup of Lagrange' s equations:

= � (x2 + y2 + z2), w e may use

Since the kinetic energy T for this problem is T m the constraint to eliminate x from the problem by writing T

= 1 m ([z.y -1] 2 + y. 2 + z.2 ) 2'

and then obtain the equations of motion by writing

4. A u niform solid sphere B of mass m and radius R rolls without slipping on a plane horizontal surface which rotates with constant angular speed {1 about a vertical axis through a fixed point P.

p

y

r-----��

x

Fig.1.7. A sphere on a turntable

e2

29

..

30

METHOD OF GENERALIZED SPEEDS

a. How many coordinates do you need to specify the configuration of the sphere? b. Derive the rolling-without-slipping constraint equations. c. Derive the equations of motion by the method of generalized speeds using the generalized speeds

where x,y refer to the position of the center of mass of the sphere relative to an inertial frame E, and where

d. Solve the resulting equations and describe in words the motion of the sphere relative to an inertial frame.

METHOD OF GENERALIZED SPEEDS

Appendix 1: Derivations Related tp the Method of Generalized Speeds Al. I Derivation of the equations for a system of particles The derivation of the method of generalized speeds , like that of Lagrange' s equations, follows from D' Alembert' s Principle, which the reader is assumed to be familiar with (see Goldstein, Chapter 1), and which may be written in the form:

(Al . l . I )

N Or.

l-zxt .

J

1

_

Fj

.

= lat· N ar . .

1

J

mj aj , i

=

I , . . . ,n.

in which Fj represents applied forces only. (Constraint forces are eliminated from (Al . l . I ) by dotting Newton' s equations for each particle with a virtual displacement

=/

and then summing over all particles.) Now since I j I qi ,t), differentiation gives Edr . n ar . � UI: · V qi + (Al . l .2) j 1 i

= at = lat -

-

0(

so that

--1= --1 avo

(Al . l .3 )

Or.

aq 1.

aq 1.

and so ( A l . l . I ) may be written

(Al . I A)

_ . --1 =� �

n avo '\'

J

aq.

1

F. J

n avo

'\' --1.

J

aq.

1

m . a. J J

Now the generalized speeds ui are assumed to be an affine function of the generalized velocities q i :

31

• METHOD OF GENERALIZED SPEEDS

32

=I n

(A I . 1.5)

uk

Rki q i

+ Ck

' k

=

I , . . . ,n.

where Rki is an nil n matrix and ck is an nil 1 column vector. The elements of Rki and ck may depend upon the generalized coordinates qi and time t. We assume that the matrix Rki is nonsingular so that (Al.1.5) may be solved for the q i' s as affine functions of the ui' s.

(Al . 1 .6)

Substituting (Al . 1 . 6) into (Al . 1.4), we obtain Nn (Al . 1 .7)

II V j,k Rki

•

j k

Fj

= II Nn

V

j k

j,k Rki . mj aj

which becomes, upon changing the order of summation, n

(Al . 1.8)

N

I Rki I k

V

j

j,k . Fj

=I

N

n k

Rki

I V j,k . mj aj j

Multiplying (A I . 1.8) by the inverse of the matrix Rki gives the desired form of the equations of motion: N

(Al . 1 .9)

I j

V

j,k .

Fj

=I

N j

V

j,k . mj aj , k

=

I , . . . ,n.

METHOD OF GENERALIZED SPEEDS

A1.2 Derivation of the rigid body equations In order to determine the contribution made by the torque T to the equations of motion ( 1.6. 1 ) , we treat T as a couple, L e. , as two forces Fp and F Q' which are equal in magnitude and opposite in direction, but which act at distinct points P and Q. The contribution of these two forces to the equations of motion will be

(A 1 . 2 . 1)

where we have used the fact that FQ = FP' L e. , that FP and F Q are equal in magn itude and opposite in direction. From kinematics we have that

-

( A1.2.2 )

where rQP is a vector from Q to P. Using ( A1.2.2 ) in ( A1 . 2 . 1 ) , we obtain ( A1.2.3 )

{) au. 1

[ E-B xr-QP ] oFp=au.E-B xr-QPoFp w

-

{)

w

-

1

( A1.2.4 )

where we have used the fact that T = rQP x Fp . This last expression is in the desired form, cf. ( 1 . 6 . 1 ) . To obtain the right-hand side of the equations of motion ( 1.6.2 ) , we utilize Newton' s equations about the center of mass c, in which we omit constraint forces ( since they

33

METHOD OF GENERALIZED SPEEDS

34

w ill eliminated by dotting w ith the partial velocities and summing ) : (A1 .2.5)

-

\' M c = \'L.F . = m -ac , L. j

J

= E-B Ic . a

=

w )( I c + E-B

. E-B w

L

where M c is the sum of the moments about c, amd where the other symbols have been defined previously. In order to relate (A1.2.5) to ( 1 . 6 . 1 ) , we refer all position vectors to the center of mass c. Let 0 be a point fixed in the inertial frame E, and let P be a point On the rigid body. Then j (A1.2.6) where rOP . is a position vector from 0 to P j' and so On. Differentiating (A1.2.6) J with respect to time t in the inerti al frame E, we get (A1.2.7)

-· v = vc J

­ w )( r cp . + E-B J

Taking the partial derivative of (A1.2.7) with respect to ui' we obtain (A1.2.8)

-·· = -v . v ,I C,I J

w·I )( -rCP + E-B j

Substituting (A1.2.8) into ( 1 . 6 . 1 ) , we obtain

METHOD OF GENERALIZED SPEEDS

L

[- . + E-B

-

. '

�

(A1.2.9)

j

v

v

e,I

w· I

e,I

2 -F. + E-w·B ·

. J J

v

-

V

e,I .

•

1

.

e,I

'

J [2

-r e . . p J J

><

j

I

J

[=

which is in the desired form, cf.(1.6.2).

B I C . EQ:

=

-FJ + -T ] = .

2 2 F. + E-B w· · M

w· · m ae + E-B I -

'

-r . F. + E-wBI. . T p e · J

><

35

e

=

+ E-WB

><

=

B I C . Ew

]

36

METHOD OF GENERALIZED SPEEDS

Appe ndix 2: M ACSYM A P rograms for Deriving Equations of M otion A2. 1 Ex ample of section 1. 7 by the method of generalized speeds. The following program, when batched into M ACSYM A, produces eqs . ( 1 . 7.23)-(1 . 7. 25) of the text : 1* rolling wheel by generalized speeds * / / * vector package * / 1* al,a2,a3 orthogonal unit vectors * / 1* define cross product * / cross(u,v):=(matrix ( [al ,a2,a3] , [ratcoef(u,al) ,ratcoef(u,a2),ratcoef(u,a3)] , [ratcoef(v,al),ratcoef(v,a2),ratcoef(v,a3) ]) , determinant(%%)) ; 1* define dot product * / dot (u,v) : =ratcoef(u,al)*ratcoef(v,al)+ ratcoef( u,a2 )*ratcoef( v ,a2)+ ratcoef(u,a3)*ratcoef(v,a3); 1* define el ,e2 ,e3 in terms of al,a2,a3 * / el : al *cos(th)+sin(th)*( cos(ps)*a3-sin(ps)*a2); e2: al *sin( th)-cos( th)*( cos(ps )*a3-sin(ps )*a2); e3: cos(ps )*a2+sin(ps )*a3; 1* set up variables * / depends( [u,x,y,th,ps,ph,al ,a2,a3] ,t); 1* define wb=angular velocity of B in E and wa=angular velocity of A in E * / wb:u[l] *al +u[2]*a2+u[3] *a3; wa:u[l]*al +u[2]*a2+ u[2]*tan(ps )*a3;

METHOD OF GENERALIZED SPEEDS

1* function to differentiate unit vectors * t diffvec: [diff( al ,t )=cross( wa,al) , diff( a2,t )=cross( wa,a2), diff( a3,t )=cross( wa,a3)] ; 1* vc=velocity of center of mass c * 1 vc:cross( wb,r*a2); 1* partial velocities * 1 for i: l thru 3 do ( wb[i] :diff(wb,u[i]) , vc[i] :diff( vc,u[i] )); 1* ac=acceleration of c* 1 ac: diff( vc,t ) ; ac: ac,diffvec; 1* al=angular acceleration of B in E *1 al:diff( wb,t); al:al,diffvec; 1* fc=force of gravity at c *1 fc:-m*g*e3 ; 1* idotal=inertia tensor dotted into angular acceleration *1 idotal:m *rA 214 *( ratcoef( al,al )*al +ratcoef( al,a2)*a2+2*ratcoef( al,a3)*a3) ; 1* idotwb=inertia tensor dotted into angular velocity *1 idotwb:m*rA 21 4 *( ratcoef( wb,al )*al +ratcoef( wb,a2)*a2+2*ratcoef( wb,a3)*a3 ) ; 1* wbxidotwb=angular velocity crossed into idotwb * 1 wbxidotwb:cross( wb,idotwb);

37

38

METHOD OF GENERALIZED SPEEDS

/ * eqs . of motion * / for i : 1 thru 3 do eq[i] : dot( vc[i] ,fc )=dot( vC[i] ,m*ac)+dot( wb[i] ,idotal+wbxidotwb)j / * simplify the eqs. of motion * / solve(makelist( eq[i] ,i, 1 ,3) ,makelist ( diff( u[i] ,t ),i,1,3))j A2. 2 Example of section 1. 7 by the Lagrange ' s equation The following program, when batched into MACSYMA, produces eqs . ( 1 . 7.38)-( 1 . 7.40) of the text : / * rolling wheel by Lagrange ' s equations * / / * set up variables * / depends([x,y ,theta,psi,phi] , t)$ xd:diff(x,t)$ yd:diff(y,t) $ thetad:diff( theta,t) $ pSid:diff(psi, t)$ phid:diff(phi, t)$ /* define w=angular velocity, i=moment of inertia, ke=kineic energy, pe=potential energy, lag=lagrangian * / w:[-psid,thetad*cos(psi) ,thetad*sin(psi)-phid]j i:m *r' 2*m atrix([1/ 4,0,0] , [0,1/ 4,0] ,[0,0,1/ 2])j ke:m/ 2*(xd ' 2 +yd' 2+r' 2*sin(psi) ' 2*psid ' 2)+ 1/ 2*w.i. Wj pe:m *g*r*cos(psi)j lag:ke-pe$

METHOD OF GENERALIZED SPEEDS

1* state constraints * / con I :expand( xd +r*( (-phid +thetad *sin(psi) )*cos( theta) +psid*cos(psi )*sin( theta))) ; con2:expand(yd +r*( (-phid+thetad*sin(psi) )*sin( theta) -psid*cos(psi )*cos( theta))); /* lagrange ' s eqs. with lagrange multipliers laml ,lam2 * / lee q) : =expand( diff( diff(lag,diff( q, t)), t )-diff(1 ag,q)= laml *coeff( conl ,diff( q,t)) + lam2* coeff( con2 ,diff( q, t ))) ; eqx:le(x); eqy:le(y) ; eqtheta:le( theta); eqpsi :le(psi ); eqphi :le(phi) ; / * eliminate laml and lam2 * / solve( [eqx,eqy] , [laml ,lam2]) ; temp I : [eqtheta,eqpsi,eqphi] , %$ /* eliminate x and y * / solve([ conI ,con2] ,[diff(x,t ) ,diff(y ,t)] ) ; diff( %,t); temp I , % ,diff$ solve( % , [diff( theta, t ,2) ,diff(psi, t ,2) ,diff(phi, t ,2)])$ /* the final equations: * / eqs:expand( trigsimp(%));

39

CHAPTER TWO LIE TRANSFORMS

2.1 Introduction Lie transforms is a perturbation method for Hamiltonian systems. The idea of the method is to produce a near-identity canonical transformation which simplifies the Hamiltonian. In contrast to other methods based on near-identity transformations, e.g. averaging, Lie transforms does not perturb the vector field itself, which consists of 2n scalar functions in an n degree of freedom system, but rather just perturbs the Hamiltonian, a single scalar function. We begin with a Hamiltonian system in the form: dp1.

(2. 1 . 1 )

8H

= - 8q at i

where the Hamiltonian H = H( qi ' Pi ' f) . Although the explicit presence of time t is excluded from H, we will be able to handle nonautonomous problems by using extended phase space. We assume that the system corresponding to f = 0 , H = H( qi ' Pi ' 0), is easy to solve. In most cases this will correspond to a system of linear oscillators. We expand H in a power series in

f:

- 40 -

LIE TRANSFORMS

(2.1.2)

H

=

H(q1. ,p1. , f)

=

, HO (q1· ,p1· ) + f H 1 (q1· ,p1· ) + f 2 H 2 (q . ,p . ) + . . . 1 1

Then we posit a near-identity transformation from (q1. ,p1. ) to (Q1. ,P 1. ) :

(2.1.3)

q1.

=

2 Q1. + f F 1I· (QJJ · ,P . ) + f F 2I· ( QJJ · ,P . ) + . . .

p1.

=

p. + f G ( Q ,P . ) + f2 G ( Q ,P . ) + . . . · · 1 2I· JJ 1I· JJ

where the F ki and G ki must be chosen so that the transformation is canonical, i.e. , so that the Hamiltonian form of the equations is preserved. Substituting (2. 1 .3) into (2.1 .2) produces a new Hamiltonian K(Q i ' P i ) : (2. 1 .4)

K(Q 1. ,P1. , f)

=

H(q1. ,p1. , f)

=

fF 11· + · · · , P 1. + fG 11· + · · . , f) H(Q+ 1

The Hamiltonian K (called the Kamiltonian after Goldstein) may itself be expanded in a power series in E: (2.1.5) where K O = HO ' since the transformation is an identity for f = 0 , and where the other K i functions depend on the Hi functions and on the functions Fki and G ki which determine the near-identity transformation. The idea of the method is then to select the near-identity transformation so that the functions Ki become as simple as possible. E.g. a typical goal is to eliminate Q i from Ki ' producing ignorable coordinates.

41

1111 LIE TRANSFORMS

42

2.2 The N ear-Identity Transformation

The heart of Lie transforms is the ingenious manner in which the near-i dentity transformation is generated, i. e. , the scheme by which the functions Fki and G ki are chosen. The transformation (2.1 .3) relating (qi ' Pi ) to (Q i ' Pi ) is a function of f, and this transformation is viewed in Lie transforms as evolving as f increases from zero. From this point of view, f plays a time-like role. The process by which the transformation evolves is itself postulated to be Hamiltonian:

(2.2. 1 ) with the initial condition f

(2.2.2)

=

0 , q1.

=

Q 1. , p1.

=

P1.

In (2.2. 1 ) the function W, which plays the role of the Hamiltonian, is called the generating function, and W = W(qi ' Pi ' f) . Like H and K, W is also expanded in a power series in f: (2.2.3)

W

=

W(q1. ,p1. , f)

=

W(Q+ fF 11· + · · · , P1. + fG 1 1· + · · · , f) 1

N ote that we now have two Hamiltonian processes: Eq. ( 2 . 1 . 1 ) governs the evolution of the original dynamical system in time, for a fixed value of f, and eq . ( 2 . 2 . 1 ) governs the evolution of the near-identity transformation itself, as f varies. Eqs.(2.2.1) and (2.2.2) may be used to generate the transform ation (2. 1 .3) by writing a Taylor series for qi and Pi as functions of f. Here is the Taylor series for qi ' in which we supress the dependence of qi on t for sim plicity:

43

LIE TRANSFORMS

(2.2.4) Using (2.2.1)-(2.2.3) we find that

(2.2.5) where the last step follows because Pi

=

P i when f

=

o.

Thus the near-id entity transformation may be written, to O ( f2 ) :

(2.2.6) Higher order terms may be obtain ed in a similar fashion. It is a remarkable fact that the transformation generated in this fashion is canonical! We do not prove this here, but a proof may be found in the original paper by Deprit. 2.3 The Kamiltonian In order to obtain the transformed Hamiltonian, we must substitute (2.2.6) into eq. (2.1 .2) for H. Each of the terms, H O ,H l ' H 2 , . · . , is a scalar function of qi and Pi ' and so the following calculation applies to each of them. Let f be an arbitrary function of qi and Pi : (2.3 . 1 )

f = f(q1. ,p1. )

=

OW l OW . -f >rrll + . . . ) f(Q 1. + f � P , . . . + ur . 1 u"'c:; 1

1

LIE TRANSFORMS

44

It is convenient to use P oisson bracket notation here, defined by:

(2.3.2)

of og of og [f,g] = Qq. w. - OF. 7K[ 1 1 l l i

l

Then (2.3 . 1 ) can be written (2.3.3) N ow eq. (2 . 1.2) becomes to O ( f2 ): (2.3.4) Identifying the terms in (2.3.4) with those of the Kamiltonian in (2.1.5), we find (2.3.5) N ote that all the K i ' s are functions of Qi and P i since they result from a Taylor series expansion about f = 0, at which qi =Q i and Pi =P j " Eq.(2.3.5) shows that K l may be simplified by choosing W I so that the P oisson bracket [H O ' W 1 ] cancels out those terms in H I which it is desired to remove. It ' s time for an example. 2.4 Example As an introductory example, we take the free vibrations of the Duffing oscillator, (2.4 . 1 )

X. . + X + f X3 = 0

which may be put in Hamiltonian form by taking Px =:ie, giving the Hamiltonian:

LIE TRANSFORMS

45

(2.4.2) In order to simplify the f = 0 system as much as possible before beginning the perturbation process, we use "action-angle" variables p,q: (2.4.3)

x = lIP sin q , Px = lIP cos q

which gives:

H = P + f P2 sm. 4q It will be convenient to trigonometrically reduce H before proceeding: (2.4.4)

(2.4.5) from which (2. 1 .2) gives (2.4.6)

H I ( q,p) = p2 [3 I cos 2q + I cos 4q] g

-

2"

g

N ext we change variables from (q,p) to (Q,P ) using the transformation (2.2.6):

(2.4.7) which by (2. 1.5) and (2.3.5) gives the new Kamiltonian: (2.4.8) where (2.4.9)

K = K(Q,P ) = K O +

f

I

K + O ( f2 )

III LIE TRANSFORMS

46

which gives

(2.4.10) N ow we may choose W 1 in any way at all. The idea is to simplify K 1 as much as possible. One strategy is to choose W 1 to kill the Q-dependent terms in K 1 by setting W 1 = A sin 2Q

(2.4. 1 1 )

+ B sin 4Q

whereupon (2.4. 10) becomes (2.4. 12)

�

so that the choice A = - p 2 and B = the Kamiltonian (2.4. 13)

K=P

k p 2 will make K 1 = � p 2 .

+ g3 f P 2 +

O( f2 )

and from Hamilton ' s equations, (2.4. 14)

dP = - oK (Jt oq

we find that (2.4. 15) where P o and Q O are arbitrary constants of integration.

This produces

LIE TRANSFORMS

Having obtained an approximate solution i n P-Q coordinates, we wish to transform back to the original variable x. From eqs.(2.4. 7) and (2.4. 11) we obtain

(2.4. 16) p = P + f p2 (

� cos 2Q - l cos 4Q) + O( f2 )

N ote that when Q = 71"/ 2, q = 71"/ 2, and thus from (2.4.3), Px = 0 and x = ..J2p . This serves as a definition of the amplitude A (i.e. , x = A when Px = 0), giving: (2.4. 17)

Since x and Px vary like the sine and cosine of q, we see from (2.4. 15) and (2.4.16) that the frequency w of the x-motion is characterized by the nonperiodic (secular) part of q, i.e. (2.4. 18) where we have used (2.4.17). Eq.(2.4. 18) describes the relationship between frequency and amplitude for eq.(2.4. 1). This phenomenon illustrates a basi c .difference between linear and nonlinear systems. 2.5 Higher Order Approximation In order to extend the method to include higher order terms in f, we must obtain versions of the near-identity transformation (2.2.6) and the Kamiltonian (2.3.5) which are valid to higher order in f. Both of thes e results will follow once we have a higher order version of eq. (2.3.3), which expresses the expansion of an arbitrary scalar function f( qi ' Pi ) in a power series in f:

47

LIE TRANSFORMS

48

(2.5 . 1 )

where

(2.5.2)

OW df = � of dqi of dPi = � of OW - of . 7JCj"":" = [f,W] + Of l Oij. Of op. Of l Oij. ""lJ"p. op . . 1 1 1 1 1 1 1

1

g'Z'

r:

' the latter due to where we have used eq. (2.2.1) ,(2.3.2) and the fact that = the Hamitonian nature of W. At f = 0 eqs. (2.5.2) and (2.5.3) become

(2.5.4)

where we have used the ex pansion for W, eq.(2.2.3). Thus (2.5. 1 ) becomes :

Eq. (2.5.5) is the O( f2 ) version of eq.(2.3.3). The near-identity transformation may be obtained from (2.5.5) by taking f=qi and f=p1. : (2.5.6)

LIE TRANSFORMS

and a similar equation for Pi ' In order to obtain expressions for the Kamiltonian terms KO ' K l and K 2 in (2.1.5), we take f= HO ' f= H 1 and f=H2 in (2. 5 . 5 ): (2. 5 . 7 ) HO (qi 'Pi ) = HO( Qi ' Pi ) + f [HO 'W1 ] + 2"1 f2 ( [[HO 'W I] ,W 1] + [HO 'W2] ) + O( f3) (2. 5 . 8 ) (2. 5 . 9 ) Substituting (2. 5 .7)�2. 5 . 9 ) into the expansion for the Hamiltonian eq.(2. 1 . 2 ), we obtain (2. 5 .10) K = H O + ( H 1 + [HO ' W 1] ) + f2 ( H2 + [H l 'W1] +� [[HO 'W1] , W1] + � [HO 'W2] ) + O(f3 ) which gives (2. 5 .11) K O = HO ' K l = H 1 + [HO ' W 1 ] , K2 = , H2 + [HI ' WI] +2"1 [[HO 'W1] ,W 1] + 2"1 [HO 'W2] Eq.(2. 5 .11) is the O( �2 ) version of eq.(2. 3 . 5 ). Still higher order approximations are best obtained by computer algebra. See Appendix 3 where a MACSYMA program is presented which derives expressions for the Kamiltonian to any order in f. f

49

LIE TRANSFORMS

50

2.6 Example We return to the free vibrations of the Duffing oscillator, eq.(2.4. 1 ) , this time working to O( f3 ) . From section 2.4 we have seen that: (2.6.1 ) Using the generating function W I given by: (2.6.2) we were able to obtain: (2.6.3) N ow we use the expression for K 2 given in eq.(2.5. 11) to determine W 2 . This is bes done using computer algebra. Working interactively in MACSYMA, for example, \1 define a Poisson bracket PB as: (2.6.4)

PB(f,g) : = diff(f,q)*diff(g,p )-diff(f,p )*diff(g, q )j

Then after assigning values to H O ,H 1 and W I ' it is a simple matter to compute K 2 from (2.5. 1 1 ) . Inspection of the result shows that W2 will be of the form (cf. (2.4. 1 1) ) : (2.6.5)

W 2 = A sin 2Q + B sin 4Q + C sin 6Q

Evaluating the expression for K 2 with W2 so assigned permits us to solve for A ,B a: C to obtain: (2.6.6)

W2 = p 3

(a sin 2Q - � sin 4Q - Ik sin 6Q)

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which results in (2.6.7) In exactly the same way, W3 and K 3 can be obtained by using eq.{A3.3) in Appendix 3 which gives an analogous expression for K 3 . The result is: (2.6.8)

£

W 3 = p 4 (- sin 2Q +

Ifik sin 4Q + � sin 6Q - � sin 8Q)

(2.6.9) Thus we obtain the Kamiltonian (2.6.10) t re

The frequency w of the resulting motion is given by: (2.6. 1 1 ) I n order to obtain a frequency-amplitude relation, we need t o express P o i n terms of the amplitude A of the x-motion. We use eq. {2.5.6) to write:

(2.6.12) which is easily evaluated in the same MACSYMA session begun earlier. As in section nd

2.4, the amplitude A = lIP when Q = 7r/2, cf.eq.{2.4. 17), which turns out to give: (2.6.13)

A 2 = 2P 0 - 45

which may be inverted to give

f

63 P o2 + '3'2"

2 P 3 + O{ f3 ) o

f

P o as a function of A by setting

51

•

LIE TRANSFORMS

52

P o = U 1 + U 2 f + U3 f2 + O( f3 )

(2.6. 14)

and substituting into (2.6. 13) . Collecting terms and solving for the Ui ' s gives : (2.6. 15) Now we may substitute (2.6. 15) into (2.6. 11) to get the frequency-amplitude relationship: (2.6.16)

_ w-

1

+

3

Sf

21 f2 A4 + 81 f3 A 6 + O( f4 ) A2 - m W48

In order to check this result, we may compare it to the exact solution of eq. (2.4. 1) which is obtainable in terms of elliptic functions. A summary of elliptic functions and their application to simple oscillator problems is given in Appendix 4. In order to apply the results of Appendix 4 to eq.(2.4. 1 ) , we set x = y/{i which transforms (2.4. 1 ) into (2.6 . 1 7)

y. . + y + y 3 = 0

which is of the form of eq.(A4.2 . 1 ) , and which has, from (A4.2.3) and (A4.2.9), the general solution:

(2.6. 18)

The amplitude of the exact solution (2.6.18) in y is a I ' and its period T in t is 4 K(k)/a2 , where K(k) is the complete elliptic integral of the first kind. Thus the amplitude A of the x-motion is a 1 /{i, and its frequency w is 27r/T, where

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( 2.6.19 )

Now eq. ( A4. 1 . 8 ) gives ( 2.6.20 )

a result obtained by using MACSYMA. Substitution of ( 2.6.20 ) into ( 2.6.19 ) gives eq. ( 2 . 6 . 16 ) obtained by Lie transforms. 2.7 Nonautonomous Hamiltonians and Extended Phase Space The discussion so far has required that the Hamitonian be autonomous, i.e. , independent of time t . In this section we use a standard trick to deal with nonautonomous systems. The procedure is related to a well-known scheme for turning a general ( not necessarily Hamiltonian ) nonautonomous system into a larger autonomous system. As an example, consider Mathieu ' s equation: ( 2.7.1 )

XO

+

( 5 + f cos t ) X = 0

This may be written as an autonomous first order system by letting y = x and z = t : x=y ( 2.7.2 )

y = - (5 +

f

cos z ) X

Alt hough Mathieu ' s equation ( 2.7.1 ) can be put in Hamiltonian form with

53

.....�-���" "���-" �"

....

'�'�"

"" �"

�'��'

''''-''�'''�'" '�" " "

54

LIE TRANSFORMS

( 2.7.3 )

the foregoing procedure of eqs. ( 2. 7.2 ) loses the Hamitonian structure of the system. Extended phase space is a scheme by which an n degree of freedom nonautonomous Hamitonian system may be converted to an n+1 degree of freedom autonomous Hamiltonian system. The procedure involves replacing the original Hamiltonian N

( 2.7.4 )

N

That is, time t is taken to be a generalized coordinate qn+1 (just as t was taken to a new phase variable z in ( 2.7.2 )) , and a corresponding momentum, Pn+ 1 ' is added to the original Hamiltonian. Now, besides the usual Hamilton ' S equations for i = l , ... ,n, we have two additional equations associated with the new degree of freedom: N

( 2.7.5 )

oH 1 q. n+1 - 8 P n+1 -

N

oH p. n+ 1 - - 8 q -

oH - - 8q n+1 n+ 1 - .;,=-

The first of ( 2.7.5 ) confirms that qn+ 1 has been identified with t . The second of ( 2.7.5 ) has no particular significance except that it is necessary in order for the system to be Hamiltonian, cf. ( 2.7.2 ) . 2.8 Example As an example, we take Mathieu ' S equation ( 2.7.1 ) with nonautonomous Hamiltonial N

( 2.7.3 ) . The extended phase space Hamiltonian H is: ( 2.8.1 )

55

LIE TRANSFORMS

We begin by replacing x and Px by action-,angle variables, cf.eq. (2.4.3): (2.8.2) which gives

(2.8.3) IV

Writing

H HO + f H I ' we find =

Now we use Lie transforms to O( f 2 ), eq.(2.2.6):

(2.8.5)

qI.

=

Q I.

+

8VVI

f (J"JJ:"" + O ( f2 ) , p1. i

which gives the new Kamiltonian K = K O (2.8.6) (2.8.7)

where we have used the result that

+

=

fK

8VV I + O ( f2 ) P . - f >l7l V ""' i 1

I+

...

, cf.eq.(2.3.5):

56

(2.8.8)

LIE TRANSFORMS

[HO ,W 1 J =

aH o OW l aHo OW l + 0Q1 7lP1 - 0PI 0Q1 OQ2 ov; - 0P2 1JQ2 aH o OW l

aH o OW l

I

In order to simplify K 1 in eq.(2.8.7) as much as possible, we take W in the form: (2.8.9) Substitution of (2.8.9) into (2.8.7) shows that we may obtain K 1 = 0 for the choices : (2.8.10)

PI

B = - ---4{b ( 2 {b 1 )

+

C=-

PI

----

4{b ( 2 {b -1)

Thus we have shown that the Kamiltonian can be reduced to K O (to O ( f 2 ) ) , which, {b t from (2.8.2) gives solutions x = , i.e. all solutions remain bounded as t goes to infinity.

��� +

"

'

i,

This conclusion does not hold, however, if 8 = since in that case there is a vanishing denominator in C in (2.8.10). This situation corresponds to the forcing frequency in Mathieu ' s eq.(2.7.1) being twice the natural frequency of the unforced equation, a situation which is called subharmonic resonance. In order to investigate this resonance, we introduce a detuning coefficient 81 : (2.8. 1 1 ) Redoing the calculation with this value of 8, we find that (2.8.12)

LIE TRANSFORMS

Thls time we take W I in the form: (2.8. 14) where (2.8.15) whlch gives (2.8.16) Note that the vanishing denominator in (2.8. 10) led to a nonremovable term in K 1 . The result of our Lie transform analysis in this resonant case is the system: (2.8. 1 7) Note that although Q 1 and Q 2 are not ignorable coordinates in (2.8.17), they only appear in the combination 2Q C Q 2 . This permits us to perform a canonical transformation which will essentially reduce the problem to an autonomous one degree of freedom system. We go from ( Q i ' P i ) to ( Xi ' Yi ) via the linear canonical transformation ( see Exercise 5):

whlch produces the Kamiltonian:

57

LIE TRANSFORMS

58

(2.8.19) In order to understand the implications of this Kamiltonian on the dynamics of Mathieu ' s eq.(2.7.1), we proceed in two ways. First we work in (Xi ' Yi ) variables, then we return to the original (x,px ) variables. Since X 2 is absent from K in (2.8. 19), it follows from Hamilton ' S eqs. that Y2 is a constant of the motion (to O ( f 2 )). Of course K is also a constant of the motion in this autonomous two degree of freedom system. Thus it follows that K - Y2 is a constant of the motion, so that : (2.8.20) Now since Y 1 =

� P I = � P I + O( f) , and since PI �O from (2.8.2), Y1 �O to O ( f). Also,

since (2.8 .20) is 2 11"-periodic in X l ' the (X I ' Y 1 ) phase space is topologically 3ICIR + . The integral curves of (2.8.20) may thus be displayed in a half�trip, 0 � X l < 2 11", Y1 �O. There are two cases, depending upon whether or not 1 81 1

>�.

See Fig.2. 1 .

o

Fig.2.1. Schematic depiction of approximate first integral (2.8.20).

LIE TRANSFORMS

In both cases, Y 1 = 0 is an integral curve which corresponds to the origin x = Px = 0 in eq. ( 2 . 7 . 1 ) . In the case that I 0I l > , all motions which begin close to Y 1 = 0

�

remain close to it for all time. However, if I 0I l < , then there exist motions which begi n near Y 1 = 0, but which eventually achieve unbounded values for Y 1 . Thus the

� �

origin is stable if I 0I l > , and unstable if I 0I l < . Since from ( 2.8. 1 1 ) , 0= 01 f+O ( f2 ) , there is a region of instability in the 8-f plane ( an Arnold tongue ) which emanates from the point o= on the ° axis. The transition between stable and

}r

�

�

unstable corresponds to I 0I l = , and so the transition curves have the approximate equation:

�

( 2.8.21 )

1 /4 Fig.2 . 2 . Tongue of instability ( 2.8.21 ) in Mathieu ' s eq. ( S=stable, U=unstable) . See Fig.2.2. Further analysis of Mathieu ' s equation shows that there are an infinite number of such tongues of instability, and that they emanate from those points on 2 the 8-axis for which ° = ' n = 1 ,2,3, . . . . In addition there is a single transition curve which goes through the origin ° = f = O. See Stoker for a lucid treatment of Mathieu ' s equation.

r

59

LIE TR ANSFORMS

60

understandi ng of both Mathieu ' s equation and Lie x,px ) variables. ns fo rm the first integral (2.8.20) back to original ( trans £,orms , we t ra . s of O( f ) , we see that Ne g1 ec t mg term

In order to supplemen t

u O r

( 2 .8.22) where we have used eqs.(2.8. 18),(2.8.5) and (2.8.2). Similarly, (2.8.23) so that (2.8.24)

Thus the term Y I cos X l in (2.8.20) can be written to O ( f): (2.8.25) Then (2.8.20) becomes: (2.8.26) For a given value of the constant on the right hand side of (2.8.26), this equation can be viewed as an approximate invariant surface in the three dimensional space with variables x , Px and t. In order to see things more easily, we think in terms of a Poincare map corresponding to surface of section at t=O (mod 27r) . See Fig.2.3.

61

LIE TRANSFORMS

t

To

o "'------

x

Fig.2.3. The flow of the phase fluid takes a point a to a point Ta, producing a Poincare map T. Setting t =O in ( 2.8.26 ) gives an approximation for the invariant curves in the Poincare map:

] [

[

]

1 1 1 61 Px2 + :{ x2 - 2 Px2 - g x2 = constant

( 2.8.27 )

which may be rearranged to give:

[

]

[

]

1 1 Px2 61 - 2 + :{1 x2 61 + 2 = constant

( 2.8.28 )

�

This represents a family of ellipses surrounding the origin in the x-Px plane, if 61 and 61 + have the same sign, i.e. , if 1 61 1 , in which case the origin is stable. On

�

>�

the other hand, eq. ( 2.8.28 ) represents a family of hyperbolas if the signs of 61 - and

�

61 +

� are opposite, corresponding to instability of t1).e origin. See Fig.2.4.

LIE TRANSFORMS

62

1 /4

�----"---

o

Fig.2.4. Invariant curves ( 2.8.28 ) in the Poincare map of Fig.2.3 displayed on the {j-f parameter plane, together with the transition curves ( 2.8.21 ) . Cf.Fig.2.2. 2.9 A Two Degree of Freedom System As a final example of Lie transforms, we take the Henon-Heiles system: ( 2.9.1 )

H

==

1 2

� Px

[

1 2 1 2 1 2 2 1 3 + � Py + � x + � y + f X Y - 3" y

]

This may be thought of as a system of two identical linear oscillators with order f nonlinear coupling. Since the uncoupled oscillators have the same frequency, we expect there to be nonremovable terms in the perturbation solution due to the 1 : 1 resonance. Before using Lie transforms , we transform to action-angle variables for the f problem:

( 2.9.2 )

x

=

v'2Pl sin q 1 '

y

==

v2P2 sin q2 '

\

=

0

63

LIE TRANSFORMS

which gives

from which we obtain: (2.9.4 ) (2.9.5) 1 - 3"

1 . . 2 [3 sm q sm 3 q2 P3/ 4 4 2 2

1]

Now we use Lie transforms to O ( (2 ) , eq.(2.2.6): (2.9.6) which gives the new Kamiltonian K = K O + ( K l +

.

.

.

, cf.eq.(2.3.5):

(2.9.7) (2.9.8)

It turns out that we may choose W 1 so that K l = O. We find:

LIE TRANSFORMS

64

where

(2.9.10)

]

[

J2 p 3/2-2P p 1/2 A-2 ' 2 1 2 -

C-

-

_

J2 p 1/2 2 P1 2 '

Thus there are no resonance terms in the solution so far. This motivates us to go to the next order in f. We try to choose W2 so as to simplify K 2 as much as possible. From (2.5. 1 1 ) we have the following formula for K 2 : (2.9 . 1 1 ) By using the relation K 1 form:

=

H 1 + [H O ' W 1 ] , eq.(2.9. 1 1 ) may b e written i n the alternate

(2.9.12) Now since H2 and K 1 are zero, K 2 becomes: (2.9.13) Now however we have a tough computation before us, that of finding [H 1 , W 1 ] ' where H 1 is given by (2.9.5) and W 1 by (2.9.9),(2.9.10). It ' s time to use computer algebra. We enter expressions for both H 1 and W 1 into our computer algebra environment , and form the Poisson bracket [H 1 ,W 1 ] . The result, after trig reduction, is:

r

65

LIE TRANSFORMS

(2.9. 14)

Substituting (2.9. 14) into (2.9.13), we see that we can choose W2 to eliminate all the trig terms in K 2 except for the cos 2(Q 2-Q 1 ) term, which is the expected nonremovable term associated with the 1 : 1 resonance in the original system ( 2 . 9 . 1 ) .

� � i

� �

W e also d o not remove the non-trig terms i n K 2 , i.e. - P + P 1 P 2 - P , since these would require that W 2 contain terms which are linear in Q 1 and Q 2 ' which leads to a generating function W = W 1 + f W2 + which is not uniformly valid as t goes to infinity. .

.

.

Thus an appropriate choice for W2 gives K 2 as: (2.9.15) Since K = K O + f 2 K 2 + O( f3 ) , we note that Q 1 and Q 2 only appear in K in the form of Q 2-Q 1 ' Thus we may essentially reduce the problem to a single degree of freedom system by transforming from (Qi ' Pi ) to (Xi ' Yi ) via the linear canonical transformation (cf. Exercise 5): (2.9.16) Thus we obtain:

(2.9 . 1 7)

LIE

66

TRANSFORMS

, it follows from Hamilto n ' s eqs. that Y2 is a Since X 2 is abs ent from K in (2.9.17) 3 constant of the motion (to O( ( )). Of course K is also a constant of the motion in this autonomous two degree of freedom system. Thus it follows that K - Y2 is a constant of the motion, so that K 2 is a first integral to O ( (3 ). In order to utilize this result , we transform K 2 in (2.9. 15) back to the original variables (x,y,px ,Py ) ' First note that by (2.9.6), the first integral may, to lowest order in t , be written in terms of qi and Pi : (2.9.18) From (2.9.2), we obtain (2.9.19) It remains to trigonometrically expand cos 2(q2-q 1 ) in (2.9. 18) and use (2.9.2) to eliminate (qi ,Pi ) in favor of (x,y,px ,Py ) ' Here we again use MACSYMA, with the result : (2.9.20)

Now the question is, how to view this ? The situation is difficult to picture because

1R4 .

the phase space is Since the original Hamiltonian (2.9 . 1 ) is conserved, H = h represents a 3-dimensional manifold which a given motion stays on for all time. The approximate first integral (2.9.20) , also a 3-dimensional manifold, intersects H=h in a 2-dimensional manifold, an invariant torus. As the constant in (2.9 .20) is varied, for a given value of the energy h, the approximate first integral foliates the energy surface H=h, giving rise to a family of invariant tori. In order to see an invariant torus, we may intersect it with a hyperplane, say x=O . The intersection will be 1-dimensional manifold, i.e. a curve, which may be viewed by projecting it onto the

LIE TRANSFORMS

y-p y plane. This projection may be effected by setting x=O in H=h, and solving for

�

p in terms of y and P y ' From (2.9.1) this gives: (2.9.21 )

px2 = 2h _ py2 _ y2 +

a,) f y3

(section at x=O)

Now the equation of the foliating invariant tori may be obtained by first setting x=O in (2.9.20), which gives: (2.9.22) and then eliminating px from (2.9 .22) by using (2.9.21). The result is: (2.9.23)

5 (p 2 + y2 ) 2 + (10 P 2 - 18 Y2 ) (2h - py2 - y 2 + "If2 f Y 3 ) ,) Y Y

This family of curves is displayed in Fig.2.5 for f = 0 . 1 and h = 1 . The region displayed is -2 < y < 2, -2 < Py < 2.

67

68

LIE TRANSFORMS

Fig.2.5. Invariant tori (2.9.23) obtained by Lie transforms. In order to check this computation, we can compare the results achieved by Lie transforms with those of numerical integration. We must numerically integrate Hamilton ' S differential equations based on the Hamiltonian (2.9. 1 ) , and obtain a Poincare map corresponding to a surface of section x=O with Px >O. This entails choosing initial conditions: x=O, y and py arbitrary, and px >0 chosen as in (2.9.21). Then we numerically integrate forward in time until x changes sign from <0 to > 0 (in which case we will have pierced x=O with Px=x>O). We use linear interpolation to estimate the time at which x=O (between two steps of numerical integration), and we use this time to interpolate corresponding values for y and P y ' which we plot. The results are shown in Fig.2.6, where a variety of initial conditions were chosen. In Fig.2.6, f = 0 . 1 , h = 1 and -2 < y < 2, -2 < Py < 2.

LIE TRANSFORMS

Fig.2.6. Poincare map obtained by numerical integration. A comparison of Figs.2.5 and 2.6 shows that although the results derived by Lie transforms bears a resemblance to the results of numerical integration, the separated region near the Py axis in Fig.2.6 is absent in Fig.2.5. Better agreement can be obtained by reducing the value of f, or by extending the Lie transform analysis to a higher order approximation.

69

70

LIE TRANSFORMS

2 . 1 0 References

Abramowitz,M. and Stegun,LA. , "Handbook of Mathematical Functions" , Dover (1965) Arnold,V.L , "Mathematical Methods of Classical Mechanics " , Springer ( 1978) Byrd,P . and Friedman,M., "Handbook of Elliptic Integrals for Engineers and Physicists" , Springer ( 1954) Coppola,V. T. and Rand,R.H. , " Computer Algebra Implementation of Lie Transforms for Hamiltonian Systems: Application to the Nonlinear Stability of L 4 " , Z.angew.Math.Mech. ( ZAMM ) , 69:275-284 (1989) Deprit ,A., " Canonical Transformations Depending on a Parameter" ,Celestial Mechanics, 1 : 1 -31 ( 1969) Goldstein,H. , " Classical Mechanics " , second ed. , Addison-Wesley ( 1980) Len,J. and Rand,R.H., "Lie Transforms Applied to a Nonlinear Parametric Excitation Problem " , Int .J.Nonlinear Mechanics, 23:297-313 ( 1988) Lichtenberg,A.J. and Lieberman,M.A., "Regular and Stochastic Motion " , Springer ( 1983) Misner,C.W., Thorne,K.S. and Wheeler,J.A. , " Gravitation" , Freeman ( 1973) Month,L. and Rand,R.H . , "An Application of the Poincare Map to the Stability of Nonlinear Normal Modes" , J.Applied Mechanics, 47:645-651 (1980) Rand,R.H. and Armbruster,D . , "Perturbation Methods, Bifurcation Theory and Computer Algebra" , Springer ( 1 987)

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Spivak,M. " A Comprehensive Introduction to Differential Geometry " , Publish or Perish ( 1970) Stoker,J. J . , "Nonlinear Vibrations in Mechanical and Electrical Systems" , Interscience ( 1 950) Vakakis,A.F. and Rand,R.H., "Normal Modes and Global Dynamics of a Two-Degree-of-Freedom Nonlinear System", Int.J.Nonlinear Mechanics, 27:861-888 ( 1992)

71

LIE TR ANSFORMS

72 2. 1 1 Exercises

�

2 1. In e qs.(2.4. 10),(2.4. 1 1 ) , we chose W I so that K l simplified to p . If we had 2 added a term to W I of the form p Q, then K l could have been made to vanish. Show that the resulting frequency-amplitude relation, eq. (2.4. 18) , would be unchanged to O( f2 ) by this choice of W I .

i

2. Appendix 5 describes the use of differential forms (wedge products) to test whether a given transformation is canonical. Use this test to show that the transformation generated by Lie transforms is canonical to O( f 3 ) (cf. eqs.(2.5.6) and (2.6. 12)):

3. Use Lie transforms to obtain the frequency-amplitude relation for the nonlinear quadratic oscillator:

a. Follow the procedure in sections 2.4 and 2.6 (which applied to x" + x + x3 and work to O( f3 ) , cf.eq.(2.6. 16).

=

0),

b. Check your result by comparing with the exact elliptic function solution. (Cf.eqs .(2.6. 17)-(2.6.20) and see Appendix 4.) 4a. In Appendix 5 it is shown that the transformation (2.4.3) to action angle variables is canonical. Show that the apparently equivalent transformation: x

=

.p;p cos q , Px

=

.p;p sin q

73

LIE TRANSFORMS

is not canonical. b. In Appendix 4 it is shown that the assumption: x = a 1 cn(u,k), produces the general solution to the Duffing oscillator: x ' + x + x3 = 0

Show that the apparently equivalent assumption: x = a 1 sn(u,k), will not produce the general solution to the Duffing oscillator. 5. Show that the transformation (2.8. 18) is canonical in two ways: a. Use the differential forms condition (A5.4.2). b. Find a classical generating function F( Q1. , Y1. ) which produces the transformation (a point transformation) via eqs.(A5.5.4) and (A5.5.5) 6. Use Lie transforms to investigate the nonlinear Mathieu ' S equation: x' +

(8 +

f. cos t) x + f. ex x3 = 0 ,

8=i+

2 81 f. + 0 ( f. )

Neglect terms of O( f.2 ) and assume ex > o. a. Follow the treatment in section 2.7 for the linear Mathieu equation, and show that there is now an additional term in eq.(2.8.20). b. Sketch the phase portraits for this equation which correspond to Fig.2. 1 .

LIE TRA NSFORMS

74

1 £ 1 1 1 Hint : There are three cases: 51 < - 2" ' - 2" < 51 < 2" , and u l > '2" ' For each case look for equilibria. Linearize about each equilibrium to determine its classification ( center or saddle) .

7. For the nonlinear nonautonomous system ( Len and Rand ) : •

•

x +

2 If

x + £ X3 cos t

=

0,

£«

1

a. Use Lie transforms to obtain a near-identity transformation and approximate � Kamiltonian, neglecting terms of O( £2 ) . By inspection of the near4dentity

transformation, find all values of w> o which are resonant.

b. Use Lie transforms to determine the ( nonlinear ) stability of the equilibrium at the Neglect terms of O ( £2 ) . Confirm your stability origin, x = x = 0 , for w =

�

.

conclusion by numerically integrating the original differential equation. c. Find all resonant values of w>O, neglecting terms of O( f 3 ) .

8. An autonomous two degree of freedom Hamiltonian system consists of two unit masses constrained to a line and restrained by three nonlinear springs ( Month and Rand, Vakakis and Rand ) , see Fig.2 . 7.

Fig.2.7. A nonlinear Hamiltonian system. In Fig.2.7, the forces in the springs, from left to right, are: x + £ k x3 , f ( x_y ) 3 , and

LIE TRANSFORMS

y+ £k

i.

The Hamiltonian is: H = 2"1 (px2 + Py 2 + x2 + Y 2 ) + 4£ [k x4 + k y4 + (x - y) 4J

a. Using Lie transforms in action-angle variables Pi ' qi for the £=0 system, perform a near-identity transformation to variables P i ,Q i and obtain an approximate expression for the simplified transformed Kamiltonian K(P I. ,QI. ). Neglect terms of 2 O(( ). b. Following the argument used on the Henon-Heiles system i n section 2 . 9 , obtain an approximate analytic expression in the original (x,y,px ,p y ) variables for the invariant curves in the Poincare map for the cut x=O, Px >O. c. Setting k=5, £=0 . 1 and h= 1 , plot on the y-Py plane the invariant curves you obtained in part b. d. Numerically integrate the equations of motion for this system and obtain a numerical Poincare map, to be compared with your result in part c.

75

LIE TRANSFORMS

76 Appendix

3: MACSYMA Program for Deriving the Lie-Transformed Kamiltonian

The following short MACSYMA program generates expressions for K i ' where i goes from ° to n: 1* program to derive kamiltonian * / n:read("truncation order ? " ) ; d(f) : =sum( e Ai*(f. w[i + 1 ] ) ,i,O,n)+diff( f,e)$ dd[i] : =d( dd[i-l] ) $ dd[O]:f$ for i: l thru n do dd[i]$ fseries :sum( ev( dd[i] ,e=O)*eAi/i ! ,i ,O,n)$ sum( (k[i]-ev( fseries ,f=h[i]) ) *e Ai ,i,O,n)$ taylor(�,e,O,n)$ solve( makelist(part(�,i) ,i, 1 ,n + 1 ) ,makelist(k[i] ,i,O,n)) $ expand(�); Here is a sample of the output, for n = 3: ( (K o

K 2

=

H

1

•

W

=

H , K 0 1 H 0

1

+

=

H +

0

H 2

•

•

W 2

W

1

•

+

W

<2 >

+

1

•

W

1 +

H 0

1 +

•

W

1

•

1

2

W 2

---------

6

+

W 2

-------

•

I

W 2

•

2

H

2

1

-------

<2>

---------

H

+

1

H 0

1

2

----------

3

W

W

•

---------

H K 3

•

H 0

=

+

H , 2 H o

+

H )] 3

•

W

<3 >

H o

1

---------

+

•

W 3

-------

3

LIE TRANSFORMS

The notation used here is that a dot represents a Poisson bracket. For example, (A3 . 1 ) Exponents are interpreted a s multiple dot products, for example, , (A3.2) Thus the foregoing output checks with eqs.(2.5.11), and gives in addition: (A3 .3) K 3 = [H 2 ,W 1 ] + +

� [[H 1 ,Wl] 'W1] + � [Hl 'W2] + � [[[HO 'Wl] 'W l ] 'W 1]

i [H O ' W3 ] + i [[HO ' W2] ' W 1] + � [[HO 'W l ] ' W2] + H 3

As Lichtenberg and Lieberman point out (p. 127), the inverse Lie transformation is obtained by inverting the order of all nested Poisson brackets and replacing Wi by ­ Wi ' Thus the expressions we obtain here differ from those given in Coppola and Rand or in Rand and Armbruster, as we use the inverse of the transformation which was used in those references. Naturally it makes no difference whether we use a transformation or its inverse, since both are canonical.

77

LIE

78

TRANSFORMS

\

Appendix 4: Elliptic Function Solutions to Two Simple Oscillator Problems A4. 1 Elliptic Functions In this Appendix we collect together some facts about elliptic functions in order to apply them to the solution of some simple oscillator problems. In particular this will permit us to check the approximate solutions obtained by Lie transforms. An excellent reference on elliptic functions is Byrd and Friedman. Just as the general solution to the linear oscillator XO + x = 0

(A4 . 1 . 1 )

can be expressed in terms of the trigonometric functions sin and cos , the general solution to the nonlinear oscillators (A4 . 1 .2)

x + x + x3 = 0 0

0

and

x + x + x2 = 0 0

0

can be expressed in terms of the elliptic functions sn, cn and dn. The sn function may be thought of as the elliptic version of the trigonometric function sin, while cn may be thought of as corresponding to cos. This association is based on the fact that sn and sin are odd functions, while cn and cos are even. Moreover, the identity (A4.1.3) reminds us of the comparable trig identity. In contrast to sin and cos, the three elliptic functions sn,cn and dn each depend on two variables , (A4 . 1 .4)

sn = sn{u,k), cn = cn{u,k), dn = dn{u,k)

where u is called the argument and k is called the modulus. {Note that in contrast to

--------

LIE TRANSFORMS

Byrd and Friedman, the book by Abramowitz and Stegun uses m = k 2 instead of k. MACSYMA also uses m instead of k, the MACSYMA notation for K(m) being ELLIPTK(M). ) The elliptic function sn reduces to sin when k=O, and cn reduces to cos when k=O. There is no trigonometric counterpart to dn, which reduces to unity when k=O. The formulas for the derivatives of sn and cn remind us of their trigonometric counterparts: (A4 . 1 .5)

{) au sn

= cn dn,

{) au cn

= - sn dn

To complete the trio of elliptic functions, we have the additional formulas ( A4 . 1 .6) The period of sn and cn in their argument u is 4 K, where K(k) is the complete elliptic integral of the first kind, whereas dn has period 2 K. As k goes from zero to unity, K(k) goes monotonically from 7r/2 to infinity. In the limit as k approaches unity, the elliptic functions take on the following simple limiting values: (A4 . 1 . 7)

sn(u, l) = tanh u, cn(u, l ) = sech u, dn(u, l) = sech u

When comparing exact elliptic integral solutions to eqs.(A4. 1.2) with approximate solutions obtained by perturbation theory, the following expansion for K(k) is useful (Byrd and Friedman, p.296, § 900.00):

A4. 2 The oscillator :ic: ' t x t x3 = 0 One way to obtain the exact elliptic function solution to (A4.2.1)

x. . t x t x3 = 0

79

LIE TRANSFORMS

80

is to use conservation of energy in the form:

(A4.2.2)

x· 2 + "2 x2 + "4 x4 = h or dt =

"2

[

dx 1/2 2h - x2 - '2"1 x4

__

--"___--,,...,.,..

]

where the last equation may be written as an elliptic integral and then inverted to obtain x as an elliptic function of t. Instead of such an approach, I think it is more instructive to look for a solution to (A4.2.1) in the form: (A4.2.3)

x = a 1 cn(u,k),

Since ( A4.2 . 1 ) is a second order o.d.e. , its general solution will possess two arbitrary constants. Since it is an autonomous o.d.e. , one of the arbitrary constants will be the phase b. Of the other three constants, a 1 ,a2 and k, only one is independent. In order to obtain the relations between these three, we substitute (A4.2.3) into (A4.2.1) and use the identities given in section A4. 1. We begin by taking the derivative of (A4.2.3) with repspect to t : (A4.2.4) where cn=cn(u,k), sn=sn(u,k) and dn=dn(u,k), and where primes represent differentiation with respect to the argument u. Differentiating (A4.2.4), we obtain

Using the identities (A4 . 1.3) and (A4. 1.6), this becomes (A4.2.6)

x. . = - a1 a22 cn ( 1 - 2 k2 + 2 k 2 cn2 )

Substituting (A4.2.6) into (A4.2. 1) and equating to zero the coefficients of cn and of

LIE TRANSFORMS

cn3 gives two equations relating a 1 ,a2 and k: (A4.2.7) (A4.2.8) These may be solved for a2 and k in terms of a 1 as follows:

(A4.2.9)

Eq.(A4.2.3) together with (A4.2.9) is the exact solution to (A4. 2.1). A4. 3 The oscillator x · ± x ± x2 = 0 This time we look for a solution to (A4.3.1)

x ± x ± x2 = 0 .

•

in the form (A4.3.2) which gives (A4.3.3) and

x = aO ± a 1 sn 2 (u,k),

81

"

LIE TRANSFORMS

82

(A4.3.4)

x. . = 2 a l a22 ( sn cn dn + sn cn dn + sn cn dn , ) I

I

Substituting (A4.3.4) into (A4.3. l) and equating to zero the coefficients of cn O ,cn2 and cn4 gives three equations relating aO ,a l ,a2 and k which may be solved for aO ,a l and � to yield: (A4.3.5) Eq.(A4.3.2} together with (A4.3.5) is the exact solution to ( A4.3. l).

\

83

LIE TRANSFORMS

Appendix 5: Differential Forms A5. I Introduction Differential forms (also known as exterior calculus) is a relatively recent addition to the tools available to the applied mathematician. Differential forms offers an elegant and efficient notation for certain kinds of calculations. In particular, it offers a convenient test for a transformation to be canonical. Although there are several excellent references on the subject (Spivak, Misner et al. , Arnold), it has still not yet entered the mainstream of courses and texts on applied math. Thus I am including some general background material as well as a section on the application to canonical transformations . We begin with a preliminary section which deals with functions on the x-y plane. A5. 2 Differential Forms in 1R2 . In discussing differential forms on the x-y plane, there are three objects we shall be concerned with: O-forms, which are just scalar functions of x and y, f(x,y), I-forms, which are differentials of the form f(x,y) dx + g(x,y) dy, and 2-forms, which are elements of area, written h(x,y) dx A dy. Here the symbol A is called the wedge product and dx A dy is read "dx wedge dy" . The wedge product, like the more familiar cross product of vector calculus, is anti-commutative: (A5.2. I )

dx A

dy = - dy A dx

and

dx A

dx = 0

Given two differential forms, call them a and {3, the exterior product a A {3 obeys the usual rules of algebra. For example, if a and {3 are general I-forms defined by

84

LIE TRANSFORMS

then (A5.2.3)

where (A5.2 . 1 ) has been used. Given a differential form w, we may associate with it its exterior derivative dw. This is obtained by taking, for each term of the differential form w, the usual differential of the scalar multiplier, and wedging the result into the rest of the term. For example, if w = f( x,y ) dx + g (x,y ) dy, then: dw = d (f dx + g dy ) = df A dx + dg A dy

(A5.2.4) =

[M

dx

+

U dy]

A dx

+

[� dX + � dy]

A

dy

Note that Green ' s theorem of the plane, (A5 . 2.5)

P M dx + N dy = II [� - �] dx

A

dy

can be written in the compact form: (A5.2.6) where S represents a region of the x-y plane, and where OS represents its boundary ( a

LIE TRANSFORMS

closed curve in the plane). As an elementary example of where the notation of differential forms can make the mathematics itself more transparent, take the question, which arises in freshman calculus courses, of changing variables from rectangular to polar coordinates. The associated formula is usually written:

II f dx dy = II f r dr d O

(A5.2.7)

The derivation of this formula usually involves a diagram showing that the element of area is r dr d O in polar coordinates. The intelligent freshman student often asks why one can ' t obtain this result by simply multiplying together the differentials dx and dy: (A5.2.8)

dx = d(r cos O) = dr cos O -r sinO dO ,

dy = dr sin O +r cos O d O

This natural move i s completely correct if (A5.2.7) i s written i n the form (A5.2.9)

II f

dx A

dy =

II f r dr

A

dO

since (A5.2. 10) dx A d y = [dr cos O -r sinO dO] A [dr sinO + r cos O d O] = r d r A d O A5. 3 Differential Forms i n 1R3 . Let x 1 ,x2 and x3 be rectangular coordinates on 1R3 . Then the differential forms which we can encounter are: O-forms: a scalar function F(xi ) I -forms: f dx 1 + f dx2 + f dx3 2 3 1

85

LIE TRANSFORMS

86

I I 2-forms: g i dx2 A dx3 + g 2 dx3 A dx + g3 dx A dx2

l where fi ' gi ,h are scalar functions of x ,x2 ,x3 . Note that no 4-forms or higher order forms are possible in 1R3 , since these require at least one repeated differential, which gives zero by virtue of eq. (A5.2.1). The integral theorem (A5.2.6) now gives both Stokes ' theorem, (A5 . 3 . 1 ) and the divergence theorem

Hw·

(A5.3.2)

n dA =

HJ V

•

w dV

In order to obtain (A5.3.1) from (A5.2.6) , take (A5 . 3.3) (which can be written 11 dr in vector notation), whereupon •

(A5.3.4)

(which is V

x

11

•

n

dA in vector notation).

In order to obtain (A5.3.2) from (A5.2.6), take

S7

LIE TRANSFORMS

(A5.3.5) (which corresponds to Vi Ii dA in vector notation) , in which case .

(A5.3.6) (which is V

dw =

•

Vi

[Ow Ow Ow ] 8x

1 !+

8x

2 2+

8x

1 3 2 3 3 dx A dx A dx

dV in vector notation) .

The discussion can easily be generalized to IRn , in which case the exterior product of a p-form and a q-form is a (p+q)-form. The exterior derivative of a p-form is a (p+ l)-form. In IRn , eq.(A5.2.6) continues to be valid and is a cornucopia of diverse high dimensional integral theorems. The three-dimensional identities (A5 .3.7)

V

x

VF

=

0

and

V · V

x

U

=

0

can be generalized to higher dimensions via Poincare ' s lemma, which states that the second exterior derivative of any (sufficiently differentiable) differential form is zero: (A5.3.S)

d(dw) = 0

The first of (A5.3.7) results from (A5.3.S) when w is taken equal to a scalar function F:

(A5.3.9)

where we have used the fact that

�Fx.l

�J F }

. The second of (A5.3.7) 8x corresponds to (A5.3.S) with w chosen as the l-form in (A5.3.3). ax 8

=

ax

.

LIE TRANSFORMS

88

A5.4 Criteria for Canonical Transformation Given an n degree of freedom Hamiltonian system with variables (qi ' Pi ) ' we are interested in making a transformation of coordinates from (q1. ,p1. ) to (Q 1. ,P1. ) , such that the Hamitonian structure of the problem is preserved. A transformation which produces a Hamiltonian system in (Q i ' Pi ) variables with the same Hamiltonian . ,P . ) ,p . (Q . ,P . )) is called a canonical transformation. H( q1. ( QJ J 1 JJ There are a number of equivalent answers to the question, "Is a given transformation canonical?" Goldstein (in Chapter 9) discusses tests for canonicalness based on a) symplectic matrices , b) Poisson brackets, and c) Lagrange brackets. My experience is that the test based on differential forms, to be described next, while equivalent to the others , is easier to remember and is easier to use. The phase space is (locally) 1R2n . We shall be concerned with 2-forms in this 2n-dimensional space. One begins with the 2-form: n

\'l

(A5.4. 1)

i=l

dq1. A dp1.

n 8q . 8q . . + � dP . and dp . by and uses the transformation to replace dq1. by l\' � dQ 1 J ur · J v,",, ; J j=l J n 8p . 8p . dQj + dPj . After collecting terms, the resulting 2-form may, or may J J j= l n not , be of the form dQ i A dPi . The transformation is canonical if and only if: i=l

l oct

or!

L

n

(A5.4.2)

n

dqI. A dpI. = \'I dQ 1. A dP1. i= l i=l

\'l

89

LIE TRANSFORMS

As an example, consider the transformation (2.4.3) from physical variables (x,px) to action-angle variables (q,p) : x

(A5.4.3)

=

lIP sin q ,

Px

=

lIP cos q

We find (A5 .4.4) dx = lIP cos q dq + _1_ sin q dp , lIP and (A5 .4.5) dx A dpx =

dpx

=

-

lIP sin q dq + _1_ cos q dp lIP

[lIP cos q dq + � sin q dP] [- lIP sin q dq + � cos q dP]

=

A

cos 2 q dq A dp - sin 2 q dp A dq

=

dq A dp

The transformation (A5.4.3) is therefore canonical. A5.5 Classical Generating Functions A classical scheme for producing a canonical transformation from (qi ' Pi ) to ( Q i ' P i ) involves choosing an arbitrary generating function F( qi ' P i ) ' and then forming the 2n algebraic equations: (A5.5.1)

p1.

=

of oa: '

qi

Q 1.

=

of

"!ITr

ur · 1

This use of an arbitrary function to satisfy a complicated condition is reminiscent of stress functions in linear elasticity and stream functions in hydrodynamics. We refer to F as a classical generating function, in contrast to the Lie generating functions W of eqs.(2.2 . 1 )-(2.2.3). Note that F depends on the old qi ' s and the new Pi ' s , whereas W depends on the old qi ' s and the old Pi ' s.

LIE TRANSFORMS

90

Differential forms can be used to show transformation in an efficient manner.

that this scheme leads to a canonical First compute the exterior derivative of

F(qi ' Pi ) : (A5.5.2)

dF

=

l OF

. 8F . dP . ur 1 1 uq 1. dq1 + "!lTl

==-

=

l p . dq. + Q . dP . 1

1

1

Next use Poincare ' s lemma (A5.3.8), which states that d(dF) (A5.5.3)

d(dF)

from which it follows that is canonical by (A5A.2).

=

l dq

� dp . A dq . + dQ . A dP . 1 1 1 1 l

i A dPi

=

=

=

1

0:

0

l dQi A dPi ' and hence the transformation

As an example, consider so-ealled point transformations ( Goldstein p. 386), for which (A5 .5A) Then eqs.(A5 . 5 . 1 ) become:

(A5.5.5) The nature of the transformation is that the Qi ' s depend only on the qi ' s , and not on the Pi ' s.

CHAPTER THREE THE METHOD OF AVERAGING

3 . 1 Introduction The method of averaging is a perturbation method which simplifies the form of a system of differential equations containing a small parameter E. The method involves a change of coordinates via a near-identity transformation, such that the given differential equations become simplified in the new coordinates. (Some general references are Sanders and Verhulst, Guckenheimer and Holmes, Rand and Armbruster. ) Perturbation methods can be classified as regular or singular. Regular perturbations involves expanding the solution in a power series in E. Singular perturbation methods cover a vast diversity of schemes, all of which involve the appearance of E in the solution in some way other than just a power series. The method of averaging is a singular perturbation method. In order to motivate it, we consider an elementary example for which regular perturbations fails. Let us consider the dynamics of a linear oscillator with small damping: (3. 1 . 1 )

x" + X = - E X

Although the exact solution of this problem is easily obtained, let ' s not refer to it just yet. Instead, let ' s look for a solution by regular perturbations, i .e. in the form of a

- 91 -

92

power series in

A VERA GING f:

(3. 1.2) Substituting (3. 1 .2) into (3.1.1) and collecting terms gives a sequence of equations, the first two of which are: (3. 1.3) (3.1 .4) The general solution of (3. 1 .3) is (3.1.5) whereupon (3.1 .4) becomes (3.1.6) which has the particular solution: (3. 1.7) Note the appearance of terms in x l which grow linearly in t. These terms are called resonance or secular terms. Assembling the solution (3. 1 .2) using (3. 1 .5) and (3. 1 . 7) gives (3.1.8) Now it is clear on physical grounds that the solution of the damped oscillator (3. 1 . 1 ) must approach zero a s t goes t o infinity. The regular perturbation solution (3.1.8), however, blows up as t goes to infinity. Thus regular perturbations fails to obtain

r

A VERA GING

93

even the correct qualitative behavior in the large t limit ! Let us compare the perturbation solution (3. 1 .8) with the exact solution to (3. 1 . 1 ) , which may b e written:

(3.1.9)

1 Et s i n 1 _ 1 f2 t x=e 4' cos � 1 - 2'

Expanding (3. 1 .9) in a power series in

[ ]

f,

and neglecting terms of O ( f2 ) gives:

(3. 1 . 10) Thus the regular perturbation solution (3.1.8) agrees perfectly with the power series expansion of the exact solution (3. 1 . 10). Nevertheless the perturbation solution is unacceptable because it fails to characterize the true nature of the behavior of (3. 1 . 1 ) . The trouble is that we have sought the solution t o ( 3 . 1 . 1 ) in the form of a power series in f, which is inappropriate for present purposes. Note that the exact solution (3. 1 . 9) can be written in the form: (3. 1 . 1 1 )

x = a( t )

[���] VJ(t )

This observation leads us to the starting point for the method of averaging. 3.2 The Method of Averaging To begin with, we generalize the example (3. 1 . 1 ) to (3.2. 1 )

x · + x = f F(x ,x)

The explicit appearance of t is omitted from (3.2 . 1 ) for simplicity of presentation; nonautonomous systems will be discussed later on. Motivated by the discussion in

F

94

A VERA GING

the previous section, and by eq.(3 . 1 . 1 1 ) in particular, we seek solutions to (3.2.1) in the form: x

(3.2.2)

=

a(t) cos rp(t),

x = - a(t) sin rp(t)

Substitution of (3.2.2) into (3.2 . 1 ) will yield equations on a(t) and rp(t). Note that this procedure is identical to the method of variation of parameters, which is widely used to obtain particular solutions to nonhomogeneous linear differential equations. Differentiating the first of (3.2.2), and requiring that the result be in the form of the second of (3.2.2), gives: a

(3.2.3)

cos rp - a Cp sin rp = - a sin rp

Differentiating the second of (3.2.2) and substituting into (3.2 . 1 ) gives (3.2.4 )

-a

sin rp - a Cp cos rp = - a cos rp + f F( a cos rp, - a sin rp)

Multiplying (3.2.3) by cos rp, and (3.2.4) by - sin rp, then adding, gives: (3.2.5)

a

=

- f sin rp F(a cos rp, - a sin rp)

Similarly, multiplying (3.2.3) by - sin rp, and (3.2.4) by - cos rp, then adding, gives: (3.2.6) Eqs.(3.2.5) and (3.2.6) on a and Cp are exact. Now we introduce the asymptotic approximation involved in the method of averaging by positing a near-identity transformation from (a, rp) to (a,�):

r

95

A VERA GING

(3.2.7)

where the generating functions w I and w2 are free to be chosen by us as desired. We substitute (3.2 .7) into (3.2.5) and (3.2.6) to obtain eqs. on a and cpo Differentiating the first of (3.2.7),

(3.2.8)

a

[

o

=a+

f

Ow l

-

8a:

0

a+

Ow l

-

81ji

oJ

cp

+ O( f 2 ) = a + 0

f

Ow l

-

81ji

+ O( f 2 )

where we have used a = O ( f ) and cp = 1 + O( f ) , which follow from (3.2.5)-(3.2.7). Similarly, o

� = cp +

(3.2.9)

f

Ow 2

-

81ji

+ O( f2 )

Thus the eqs. on a and cp become: o

(3.2.10)

a=

(3.2 . 1 1 )

. cp = 1 +

f

[

f

-

[

Ow 1 81ji

- . - sin cp F( a cos cp, - a sm Iji)

J

+ O( f2 )

Ow - _2 - � F(a cos cp, - a sin Iji) a 81ji

J

+ O ( f2 )

where we have used cp = cp + O( f) , so that sin cp = sin cp + O ( f ) , etc. Now the idea is to choose w I and w2 so as to simplify the transformed eqs.(3.2. 10),(3.2.11). In order to illustrate how this is done, we return to example (3. 1 . 1 ) of the damped linear oscillator.

A VERA GING

96

3.3 Example When eq.(3. 1 . 1 ) is cast in the general form (3.2.1), we find F(x,:ic) = - :ic

(3.3. 1 ) so that

F(a cos
(3.3.2)

whereupon eq. (3.2.10) becomes:

(3.3.2)

= f

[ Owl -

-]

a {)� - a '2" + '2" cos 2
Now we choose w I so as to kill the trig term

+ 0 ( f2 )

� cos 2� in (3.3.2):

-

(3.3.3)

wI =

l sin 2� + K 1 (a)

where K 1 (a) is an arbitrary function of a. Note that we do not kill the �independent term,

- � , in (3.3.2), since this would require that w I contain a term

which is linear in �. Since � -+ W as t -+ w , such a term would cause the near-identity transformation (3.2.7) to become unbounded in the large t limit. No matter how small f were to be chosen, the appearance of such a term in w I would produce in the

Q

97

A VERA GING

near-identity transformation (3.2.7) an t Cj; term which would become arbitrarily large, i.e. , the series (3.2.7) would cease to be uniformly valid in the large t limit.

With the choice (3.3.3) for w I ' the eq.(3.3.2) on a becomes: (3.3.4) In a similar way, the choice for w2 as (3.3.5)

w2

=

1 cos 2cp + K 2 ( a)

4"

-

where K 2 (a) is an arbitrary function of a, produces from (3.2. 11) the following eq. on cp :

(3.3.6) Solving (3.3.4) and (3.3.6), we obtain to O ( t2 ):

(3.3.7) where the arbitrary constants c 1 and c2 will, in general, depend on t. Now since a = a + O( t) and cp = Cj; + O( t), the resulting expression (3.2.2) for x can be written: (3.3.8)

x = a cos cp = [a + OC t)] cos [Cj; + OC t)]

pi

98

A VERA GING

This expression agrees with the exact solution (3.1.9) to O{ f), which may be written:

(3.3.9) 3.4 Second Order Averaging The asymptotic approximation (3.2.7) may be extended to include terms of higher order in f. E.g. , to O( f 3 ) we can write:

(3.4. 1 )

where w 1 and w 2 are chosen as i n first order averaging, and where the generating functions v 1 and v 2 are to be chosen so as to simplify the O( f 2 ) terms in the and equations.

i

�

Let us illustrate the procedure by continuing the treatment of the previous example, eq. ( 3 . 1 . 1 ) . If we select w 1 and w2 as in eqs.(3.3.3) and (3.3.5), and substitute the near-identity transformation (3.4. 1) into the exact eqs . {3.2.5) and (3.2.6), with F as in (3.3. 1 ) , we will obtain O( f 3 ) versions of eqs. {3.3.4) ,{3.3.6): a f2 a. = - f � +

(3.4.2)

[

8V - ffqJ1

1

+ ?11 +

O( f 3 )

Finding the ??? terms is straightforward but algebraically complicated, a perfect job for computer algebra. Once these terms are known, v 1 and v 2 can be chosen to kill the trig terms, as was done for w 1 and w2 in the previous iteration.

-

A VERA GING

99

Here are some comments on how you could use computer algebra to handle this problem. After setting up the exact eqs.(3.2.5),(3.2.6) on a and Cp, and the transformation (3.4. 1) with w I and w2 defined as in (3.3.3) and (3.3.5), the key steps are as follows: 1 . Substitute the transformation (3.4. 1) into the exact eqs. (3.2.5),(3.2.6). 2. Solve the result for a and q; . 3. Taylor expand the result in f, about f = 0, up to and including terms of O( f 2 ) . 4. Trigonometrically reduce the result. This will produce a result which i s equivalent to eqs . (3.4.2). 5 . Isolate the O ( f2 ) terms and integrate them with respect to q;. 6. Solve the result for v I and v2 ' and remove any terms from v I or v 2 that are linear in q; (in order for the transformation (3.4.1) to be uniformly valid in the large t limit) . 7. Substitute the expressions for v I and v2 into the results of step 4. This is a check on the calculations in steps 5,6. The resulting eqs. on a and q; should have no trig terms . 8 . Finally, substitute the expressions for w 1 ,w2 ,v 1 and v2 into the transformation (3.4. 1 ) . The expression for x

=

a cos cp can then be written in terms of a and q;.

The equations on a and q; which result from step 7 are called the averaged equations, or the slow flow equations. If the averaged equations are able to be solved in closed form, then an explicit expression for x as a function of t can be obtained by using the result of step 8 . In the case that the averaged equations cannot be solved in closed form, it may still be possible to obtain closed form expressions for the equilibria of these equations . These equilibria represent approximations for periodic motions x(t ) i n the original equations. A n expression for such a periodic motion may b e obtained by substituting the expression for an equilibrium of the averaged equations into the result of step 8 .

A VERA GING

100

Returning to the example (3. 1 . 1 ) , there is in Appendix 6 a MA CSYMA program which accomplishes the foregoing steps, producing the following result s. With w I and w 2 chosen as in (3.3.3) and (3.3.5), and with v I and v2 chosen as: (3.4.3)

vI

=

k a cos 4� + i K 1 sin 2� + � K2 a cos 2� + K 3 v2

(3.4.4)

=

. 1 1 sm · 4tp - 2" K sm 2 tp + K - 32 2 4

where K 3 and K 4 are arbitrary functions of a, the averaged eqs.(3.4.2) become: .:.a = - f a 1 2 3 2" + 2" f K ( a ) + 0 ( f ) 1 -

(3.4.5)

. � = 1 - B"1 f2 + O( f3 ) Note that the form of the a equation in (3.4.5) depends on the choice of the arbitrary function K 1 (a). The obvious choice for K 1 is zero, since this simplifies the a equation by eliminating the f2 term. With K 1 = 0, the general solution to (3.4.5) becomes:

(3.4.6) where the arbitrary constants c 1 and c2 may depend on f . In order to connect the constants c 1 and c2 to initial conditions on x and X, we must use x = a cos tp and

x = - a sin tp, together with the near-identity transformation (3.4. 1 ) , and the expressions derived for the wi and vi . We expand c 1 and c2 in power series in

(3.4.7)

c1.

=

c·1 O +

f

c·1 1 + f2 c·1 2 + O( f3 ) ,

i

=

1,2

f:

A VERA GING

The constants cij may be found by plugging the given initial conditions on x and x into the derived second order averaging solution, expanding in f, and collecting terms . Appendix 6 contains a MACSYMA program which continues the calculation which produced eqs.(3.4.3)-(3.4.5) and obtains for the initial conditions x(O)= I , x(O)=O the following coefficients c1. : (3.4.8)

c1 = 1 +

[7

'64 -

]

K 3 f2 + O( f3 ) ,

The form of the solution will obviously depend on our choice of the arbitrary coeffcients Ki . The question of how to best pick these coefficients is still an open research topic (see work by P .Kahn and his associates. ) Two schemes which suggest themselves are to choose K 1 =0 to simplify (3.4.5), and to choose: i) K 2 =-1/4, K 3 =7/64, K4 =0 to simplify the expressions (3.4.8) , or ii) K 2 =0 , K 3 =0, K 4 =0 to simplify the expressions for wi ' vi ' in eqs.(3.3.3) , (3.3.5),(3.4.3),(3.4.4) , which simplifies the near-identity transformation (3.4. 1 ) . Each o f these schemes i s evaluated i n Appendix 6. Scheme i) leads t o the following expression for x( t ) :

where Tp = ( 1

2

- .g-) t.

Scheme ii) leads to the following expression for x(t) :

101

.

...

-���-�

102

( 3.4. 10 )

A VERA GING

[

- '2"f t 7 f2 x( t ) = e 1 + '64

] [1 + 1

4"

•

f

. 1 E2 cos 4tp - + 0 ( f3 ) tp + '64 sm 2-

]

•

2 f ) where tp = ( 1 - '8 t - 4"f . These perturbation results should be compared to the exact solution (3.3.9), which for the same initial conditions x(O)=l, :ic(0)=0 be�omes:

�

[

- t xexact (t) = e cos wt +

(3.4. 1 1 ) where w = �1 1 - 14" f2

b sin wt

]

•

For small values of f, there is excellent numerical agreement between the approximate solutions based on either scheme, and the exact solution. See the following Table: scheme

i)

scheme

ii)

exact

x

x

x

x

x

. x

0.1

. 555

-.801

.555

-.801

. 555

-.801

1.0

.657

-.570

.638

-.521

.660

-.534

Table of exact and approximate values of x( t) for t =1 . Note that although the close numerical agreement shows that the method of averaging works well for small values of f, the forms of the approximate solutions

A VERA GING

103

( 3.4.9 ) and ( 3.4.10 ) are not identical to the form of the exact solution ( 3.4. 1 1 ) . It is clear that extending the method to higher order terms will not change the form of the solution. This is an important feature of the method of averaging, and of perturbation methods in general: They produce a solution which is asymptotic to the exact solution, i.e. , which differs from the exact solution by a quantity which is O( En ) , but they do not in general produce the exact solution itself.

3.5 Limit Cycles and van der Pol ' s Equation In this section we shall use the method of averaging to investigate the dynamics of van der Pol ' s equation for small E: ( 3.5.1

..

)

( 2) x.

x + x = E 1-x

Numerical integration of ( 3.5.1 ) shows that, besides the rest solution x = x = 0 , it contains a unique periodic orbit which attracts all other motions. When viewed in the x-x plane, this orbit looks like a closed curve, which, for small enough values of E, is nearly a circle of radius See Fig.3. 1 , which displays the region -3 < x < 3 ,

2.

-3

<

x

<

3.

104

A VER A GING .

X

Fig.3 . 1 . The limit cycle L in van der Pol ' s eq. for

f

= 0.1.

A motion which i s periodic, and which i s the only periodic motion i n its neighborhood, is called a limit cycle. The closed orbits in the simple harmonic oscillator, x · + x = 0, are not limit cycles since they fill the x-x plane. Physical examples of limit cycles include the bowing of a violin string and the aeroelastic flutter of a venetian blind (or of the Tacoma-Narrows bridge disaster). In order to approximate the dynamics associated with the limit cycle in van der Pol ' s eq.(3.5.1) for small f, we follow the procedure for the method of averaging as given in sections 3 . 2 and 3.4, and set : (3.5.2)

x = a cos cp ,

x = - a sin cp

A VERA GING

105

which gives, upon substitution into ( 3.5.1 ) (cf. eqs. ( 3.2.5 ) , ( 3.2.6 )) the following exact eqs . : ( 3.5.3 )

a

= - f F sin cp ,

f cp = l - -a F cos cp •

where F = ( 1-x2 ) :ic = - ( 1 - a2 cos 2 cp) a sin cp . We shall use third order averaging, which requires a near-identity transformation of the form ( cf. eqs. ( 3.4.l )) : ( 3.5.4 )

4 -) + f 3 u ( -a, a = a + f w I ( -a, � cp, + f2 v I ( -a,cp l cp) + 0 ( f )

where the generating functions wi ' vi and ui are to be chosen so as to simplify the transformed differential equations on a and cpo Appendix 7 contains a MACSYMA program to accomplish third order averaging on van der Pol ' s eq. It is similar to the second order averaging program given in Appendix 6, except that the process is continued to third order terms in which u l and u2 are chosen to kill all trig terms of O( f 3 ) . As a time-saving short--eut, note that for the final round of averaging, we may avoid solving for the corresponding generating functions. E.g. , for third order averaging, the functions u l and u 2 need not actually be computed, but rather may be taken into account by simply removing the O( f3 ) periodic terms. If we let K l ,K 2 ,K 3 and K4 respectively represent the arbitrary functions of a which are associated with w l ' w2 ,v l and v 2 ( cf. eqs. ( 3.3.3 ) , ( 3.3.5 ) , ( 3.4.3 ) and ( 3.4.4 )) , then the program in Appendix 7 results in the following averaged eqs . :

jb

106

A VERA GING

K 1 (4 - 3 -2a ) -a. = f B"-a ( 4 - -2a ) + f2 8

(3. 5 . 5 )

(3. 5 . 6 ) cp = 1 + f2 [-B"1 + 3 -2a - 11 -4]a + f3 K 1 -a [B"3 - 11 -2a ] + O{ f4) As expected, the form of the averaged equations depends upon the choice of the arbitrary functions K {a). E.g., a judicious choice of the Ki can make these equations easier to handle: if wei take K 1 = 0 and K3 = 43 -a?6144312a2a+5 +8192576 (3. 5 . 7 ) then the averaged eqs. { 3. 5 . 5 ) and (3. 5 . 6 ) become: (3. 5 . 8 ) .:.a = f aB" (4 - -2a ) + ( f4) , The limit cycle in van der Pol 's eq.{3. 5 .1) corresponds to the non-zero equilibrium of the i eq.{3. 5 . 8 ), namely a = 2 + O{ f3). That is, in the phase plane corresponding to the transformed a,� coordinates, the limit cycle is a circle of radius 2. Of course in order to obtain the shape of the limit cycle in the original x-x phase plane, we would have to use the near-identity transformation (3. 5.4), which, for small f, would give a slightly non-circular shape, cf. the numerical integration of Fig. 3 .1. From this point of view, the near-identity transformation may be seen as geometrically stretching the x-x phase plane so that the limit cycle is deformed into a circle to O{ f3). .:.

16

m

64

a3 -==--=--=-=::...=-----!---=...:....::.... -=_

0

Although eq.{3. 5 . 5 ) on a, for general values of K 1 and K3 , would be too difficult to solve in closed form, an analytical solution for a{t) can be obtained for the simplified

107

A VERA GING

eq.(3. 5 . 8 ) on a. Separating variables and using partial fractions gives: ft a(t) = 2 e (3. 5 . 9 ) e ft - 1 + a(0)4 2 As t a(t) 2, showing that the limit cycle is asymptotically stable. For large t, eq.(3. 5 . 9 ) becomes ] + ... a(t) 2 + e-ft [1 - � (3. 5 .10) a(O) showing that the approach to the limit cycle goes like e-ft . Note that for t a(t) -1 0 for a(0)<2, i. e ., motions starting inside the limit cycle approach the origin asymptotically as time is run backwards, cf. Fig. 3 .1. However, for a(0» 2, a(t) escapes to infinity when the denominator of (3. 5 . 9 ) vanishes, i. e ., for t = .!.f in[ 1--4/a(0)2] < Thus motions starting outside the limit cycle escape to infinity in finite time as t is run backwards! This escape to infinity in finite time is not surprising when compared to a simple example like (3. 5 .11) ar = x2 which has the general solution x(t) = X(OJ 'rr---'--l­ (3. 5 .12) -t "2

-l ID ,

-I

�

-I -m ,

o.

dx

and which sends a motion starting at x(O) when t = 0 out to infinity as t l /x(O ). -I

A VERA GING 108 3. 6 Hopf Bifurcations One of the features of computer algebra which is absent from conventional languages like FORTRAN, BASIC, PASCAL or C, is that computations may be conducted in which parameters are left in unevaluated form. This permits us to search for special relationships between the parameters, i.e., bifurcation theory. E.g., in a many-dimensional parameter space, we may seek a relationship between the parameters (geometrically, a codimension one surface) which separates systems which possess a limit cycle from those which do not, a scenario known as a Hopf bifurcation. In order to use averaging to compute the condition 'for a Hopf bifurcation, we consider the following generalization of van der Pol s equation (3. 5 .1): z. . + z c z. + a1 z2 + � z z. + a3 z·2 (3. 6 .1) + f31 z 3 + f32 z 2·z + f33 z z·2 + f3 z·3 where c, a f3 are parameters. In order to use averaging, we introduce a small parameteri ' intoi (3. 6 .1) by the scaling z x which gives: =

4

f

(3. 6 . 2 )

= f

x. . + x c x. + [ a1 x2 + � x x. + a3 x.2] + 2 [f31 x3 + f32 x2·x + f33 x x·2 + f3 x.3] f

=

4

f

:i

There remains the question of how to scale the coefficient c of the linear term. Let us expand c in a power series in f:

In order to perturb off of the simple harmonic oscillator, we must take O . Next consider c 1 . As we shall see, the O( ) quadratic terms make no contribution to the Co

f

=

r

A VERA GING

109

f) averaged equations, their first effect being felt at O( f2 ). Thus if c1 were not zero, the only contribution to the averaged equations at O( f) would be from the c x term, which of course does not support a limit cycle. Physically, the damping would be too strong relative to the nonlinearities for a limit cycle to exist. Thus we scale the coefficient c to be O( f2 ), and we set c = f2 J.£ : (3. 6 . 3 ) x. . + x = f [ 0:1 x2 + 0:2 x x. + 0:3 x.2] fJ fJ fJ fJ f« 1 + f 2 [J.£ x. + 1 x3 + 2 x2·x + 3 x x·2 + 4 x.3] , As in the case of van der Pol's equation (3. 5 .1), eq.(3. 6 . 3 ) may exhibit a periodic limit cycle solution for small f. We shall be interested in understanding how such a periodic solution comes to be born as the parameters are varied. We will use second-order averaging on eq.(3. 6 .3). As in (3. 2 . 2 ), we set (3. 6 . 4 ) x = a cos cp , x = - a sin cp and apply a near-identity transformation (3.4.1) to the exact eqs.(3. 2 . 5 ),(3. 2 . 6 ). The computation may be carried out by using a version of the MACSYMA programs presented in Appendices 6 and 7. The only essential modification is to specify the right-hand side of (3. 6 . 3 ) by defining the variable f to be: a1*xA2 + a2*x*xd + a3*xdA2 (3. 6 . 5 ) + e * (mu*xd + b1*xA3 + b2*xA2*xd + b3*x*xdA2 + b4*xdA3) This results in the averaged eqs.: O(

f

(3. 6 . 6 )

...

110

A VERA GING

Note that there is no O( f) contribution to the averaged eqs. ( 3. 6 . 6) , ( 3.6.7) . Such a contribution could only come from the O( f) terms in ( 3. 6 . 3) , which are the quadratic terms. But from ( 3. 6 . 4) and ( 3. 6 . 5) these quadratic terms produce trig terms in f which are of the form cos 2rp, sin 2rp or constant in rp. When such terms enter the a and ip eqs. ( 3. 2 . 5) , ( 3. 2 . 6) , they become multiplied by sin rp or cos rp, and produce trig terms with frequency 3 or 1 in rp, all of which average to zero. This is why the quadratic terms in ( 3. 6 . 3) produce no contribution at O( f) . A limit cycle solution to eq. ( 3. 6 . 3) will correspond to a non-zero equilibrium solution to ( 3. 6 . 6) , i. e. to a real root a of the equation: ( 3. 6 . 8 ) Let us define the quantity S to be: ( 3. 6 . 9 ) Then the value of a on the limit cycle is (

3. 6 .10 ) Thus a limit cycle is predicted to occur if S 0 and if S has the opposite sign of There are two cases, depending upon the sign of S. If S 0, then a limit cycle is predicted to exist only for 0, in which case the origin is an unstable equilibrium, and the limit cycle is stable. This is called the supercritical case, see Fig. 3 . 2 . Imagine to be quasistatically increased from a negative value to a positive value. As passes through zero, two things happen: i ) the origin changes from a stable to unstable equilibrium, and ii) a stable limit cycle is born, emerging from the origin #

J1. >

J1.

J1..

<

J1.

an

111

A VERA GING

with an amplitude which grows like fP,. Physically this corresponds to a lightly damped, stable equilibrium being replaced by a steady state vibration (the limit cycle), in response to a change in the parameter J.I..

x

x

--+-*"++-+-+-+-- x

a

fL > O

UNSTABLE EQUILIBRIUM STABLE EQUILIBRIUM -----�--------�--------------- fL

Fig. 3 . 2 Supercritical Hopf bifurcation. On the other hand, if S > 0 , then a limit cycle is predicted to exist only for 0, in which case the origin is a stable equilibrium, and the limit cycle is unstable. This is called the subcritical case, see Fig. 3 . 3 . In this case, as decreases through zero, an unstable limit cycle is born from the origin with an amplitude which grows like H . J.I.

J.I.

<

paz

112

A VERA GING

x

x

---+--+-++--+- X

--���-+--- X

J.L > O

a

STABLE EQUILIBRIUM

UNSTABLE EQUILIBRIUM

------�--------L---------------- J.L

Fig. 3 . 3 . Subcritical Hopf bifurcation. 3. 7 Nonautonomous Systems So far in this Chapter we have applied the method of averaging to autonomous�> systems of the form of eq.(3. 2 .1). Now we consider nonautonomous systems of the form x" + x F(x,x, t ) (3. 7.1) We will follow the same approach of using variation of parameters and a near-identity transformation, but with r,o(t) t + .,p( t), cf.eq. ( 3. 2 . 2 ): (3.7. 2 ) x a(t) cos(t + .,p( t)), x a(t) sin(t + .,p( t)) In this case the exact eqs. (3. 2 . 5 ),(3. 2 . 6 ) become: =

f

=

=

=

-

113

A VERA GING Ii.

(3. 7 . 3 ) - f sin(H'¢') F(a cos(H'¢') , - a sin(H'¢') , t) (3.7.4) ip - i cos( H'¢') F( a cos( H'¢') , - a sin( H'¢') , t) Now we apply a near-identitity transformation, but we permit the generating functions to explicitly depend on t, cf.eq.(3.4.1): =

=

(3.7. 5 ) Substituting (3.7. 5 ) into (3.7. 3 ),(3.7.4) gives eqs. on the transformed variables a and 1; , which in this case become, to 0 (f2) (cf.eqs.(3.2.10),(3. 2 .11)): (3. 7 . 6 ) i: , [ - :1 - sin(H;l) F(i cos(t +;l), - i sin(t +;l), t) 1 O( ,2 ) (3. 7 . 7 ) � f [ - ataw2 - cos£aH1;) F(a cos(H1;), - a sin(H1;), t) 1 + O( f2 ) Now we choose WI and W2 to simplify eqs.(3. 7 . 6 ),(3.7. 7) as much as possible. Our strategy will be to trigonometrically reduce these equations, and then use the W to kill all trig terms in t. Once WI and w2 are known, we may proceed to simplifyithe O( f2) terms by similarly choosing v I and v2 ' and so on for higher order terms. There is an important difference between the results of averaging an autonomous system versus a nonautonomous system. The autonomous system (3. 2 .1) results in averaged equations of the form: �

=

(3. 7 . 8 )

+

1 14

A VERA GING

(see eqs.(3.5.5) and (3.5.6), for example.) The nonautonomous system (3. 7 .1) results in averaged equations of the form: . (3. 7 . 9 ) a = f G1 (a,�, f) , Eqs. ( 3.7. 8 ) corresponding to the originally autonomous system are much easier to solve (in principl�) than eqs. (3.7. 9 ) for the nonautonomous system. Since the a eq. of (3. 7 . 8 ) is uncoupled from the ifJ eq., the averaged eqs. ( 3. 7 . 8 ) are essentially one-dimensional, i. e ., they can be reduced to a flow on the a line. No such simplification is available in general for the pair of coupled eqs.(3. 7 . 9 ). 3. 8 Example As an example of nonautonomous averaging, we take Mathieu 's equation: d2 + ( 6 + f cos r) x = 0 (3.8.1) � dr In order to cast (3. 8 .1) into the form of (3. 7.1), we stretch time: (3. 8 . 2 ) r = -t .[0 which gives: (3. 8 . 3 ) F(x,x. ,t ) = - ox1 cos -t �

Following the treatment in section 3. 7 , eqs. ( 3. 7 . 6 ),(3. 7.7) become

.[0

A VERA GING

115

(3. 8 . 4 ) (3. 8 . 5 ) Trig reduction of (3.8.4) and (3. 8 . 5) gives: (3. 8 . 6 ) Ii: = , [ - :1 + H sin [(� + 2)1 + 2W] - H sin [(� - 2)1 - 2W] 1 + 0 ( ,2) (3. 8 . 7 )

�= ,

[ - :2 + h cos [(� + 2)1 + 2W] + h cos [(� - 2)1 - 2W]

The O ( E) terms in (3. 8 . 6 ) and (3. 8 .7) can be completely removed by choosing w I and w2 as follows: (3. 8 . 8 ) (3. 8 . 9 )

where K I and K2 are arbitrary functions of a and �, cf.eqs. ( 3. 3 . 3 ),(3. 3 . 5 ). For this

I

!

p

116

A VERA GING

choice of wI and w2 ' eqs.(3. 8 . 6 ) and (3. 8 . 7) become .:.a .. 0 + O( f2 ) and 7p' 0 + O( f2). Thus to O(f2 ), a{t) and 7fj(t) are constants, and eqs.(3. 7 . 2 ),(3. 7 . 5 ) show that x(t) is quasiperiodic for a general value of o. In particular, all solutions remain bounded as t goes to infinity. The choice (3. 8 . 8 ),(3. 8 . 9 ) for wI and w2 fails, however, if .J.IS - 2 0, i. e ., if 0 i , due to vanishing denominators. This case must 'be investigated separately. It corresponds to the forcing frequency in Mathieu s eq.(3. 8 .1) being twice the natural frequency of the unforced equation, a situation which is called subharmonic resonance. In order to investigate this case to O( fn), we expand 0 in a power series in f about the resonant value 0 i, corresponding to detuning the resonance: (3. 8 .10) o 1 + 01 f + O2 2 + . . . In this case, we set 2 t in eq.(3. 8 .1) in order to obtain an equation ofthe form (3. 7 .1): =

=

=

=

= 4"

f

T=

(3. 8 .11) which gives (3. 8 .12) F(x,x,t) 4 [ 01 + 02 f + . . . + cos 2t] x so that eqs. ( 3. 7 . 6 ),(3. 7. 7 ) become =

-

=

A VERA GING

Trig reduction of (3. 8 .13) and (3. 8 .14) gives: (3. 8 .15) (3. 8 .16)

-:;P.

= E

[ -_8Owt 2 +

COS (

4t +21/i) + 261 cos(2t + 21/i)

Now we choose w I and w2 to kill the trig terms in t in (3.8.15) and (3. 8 .16): (3. 8 .17) wI - £ cos(4t + 21/i) - 61 a cos(2t +21/i) + K 1 (a,-:;p) (3. 8 .18) w2 i sin(4t +21/i) + 61 sin(2t +21/i) + sin 2t + K2 (a,-:;p) where K 1 and K2 are arbitrary functions of a and -:;p. Note that we do not kill the t-independent terms in (3. 8 .15) and (3. 8 .16), since this would require that the wi contain terms which are linear in t, thereby ruining the uniform validity of the near-identity transformation (3. 7 . 5 ) in the large t limit. The averaged eqs.(3. 8 .15) and (3. 8 .16) then become: =

=

(3. 8 .19) (3. 8 . 2 0) Eqs.(3. 8 .19) and (3. 8 .20) are easier to solve when a and -:;P are replaced by x and y, where

117

118 (3.8.21)

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x = a cos � ,

y = - a sin �

Geometrically, it and � can be thought of as polar coordinates corresponding to the rectangular coordinates x and y, d.eqs.(3.7.2). Under the transformation (3.8.2 1 ) , eqs.(3.8.19),(3.8.20) become:

(3.8.22) which can be written in the more familiar form: •

(3.8.23)

•

x + £ 2 (4 61 2 - 1) x = 0

� � � �

The nature of the solution x( t ) depends upon whether 1 61 1 < , = or > . In particular, all solutions to (3.8.23) will be bounded if 1 61 1 > , corresponding to

�

quasiperiodic solutions of Mathieu ' S eq.(3.8.1). If 1 61 1 < , on the other hand, then (3.8.23) will exhibit solutions which grow exponentially in t , as will Mathieu ' S eq.(3.8.1). Note that the transition between bounded ( "stable") and unbounde�

�

("unstable" ) behavior is given by the condition 1 61 1 = , corresponding to a zero eigenvalue of (3.8.22) or (3.8.23) , which represents two transition curves in the 6-£ plane: (3.8.24)

6 = :{1 - "2"1 £ + O( £2 )

From (3.8.23) we see that on the transition curves (3.8.24) , one solution x grows linearly in t , as does the corresponding solution to Mathieu ' S eq. (3.8. 1 ) . The other linearly independent solution of (3.8.23) is constant on the transition curves, leading to a periodic solution of Mathieu ' S equation. See Fig.3.4.

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119

1 /4 Fig.3 .4. Transition curves ( 3.8.24 ) in Mathieu ' s eq. ( S=Stable, U=Unstable ) . If our treatment of Mathieu ' s eq. for general 0, eqs. ( 3.8.1 )-( 3.8.9 ) , were extended to include terms of O ( E2 ) , we would find another resonance which produced a pair of transition curves which cross the 6-axis at 0 = 1 . Similarly, at O( fn ) , a resonance occurs at which a pair of transition curves bounding a region of instability emanate 2 from the point 0 = on the 6-axis.

r

In Appendix 8 we present a MACSYMA program which automates the method of averaging on Mathieu ' s equation. The results , neglecting O( f4 ) , may be expressed in the form of eqs. ( 3 .8.22 ) on x and y: ( 3.8.25 )

( 3.8.26 )

The transition curves ( 3.8.24 ) correspond to values of 0i which give nontrivial

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equilibria in ( 3.8.25 ) , ( 3.8.26 ) , i.e. zero eigenvalues. These turn out to be: 8 = 41

( 3.8.27 )

+1

2"

f

-

1 f2

8'

-

1

�

3

f

+ O( 4) , f

Although even higher order approximations for these transition curves can be found by the method of averaging, a more efficient approach is offered by regular perturbations. We have seen that on the transition curves, Mathieu ' s equation exhibits a periodic solution. Thus we look for a periodic solution to x. .

( 3.8.28 )

+ ( 1 + 81 + 4

f

in the form of a power series in

v2 f:

2

f

+ ... +

f

cos t ) x = 0

f:

( 3.8.29 )

where each of the transition curves ( 3.8.27 ) corresponds respectively to the choice

Xo =

i

i·

Substituting ( 3.8.29 ) into ( 3.8.28 ) and collecting terms sin and Xo = cos gives the following equations on x l and x2 : ( 3.8.30 ) ( 3.8.31 )

Taking Xo = sin

i ' eq. ( 3.8.30 ) becomes x. l.

( 3.8.32 )

+ 1 xl 4

=(

-

vI f:

+ 1 ) sm. t 2"

2"

-

. 3 1 sm 2" t

2"

For a periodic solution we require there be no resonance terms, i.e. , 81 = case x l = sin t . Then ( 3.8.31 ) becomes

i �

� , in which

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121

(3.8.33)

�

For a periodic solution we require 62 = - , in agreement with the first of (3.8.27). This process may be continued using computer algebra ( see Rand 1984), giving results such as: (3.8.34)

Note that although regular perturbations works much more efficiently than averaging in finding the transition curves in Mathieu ' S equation, regular perturbations only works on the transition curves ( where one of the two linearly independent solutions is periodic ) . In contrast , averaging provides a general solution to Mathieu's equation on or off the transition curves.

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3. 9 Related Topics This section describes so me related topics which are treated in Appendices 9 , 1 0 and 11. I n this Chapter w e have s o far considered systems which reduce t o x · + x = 0 when f = 0, for example eq.(3.2.1). These all involve perturbing off of solutions which involve the trig functions sin and cos. Another approach is to scale the given equation so that when f = 0, the equation reduces to x · + x + x2 = 0 or x · + x + x3 = 0, in which case we will perturb off of solutions which involve the elliptic functions sn,cn and dn. This approach is treated in Appendix 9.

The near-identity transformation upon which the method of averaging is based is a power series in f , as are expressions for transition curves (cf.(3.8.34» and frequencies of periodic motions (d.(3.5.8». This raises questions of convergence of such series . Pade approximants are a technique for accelerating the convergence of power series, and are treated in Appendix 10. We looked at the birth of a periodic motion through a Hopf bifurcation in section 3.6. Another way that limit cycles can occur during a perturbation process was investigated by Andronov and his colleagues, and is treated in Appendix 1 1 . We refer to it as the Andronov bifurcation.

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123

3.10 References Andronov,A. A . , Leontovich,E. A . , Gordon,I.J. and Maier,A.G. , "Theory of Bifurcations of Dynamic Systems on a Plane" , Israel Program for Scientific Translations, Jerusalem (1971) Belhaq,M. and Fahsi,A., " Subharmonic Vibrations Close to Degenerate Poincare-Hopf Bifurcations " , Mech.Research Comm. 20:335-341 ( 1993) Bender,C.M. and Orszag,S.A., " Advanced Mathematical Methods for Scientists and Engineers" , McGraw-Hill (1978) Byrd,P . and Friedman,M . , "Handbook of Elliptic Integrals for Engineers and Scientists" , Springer ( 1954) Chakraborty,T. and Rand,R.H., "The Transition from Phase Locking to Drift in a System of Two Weakly Coupled Van der Pol Oscillators" , International J. Nonlinear Mechanics 23:369-376 (1988) Coppola,V.T. and Rand,R.H. , " Averaging Using Elliptic Functions: Approximation of Limit Cycles" , Acta Mechanica 8 1 : 125-142 (1990) Coppola,V.T. and Rand,R.H. , "MACSYMA Program to Implement Averaging Using Elliptic Functions" , in Computer Aided Proofs in Analysis, eds. K . R. Meyer and D.S.Schmidt, pp.71--89, Springer Verlag (1991) Guckenheimer,J. and Holmes,P . , "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields" , Springer ( 1983) Hunter, C . and Guerrieri,B . , "Deducing the Properties of Singularities of Functions from their Taylor Series Coefficients " , SIAM J.Appl.Math. 39:248-263 (1980) also see erratum in SIAM J.Appl.Math. 41:203 ( 1981)

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Kahn,P . B . , Murray,D. and Zarmi,Y. , " Freedom in Small Parameter Expansion for Nonlinear Perturbations" , Proc.R.Soc.Lond.A 443:83-94 (1993) Rand,R.H. " Computer Algebra in Applied Mathematics: An Introduction to MACSYMA " , Pitman ( 1984) Rand,R. H. , "Using Computer Algebra to Handle Elliptic Functions n the Method of Averaging" , in Symbolic Computations and Their Impact on Mechanics, eds. A.K.Noor, I.Elishakoff, G.Hulbert, pp.311-326, Amer. Soc.Mech.Eng . , PVP-Vo1.205, (1990) Rand,R.H. and Armbruster,D . , "Perturbation Methods, Bifurcation Theory and Computer Algebra" , Springer (1987) Sanders,J.A. and Verhulst,F., "Averaging Methods in Nonlinear Dynamical Systems" , Springer (1985) Stoker,J.J. , "Nonlinear Vibrations" , Wiley Interscience (1950) van Dyke,M. , "Analysis and Improvement of Perturbation Series" , Q.J.Mech.Appl.Math. 27:423-450 ( 1974)

a

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125

3.11 Exercises 1. Use first order averaging to obtain a slow-flow for the following system of two coupled van der Pol oscillators ( Chakraborty and Rand) :

(3. 1 1 . 1 )

Show that the four dimensional slow-flow on a:1 ,a:2'�1 '�2 can be reduced to a three dimensional slow-flow by taking �C�2 as a new coordinate. 2. This exercise concerns the system: (3. 1 1 . 2)

X. . + X = f. (-3 x2x. + x x. 2 + x. 3 ) + f.2 A x. ( 3 x2 - A )

where f. = 0.5 and A = 0 . 1 . a. Use second order averaging to investigate the number and location of any limit cycles which occur in (3. 11.2). Treat f. as the usual perturbation parameter and A as an "unfolding" parameter, independent of f.. b. Check your result by numerically integrating (3. 11.2). Note that the quantity S of eq.(3.6.9) (after rescaling (3.6.3) with f. -I {i ) is zero, since III = � = 113 = f31 = 0, f32 = -3, f33 =f34 = 1 . Thus eq.(3. 1 1.2) is an example of a degenerate Hopf bifurcation. 3. Use first order averaging to investigate the forced Duffing equation: (3. 11. 3) Obtain expressions for the equilibria of the averaged equations, and by eliminating 1jj, show that there results a cubic equation on a:2 .

pt

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4. Use third order averaging to investig ate the nonlinear M athieu equation (Belhaq and Fahsi) : (3. 11.4) Obtain expressions for the equilibria of the averaged equations, and by eliminating 1{;, show that there results a quadratic equation on a2 . 5. This exercise concerns the system: · x. . + x + 0.035 x. + x3 - 0.6 x2 x. + 0 . 1 x3 = 0

(3. 11. 5)

We are interested in the number and location of any limit cycles which occur in (3. 11. 5) . Investigate this question by using the following four approaches, and compare results: a. First order trig averaging. Rewrite (3.11 .5) in the form: (3 . 1 1 .6)

x. + x = .

f

(- 0.35 x. - 10 x3 + 6 x2 x - x. 3 ) , .

where f = 0 . 1

and apply first order averaging, d. section 3.5 o n van der Pol's eq. b. Second order trig averaging. Using the same scaling, treat eq.(3 .1 1 .6) by second order averaging. c. First order elliptic averaging. Rewrite (3.11.5) in the form: (3. 1 1 . 7)

x. . + x + x3 =

f

(- 0.35 x + 6 x2 x. - x. 3 ) , .

where

f

= 0. 1

and apply first order averaging using elliptic functions as in Appendix 9 , d. eq. (A9.3.4). Note that eq. (A9.3.5) applies to this case. d. Numerical integration.

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6. This exercise concerns the system: x + x - 0.014 x + x2 - 0 . 1 x x = 0 •

(3. 1 1 . 8)

•

•

•

Eq.(3 . 1 1 .8) exhibits a limit cycle. Obtain approximations for the limit cycle by using the following three approaches, and compare results by plotting the approximations in the x-x plane a. Second order trig averaging. Rewrite (3. 1 1.8) in the form: (3. 1 1 .9)

" x" " + x = f (- 10 x2 + x x" ) + 1.4 f2 x,

where f = 0 . 1

and apply second order averaging, cf.eq.(3.6.3). Note that eq. (3.6. 1O) applies t o this case. Also note that first order averaging gives no contribution to the averaged equations. b. First order elliptic averaging. Rewrite (3.11.8) in the form: (3. 1 1 . 10)

x " + x + x2 = f x X + 0. 14 f x,

where f = 0 . 1

and apply first order averaging using elliptic functions a s i n Appendix 9 , d. eq.(A9.2 . 1 1 ) . Note that eq.(A9.2.12) applies to this case. c. Numerical integration.

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7. In Appendix 10 P ade approximants are used to improve the convergence of a transition curve (AI0.2.2) in Mathieu's equation (AI0.2 . l). The goal of this exercise is try the same treat ment on the transition curves (3.8.34) .

a. Use regular perturbations (cf. A ppendix 10) to derive eqs.(3. 8.34). b. Use MACSYMA to obtain Pade approximants for eqs . (3.8.34). c. By finding the zeros of the denominator of the Pade approximant, estimate the radius of convergence in f for each of eqs.(3.8.34). d. Plot each of eqs.(3.8.34) together with their respective Pade approximant , as in Fig.3.5 in Appendix 10. e. Numerically integrate Mathieu's eq. (AI0.2. l) for a variety of values of 8 and f, observing in each case whether the behavior is stable or unstable. In this way obtain numerical versions of the transition curves (3.8.34) . Plot your results along with the perturbation series and the Pade approximants of part d. 8. In Appendix 1 1 it is shown that the equation (3. 1 1 . 1 1 )

· 1 X. 4 ) = 0 x + X + f X ( I - a2 x2 + IlJ .

.

.

is singular in that it possesses a semi-stable limit cycle. (See Fig.3.6 in Appendix 1 1 . ) Unfold this singularity, i.e. , add some small additional terms to (3. 1 1 . 1 1 ) , proportional t o a parameter A say, so that when A = 0 your equation becomes eq.{3 . 1 1 . 1 1 ) , and when A > 0, your system exhibits two limit cycles (all for small f). Use the Andronov criterion to design your unfolding, and check your result by numerically integrating your system.

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129

Appendix 6: MACSYMA Program for Second Order Averaging The following MACSYMA program applies second order averaging to the linear damped oscillator of example (3. 1. 1 ) . The program is based on the strategy given in section 3.4. The variable "ws6" produces eqs . (3.3.3),(3.3.5). The variable "vs6" produces eqs.(3 .4.3),(3.4.4). The averaged eqs. (3.4.5) correspond to the variable "de7" . The slow flow solutions (3.4.6) correspond to the variables "asoll " and "phisoll " . The variable "cons" generates the initial condition coefficients in (3.4.8). The variables "xscheme1" and "xscheme2" respectively produce the approximate solutions (3.4.9) and (3.4. 10). The exact solution (3.4. 1 1 ) corresponds to the variable "exactsol " . Finally, the Table in section 3.4 is generated by the "numer" statements at the end of the program. /* damped linear oscillator: second order averaging * / /* setup * / depends( [a,phi] , t) j depends([w1 ,w2, v1,v2] ,[abar ,phibar]) j depends([abar,phibar] ,t) j x:a*cos(phi ) j xd:-a*sin(phi) j f:-xdj del : [diff( a,t )=�*sin(phi )*f,diff(phi,t)= 1�/ a*cos(phi)*fj j transf: [a=abar+e*w1+eA2*v1 ,phi=phibar+e*w2+eA 2*v2]j de2: de1,transf,diff$ de3:solve(% , [ diff( abar,t ),diff(phibar,t)] )$ de4:taylor( de3 ,e,0,2)$ /* first order averaging * / wS1:ratcoef(% ,e)$ ws2:expand( trigreduce( expand( wS1)))$ wS3:integrate( ws2 ,phi bar )$ ws4:ws3, %integconst 1 :k1, %integconst2:k2$ wS5:ws4-ratcoef( ws4,phibar )*phibar$ ws6:solve(part( ws5, 1 ) , [w1, w2]) j

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de5: de4,ws6,diff$ /* se cond order averaging * / vs l :ratcoef( de5 ,e,2) $ vs2:expand( tri greduce( expand( vSl» )$ vs3:integrate( vs2,phibar)$ vs4:vs3, %integconst3:k3, %i ntegconst4:k4$ vs5:vs4-ratcoef( vs4,phibar )*phibar$ vs6:solve(part( vs5, l ) , [vl, v2]); de6:de5, vs6,diff$ de7:expand( trigreduce( expand( de6» ) ; 1* solve averaged eqs * / de8:part( de7, l , l) ; de9:part(de7,1,2); asol:ode2( de8,abar, t ) ; asoll : asol, %c=conl ; phisol:ode2( de9,phibar,t); phisoll :phisol, %c=con2 ; 1* initial conditions * / transfl :transf, vs6, ws6$ sol: [x,xd) ,transfl$ soll :taylor(sol,e,O,2)$ soI2:expand( trigreduce( expand(soll» )$ so13:s012,asoll,phisoll$ solteqO: so13,t=O$ solteqOl :taylor( sol teqO,e,O,2); con l O : l ; con20:0; 1* i.c. t=O, x=l , x'=o * / solteq02:solteqO 1 ,conI =conlO+conll *e+con12*e A2, con2= con20+con2 1 *e+con22*eA 2$ solteq03:taylor(solteq02,e,O,2);

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for i: l thru 2 do eq[i] :ratcoef(solteq03,e,i); solteq04:makelist(part( eq[i] , l ),i,1,2)$ solteq05:append(solteq04,makelist(part(eq[i] ,2),i,1,2)); cons:solve( solteq05,[ con I 1 ,con12,con21 ,con22]) ; /* compare with exact soln * / exactsol:%e' (-€*t/2)*( cos(sqrt( 1-€'2/4)*t) +e/2/sqrt( l-€' 2/4 )*sin( sqrt( l-€ ' 2/4 )*t)); 1* final perturbation soln * / pert: [x,xd] ,transf1$ pert l: pert,asoll ,phisoll $ pert2:pertl , conl=conlO+conll *e+con12*eA 2, con2=con20+con21 *e+con22*eA 2$ pert3:pert2,cons$ exact: [exactsol,diff( exactsol,t )]$ /* comparison * / cf: [pert3,exact ] $ /* scheme 1 : kl=O,k2=-1/4,k3=7/64,k4=O * / schemel : cf,kl= O,k2=-1/4,k3=7/64,k4=O$ xschemel:part(schemel , l , l) ; schemel ,e= . l ,t=O,numer; schemel,e= . l ,t= l,numer; schemel ,e= l ,t=O ,numer; schemel ,e= l ,t=l ,numer; 1* scheme 2 : ki=O * / scheme2: cf,kl=O,k2=O ,k3=O,k4=O$ xscheme2:part(scheme2, 1 , 1 ) ; scheme2,e= . 1 ,t=O,numer; scheme2,e= . 1 ,t = 1 ,numer; scheme2,e= 1 ,t=O,numer; scheme2,e= 1 ,t=1 ,numer;

131

p

c

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Appendix 7: MACSYM A Program for Thi rd Order A veraging The following MA CSYMA program applies third order averaging to van der Pol ' s eq. (3.5. 1). The program is very similar to the second order averaging program given in Appendix 6 , an d comparison of the two shows how to modify the program to accomodate still higher order averaging. Note th at the generating functions u 1 and u2 of eq. (3.5.4) need not actually be computed, but rather may be taken into account by simply removing the O( (3 ) periodic terms, a step which is accomplished by replacing "sin" and "cos" by a "null" function at the end of the program. The program produces the averaged eqs. (3.5.5) of the text. /* van der Pol ' s equation: third order averaging * / /* setup * / depends([a,phi] ,t ) ; depends( [w1 , w2 ,vl, v2] ,[ abar ,phi bar]) ; depends([abar,phibar] ,t) ; x: a*cos(phi ) ; xd:-a*sin(phi) ; f:( 1-xA2)*xd; de1 : [ diff( a,t )=-e*sin(phi )*f,diff(phi,t )= l-e/ a*cos(phi )*fJ ; transf: [a=abar+e*w1+eA 2*v1,phi=phibar+e*w2+eA2*v2j ; de2:del,transf,diff$ de3:solve(% , [ diff( abar,t ) ,diff(phibar,t)] )$ de4:taylor( de3 ,e,O,3)$

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/* first order averaging * / ws1 :ratcoef( % ,e )$ ws2:expand( trigreduce( expand( ws1)))$ ws3:integrate( wS2 ,phibar)$ ws4:ws3, %integconst1 :k1, %integconst2:k2$ wS5:ws4-ratcoef( ws4,phibar )*phibar$ ws6:solve(part( ws5, 1 ) , [w1, w2] ) ; de5:de4, ws6,diff$ /* second order averaging * / vs1 :ratcoef( de5 ,e,2)$ vs2:expand( trigreduce( expand( vs1)))$ vs3:integrate( vS2,phibar)$ vs4:vs3, %integconst3:k3, %integconst4:k4$ vs5:vs4-ratcoef( vs4,phibar )*phibar$ vs6:solve(part( vs5 , 1 ) , [v1, v2]) ; de6:de5, vS6,diff$ de7:expand( trigreduce( expand( de6))); /* third order averaging * / /* define null function to zap sin and cos * / /* a short cut valid only for the final iteration * / null(anY) : =O$ de8:subst(null,sin,de7)$ de9:subst(null,cos,de8)$ de10:ev( de9 ) ;

133

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134

Appen dix 8: M ACSYMA Program for Averaging in a Nonautonomous System A8. l Program for thi rd order averaging The following MACSYM A program applies third order averaging to Mathieu'S eq.(3.8. 1 1 ) . The program is a slight modification of the programs given in Appendices 6 and 7 which were applicable to autonomous systems. The main change here is that the generating functions are permitted to be explicit functions of t , see section 3 . 7. The arbitrary functions K i (a,1P) associated with the generating functions (cf.eqs. (3.8.8) ,(3.8.9)) have been taken as zero for convenience (and are referred to as %integconst l , etc. in the program). The averaged eqs . on a and ?p are stored in the variable "neweqs" . The last portion of the program executes a transformation from a,?p to x,y variables, cf.(3.8.2l) . The program produced eqs. (3.8.25) and (3.8.26) of the text . /* Mathieu'S equation: third order averaging * / /* setup */ depends([a,psi] ,t ); depends([wl , w2, vl ,v2,ul, u2] ,[abar ,psibar, t]) ; depends([abar,psibar] ,t); x:a*cos( Hpsi ) ; xd: -a*sin( Hpsi); f:-4*x*( dell +e*de12+eA 2*de13+cos(2*t ) ) ; del : [diff( a , t )=-e*sin( t +psi )*f,diff(psi, t )=-e/ a*cos( Hpsi )*fj ; transf: [a=abar+e*wl+eA 2*vl+eA 3*ul ,psi=psibar+e*w2+eA 2*v2+eA 3*u2] ; de2:del ,transf,diff$ de3:solve(%, [diff( abar,t ) ,diff(psibar, t )]) $ de4:taylor( de3 ,e,O,3)$ for i:O thru 3 do eq[i]: coeff( de4,e,i ) ; neweq[O] :eq[O] ; kill ( abar,psibar);

A VERA GING

/* first order averaging * / wSl:eq[I] $ ws2:expand( trigreduce( expand( wsl)))$ wS3:integrate( ws2 , t)$ ws4:ws3, %integconst 1:0, %integconst2: 0$ wS5:ws4-ratcoef( ws4,t )*t$ ws6:solve(part( ws5, 1 ) , [wl ,w2]) ; ws7:wsl ,ws6,diff; neweq[I] :expand( trigreduce( expand ( ws7) )); /* second order averaging * / vSl :eq[2] ,ws6,diff$ vs2:expand( trigreduce( expand( vSl)))$ vs3:integrate( vS2,t)$ vs4:vs3, %integconst3:0, %integconst4:0$ vS5:vs4-ratcoef( vs4, t )*t$ vs6:solve(part( vs5 , 1 ) , [vl, v2]) ; vs7:vsl ,vs6,diff$ neweq[2] :expand( trigreduce( expand( vs7))); /* third order averaging * / usl :eq[3] ,ws6,vs6 ,diff$ us2:expand( trigreduce( expand( uSl)))$ us3:integrate( uS2,t)$ us4:us3, %integconst5:0, %integconst6:0$ us5:us4-ratcoef( us4,t )*t$ us6:solve(part( us5, 1 ) , [ul,u2]) ; us7:us l ,us6,diff$ neweq[3] :expand( trigreduce( expand( us7))); /* assemble results * / neweqs: sum(neweq[i]*eAi,i,0,3);

135

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1* transform to xbar,ybar variables */ depends([xbar,ybar] ,t) ; depends([abar,psibar] ,t)j [xbar = abar*cos(psibar) ,ybar = -abar*sin(psibar)] j diff( %,t ) ; ev(%,neweqs ) j trigexpand(%)j expand(% )j trigsimp(% ) ; taylor(%,e,O,3)j ev(%,sin(psibar) = -ybar/abar,cos(psibar) = xbar/abar)j taylor(%,e,O,3)j AB. 2 Program for nth order averaging Inspection of the foregoing program for third order averaging shows that the sections marked "first" , "second" and "third order averaging" are essentially identical. This IIwtivates the following nth order averaging program. The variable "trunc" is read in from the keyboard and determines the truncation order. The generating functions w1,v1,u1 of the previous program are now referred to as wl[1] ,wl [2] ,wl[3] , respectively. The program terminates with the production of the averaged eqs. on a and Vi in the variable "neweqs" . The last portion of the preceding program (the

transformation i,y variables) is omitted here, although it may still be found useful for values of trunc > 3. 1* Mathieu ' S equation: nth order averaging * / 1* setup */ trunc:read("input truncation order")$ depends( [a,psi] ,t)j depends( [wl ,w2] , [abar,psibar,t]) j depends([abar,psibar] ,t) j

-

A VERA GING

x:a*cos(Hpsi)j xd:-a*sin( t +psi)j delta:sum( d[i]*eA (i-1 ) ,i , 1,trunc) j f:-4 *x*( delta+cos( 2*t)) j de1 : [diff( a, t )=-e*sin( t +psi )*f, diff(psi ,t )=-e/a*cos( Hpsi)* � j transf: [a=abar+sum(w1[i]*eAi,i , 1 ,trunc), psi=psibar+sum( w2[i]*e Ai,i , 1 , trunc)] j de2:de1,transf,diff$ de3:so1ve(%, [ diff( abar,t ) ,diff(psibar, t)])$ de4:taylor( de3,e,O,trunc)$ for i:O thru trunc do eq[i] :coeff( de4,e,i)j neweq[O] :eq[O] j kill( abar,psibar) j %integconstl :O$ %integconst2:0$ %integconst3:0$ %integconst4:0$ %integconst5:0$ %integconst6:0$ %integconst7:0$ %integconst8:0$ %integconst9:0$ %integconst lO:O$ %integconstl l:0$ %integconstl2:0$ results : [ ]j

137

p 138

A VERA GING

/* averaging loop * / for i : 1 thru trun c doe wsl :ev( eq[i] ,results ,diff) , ws2 :exp and( tri greduce ( expand( ws 1 ) ) ) , ws4 :integrate( ws2, t ) , wS5:ws4-ratcoef( ws4,t )* t , ws6:solve(part( ws5, 1 ) , [wl[i] , w2[i]]) , ws7:ev( wS1 , ws6 ,diff) , results:append(results ,ws6), results:ev(results), neweq[i] :expand( trigreduce( expand( ws7»), print ( "result of step" ,i, " : " ) , print( neweq[i]) )$ /* assemble results * / neweqs:sum(neweq[i]*eAi,i,O,trunc)j

&

139

A VERA GING

Appendix 9: Averaging with Elliptic Functions A9. 1 Introduction In this Appendix we will consider the problem of approximating the location of limit cycles in the two systems: ( A9. l . 1 )

x . + x + x2 = f F (x,x )

( A9. 1 . 2 )

x. . + x + x3 = f F (x,x. )

•

.

When f = 0, these reduce to the eqs. x ' + x + x2 = 0 and x' + x + x3 = 0 , respectively. Their general solutions have been given i n Appendix 4, and are summarized here: For eq. ( A9. 1 . 1 ) when f = 0, we have: ( A9 . 1 .3 )

x + x + x2 = 0 , •

.

x = aO + a 1 sn2 ( u,k )

where ( A9. 1 .4 ) ( A9 . 1 .5 ) ( A9. 1 .6 )

- 2 1 3m a = aO = 1+m2A ' a 1 = - :-:2 ' 2 2A 2A 2A A4 = m2-m+ 1 , m = k 2

Note that we may take m and b as arbitrary constants in the solution ( A9. 1. 3 ) , since aO ,a 1 and a2 are known functions of m. For eq. ( A9. 1 . 2 ) when f = 0, we have:

140 (A9 . 1 . 7)

A VERA GING

x. . + x + x3 = 0 ,

x = a 1 cn(u,k)

where (A9 . 1.8) (A9. 1.9)

2m a 1 2 = 1-2m '

1 a2 2 = I-2m

2 'm=k

Note that we may take m and b as arbitrary constants in the solution ( A9. 1 . 7) , since a 1 and a2 are known functions of m. In both cases, the f = 0 solutions involve the two arbitrary constants m and b, where b determines the phase of the solution and m (where 0 � m � 1) determines its amplitude. We shall use first-mder averaging on eqs.(A9 . 1 . 1 ) and (A9. 1.2). We begin by using variation of parameters to obtain an eq. on the slow evolution of m(t). This equation corresponds to eq.(3.2.5) on a(t) in the trigonometric case of eq.(3.2. 1). We look for a solution to eqs.(A9 . 1 . 1 ) and (A9. 1.2) respectively in the form of eqs. (A9. 1.3) and (A9. 1.7), in which the two arbitrary constants m and b are allowed to vary in time. This results in first-order differential equations on m(t ) and b(t ) . For brevity, we shall consider only the equation on m(t), the equilibria of which will correspond to limit cycle periodic motions in eqs.(A9 . 1 . 1 ) ans (A9. 1.3). Let us write the general solution, eq.(A9. 1.3) or (A9.1.7) , respectively, in the abstract form: (A9 . 1 . 10) Differentiating eq. (A9. 1. 10) gives (A9 . 1 . 1 1)

x = x(t,m,b)

A VERA GING

141

Here we use the notation that a partial derivative in t holds m( t) and b( t) fixed. As usual in variation of parameters, we require that (cf.eq. (3.2.2) in the trig case) : (A9. l. 12)) giving

dx Ox at = ar

(A9 . l . 13)

ax dm Ox db 0 arn OT + oo at -

Differentiating eq. (A9. l. 12) gives

(A9. l . 14)

�2 d2 x trx + a 2 x dm + a2 x db 2 dt - 8t2 &nat dt {)bat dt _

Now the unperturbed solution satisfies one of eqs.( A9. l .3) or (A9 . l . 7), now written as

(A9. l .15)

a2x n ::2 + x + x = 0 , at

£=0

where n = 2 or 3. The second partial derivative in t appears in (A9. l. 15) because m and b are constants when £ = O. In the £ > 0 eqs. (A9. l. 1 ) and (A9. l .2), however, m and b are permitted to depend on t , so that

(A9 . l . 16) Substituting eqs. (A9. l .15) and (A9. l. 16) into eq. (A9. l . 14) gives

(A9 . l . 17)

a 2 x dm a2 x db = £ F + &n at d t {)b at dt

Eqs.(A9. l . 13) and (A9. l. 17) may be solved simultaneously for dm/dt as follows:

(A9. l. 18)

142

A VERA GING

where subscripts represent partial differentiation. We omit reference to the associated b equation (which corresponds to the � eq.(3.2.6) in the trig case), since it is not required in order to obtain the amplitude of the limit cycle. Our treatment so far has dealt in a common way with both eqs.(A9 . 1 . 1 ) and (A9 . 1 . 2) , resulting i n eq. (A9 . 1 . 18) which i s valid for both cases. Now, however, we will treat n = 2 and n = 3 in (A9. 1 . 16) separately. A9. 2 The system x. . ± x ± x2 =

f.

F (X,J<;. )

The following MACSYMA program computes the partial derivatives in eq.(A9 . 1 . 18) when x(t) is given by (A9 . 1 .3)-(A9.1.6) , and simplifies the result: depends( [cn,sn,dnj ,[u,m] ) ; depends ( u,[t ,a2 ,b] ) ; depends([aO,a1,a2j ,m); derivs: [ ' diff( cn,u )=-sn*dn, ' diff( sn, u )=cn*dn, ' diff( dn, u )=-m*sn *cn, ' diff(cn,m)=sn*dn*f, ' diff(sn,m)=-cn*dn*f, ' diff( dn,m)=m*sn*cn*f-sn · 2/(2*dn)j ; idents: [sn=sqrt( l-cn · 2) , dn= sqrt(l-m±m*cn ·2)j; x:aO±a1 *sn · 2 ; xt :diff(x,t) ; xt :xt,derivs; xbt :diff(xt ,b); xbt:xbt,derivs; xb:diff(x,b ) ; xb:xb,derivs; xm:diff(x,m); xm:xm,derivs; xmt: diff(xt ,m); xmt:xmt ,derivs; dmdt :-xb*e*F /(xm*xbt-xmt*xb); dmdt:ratsimp( dmdt) ;

A VERA GING

dmdt:dmdt ,idents,diff; dmdt:dmdt ,derivs; dmdt:dmdt,u=a2*t+b,diff; dmdt :dmdt, aO = -(sqrt(m' 2-m+ 1)-m-1)/(2*sqrt(m' 2-m+ 1)), a1 = -3*m /(2*sqrt(m' 2-m+ 1)),a2 = 1/(2*(m' 2-m+ 1) ' (1/4)),diff; dmdt :faetor( ratsimp( dmdt)); In writing the above program, the following identities were used (see Byrd and Friedman, p. 283, formulas 710):

(A9 . 2 . 1 )

ad n a s n = - en dn f, Oiil a e n = en dn f, Oiil s n2 = m sn en f - 2"1 an

Oiil

where (A9.2.2)

) -(1-m)u - sn en f - f( u,m ) - E(u2 m(1-m 2 (1 -m)dn )

The final expression produced by the program may be written: 7 m = - �8 A1-m sn en dn f F (A9 .2.3) •

The right hand side (rhs) of eq.(A9.2.3) is periodic in t and thus in the appropriate form for averaging. For first-order averaging, we may replace the rhs of eq.(A9.2.3) by its average value taken over one period of length 4K(k) :

(A9.2.4)

J

4K 7 1 8 A sn en dn F du m = - � I-m f 4K •

o

where F = F(aO +a 1 sn2 ,2a 1 a2 sn en dn) and sn = sn(u,k), en = en(u,k) and dn = dn(u,m) .

143

•

144

A VERA GING

If F(x,x) is a polynomial in x and X, then the evaluation of the integral in eq.( A9.2.4) may be readily accomplished. Terms in F of the form xN xM for M even lead to integrals of the form

Io

(A9.2.5)

4K

cn P sn dn du

where P is an integer, which vanish due to the oddness of the integrand. On the other hand, terms in F of the form xN xM for M odd lead to integrals of the form

I

(A9.2.6)

4K

cn2P du

o

which may be evaluated by using the following results from Byrd and Friedman, pp. 192-3 , formulas 312: Define

C 2P :::

(A9.2.7) Then

cn2P du

4 [E-(1-m)K] C o ::: 4K, C 2 ::: m

(A9 .2.8) and (A9.2.9)

Io

4K

C 2P+2

:::

2 P 2m-1 C + 2P-1 1 -m C 2P 2PTI III 2P _2

2PTI --m-

where E ::: E(k) and K ::: K(k) are complete elliptic integrals. An equilibrium point of the averaged m equation, eq.(A9.2.4), corresponds to a limit cycle in the original equation (A9. l .1). Thus if m ::: mO is a root of the equation

A VERA GING

(A9.2. 10)

J

4K sn cn dn F du = 0

o

then the averaged equation predicts that for small f, a limit cycle coincides with the solution (A9. 1.3) associated with a value of m = mO . For an arbitrary polynomial F(x,x) , the limit cycle integral condition, eq.(A9.2. 10), may be efficiently evaluated by using the following MACSYMA program: F:read("Enter F(x,y), where y=dx/dt " ) ; 1* eliminate even powers of y * / F1:ev(F,y=-y); F2: (F-F1)/2; 1* form integrand of limit cycle integral, eq.(A9.2.10) * / integrand:F2*sn*cn *dn,x=aO+a1 *sn A 2,y=2*a1 *a2*sn*cn *dn$ 1* use identities to express sn,dn in terms of cn * / idents: [sn=sqrt ( 1-cnA2), dn= sqrt(1-m+m*cnA2)]; integrand1 :expand( ev(integrand,idents))$ 1* set up rules for evaluating the integral of cn Ar * / 1* KK = K(k) , EE = E(k) */ c[O] :4*KK; c[2] :4*(EE-(1-m)*KK)/m; c[r] : =(r-2)*(2*m-1)/((r-1)*m)*c[r-2]+(r-3 )*(1-m)/((r-1)*m)*c[r-4] ; 1* perform the integration by replacing cn A r by c[r] * / max:hipow(integrand1 ,cn); for i:O thru max step 2 do b[i] :coeff(integrand1,cn,i); result :sum(b[2*i]*c[2*i] ,i,O,max/2)$ 1* substitute the parameters aO,a1 ,a2 in terms of m * / result 1:result, aO = -(sqrt(mA 2-m+ 1)-m-1 )/(2*sqrt(mA 2-m+1 )), a1 = -3*m/( 2*sqrt(mA 2-m+1)), a2 = 1/(2*(m A 2-m+ 1 r (1/4))$ /* simplify the result * / result2:ratsimp( result ! ) ;

145

..

A VERA GING

146 As an example let us take (Rand 1990) (A9.2.11)

in which k0 1 and k l l are given parameters. The result of the foregoing program may be expressed in a convenient form by solving for k Ol /k l l : (A9.2. 12)

k0 1

XU -

_

1 + 5 (K-2E)m 3 + (K+ 3 E ) m 2 + ( -4K + 3 E )m+2 ( K-E) 2" 14 .x 2 [ ( K-2E ) m 2 + ( - 3 K+2E ) m+2 ( K-E ) ]

where .x is given by eq.(A9.1.6) and where K and E are complete elliptic integrals. Numerical evaluation of eq.(A9.2.12) shows that averaging predicts no limit cycle if k0 1 /k l l > 1/7 or if k 01 /k l l <0, and a single limit cycle for 0
f

F (x,x. )

The following MACSYMA program computes the partial derivatives in eq.(A9 . 1 . 18) when x(t) is given by (A9 . 1 .7)-(A9.1.9), and simplifies the result: depends( [cn,sn,dn] , [u,m] ) ; depends(u,[t,a2 ,b]) ; depends([a1 ,a2] ,m); derivs: ['diff(cn,u)=-sn*dn,'diff(sn,u)=cn*dn,'diff(dn,u)=-m*sn*cn, ' diff(cn,m)=sn*dn*f, ' diff(sn,m)=-cn*dn*f, ' diff( dn,m)=m*sn*cn*f-sn A2/(2*dn)] ; idents: [sn=sqrt ( 1-cnA2), dn= sqrt( 1-m+m*cnA2)]; x:a1*cn; xt :diff(x,t ) ;

A VERA GING

147

xt :xt ,derivs ; xbt:diff(xt,b ) ; xbt :xbt ,derivs; xb:diff(x,b ) ; xb:xb,derivs ; xm:diff(x,m); xm:xm,derivs; xmt :diff(xt ,m); xmt:xmt,derivs; dmdt: -xb*e*F/(xm*xbt-xmt*xb); dmdt:ratsimp( dmdt) ; dmdt:dmdt ,idents,diff; dmdt:dmdt ,derivs; dmdt:dmdt ,u=a2*t+b,diff; dmdt:dmdt, a1 = (2*m/( 1-2*m) f ( 1 /2) ,a2 = ( 1/(1-2*m)t ( 1/2),diff; dmdt:factor( ratsimp( dmdt)); The final expression produced by the program may be written: ill

(A9.3. 1)

= - /liIi ( 1-2m) 2 f F sn dn

where F = F(x,x) = F(a 1 cn, - a 1 a2 sn dn) . For first-<>rder averaging, we may replace the rhs of eq.(A9.3.1) by its average value taken over one period of length 4K(k):

(A9.3.2)

ill

= - /liIi ( 1-2m) 2

4K

f

kJ

sn dn F du

o

An equilibrium point of the averaged ill equation, eq.(A9.3.2) , corresponds to a limit cycle in the original equation (A9. 1 .2). Thus if m = mO is a root of the equation

148

(A9.3.3)

A VERA GING

J °

4K sn dn F du = °

then the averaged equation predicts that for small f, a limit cycle coincides with the solution (A9 . 1 . 7 ) associated with a value of m = mO . For an arbitrary polynomial F(x,:ic) , the limit cycle integral condition, eq.(A9.3.3), may be efficiently evaluated by using the following MACSYMA program: F:read("Enter F(x,y), where y=dx/dt" ) j /* eliminate even powers of y * / Fl :ev(F,y=-y) j F2: (F-Fl)/2j /* form integrand of limit cycle integral, eq.(A9.2.10) * / integrand:F2*sn*dn,x=al *cn,y=-al *a2*sn*dn$ /* use identities to express sn,dn in terms of cn * / idents: [sn=sqrt( l-cn A 2), dn= sqrt(l -m+m*cn A2)]j integrandl :expand( ev(integrand,idents))$ /* set up rules for evaluating the integral of cnAr */ 1* KK = K(k) , EE = E(k) * / c[O] :4*KKj c[2] :4*(EE-{ 1 -m)*KK)/mj c[r ] :=(r-2)*(2*m-l ) / ((r-l )*m )*c[r-2]+( r-3)*(1 -m) / (( r-l )*m )*c[r-4] j /* perform the integration by replacing cn Ar by c[r] * / max:hipow(integrandl ,cn)j for i:O thru max step 2 do b[i] :coeff(integrandl ,cn,i)j result :sum(b[2*i]*c[2*i] ,i,O,max/2)$ /* substitute the parameters al ,a2 in terms of m * / result l:result, al = (2*m/(1-2*m) r (1/2) ,a2 = ( 1/(1-2*m) r ( 1/2),diff$ /* simplify the result * / result2:ratsimp( resultl ) j

A VERA GING

149

As an example let us take (Coppola and Rand) (A9.3.4)

x. . + x + x3 = f ( k01 x. + k2 1 x2x· + k03 x. 3 )

in which k 0 1 ,k2 1 and k03 are given parameters. The result of the foregoing program may be expressed in a convenient form by solving for k01 : (A9.3.5)

where K and E are complete elliptic integrals. Eq.(A9.3.5) is the condition for a limit cycle of the form of eqs.(A9. 1 .7)-(A9.1 .9) to occur in the system (A9.3.4).

p

150

A VERA GING

Appendix 10: Pade Approximants AIO . I Pade approximants Pade approximants are a tool for improving the convergence of perturbation series (see Bender and Orszag, van Dyke). Given a truncated power series of the form (AIO . 1 . I ) a Pade approximant is a rational fraction which has the same truncated Taylor series as f up to O( fn ) . For example, the truncated series

(AIO. 1.2)

1+

2

� - .g-

+ 44 has the Pade approximant 3 ff +

since a Taylor expansion gives

The process of finding a Pade approximant involves solving a system of linear equations (Bender and Orszag) , but MACSYMA can do this for you automatically. As a model example, suppose that the function /f+f plays the role of an exact solution to a perturbation problem. We normally do not have access to the exact solution, but rather we solve for its perturbation expansion, the role of which in this case is played by the following Taylor series:

(AIO . 1 .4) Now this series diverges for I f l > 1 (since /f+f has a branch point at f = -1 ) . For

example, at f = 2, the "exact solution" gives /f+f = .fJ = 1. 7320, while the

A VERA GING

�

151

Ifik

f 6 =0.9375. Now we attempt to "perturbation solution" gives 1+ _ 0 0 0 _ improve the convergence of the series by using a Pade approximant. In MACSYMA, the command is PADE ( taylor series, m, n ) ;

(A10. 1.5)

where m is the degree of the polynomial in the numerator of the Pade approximant , and n is the corresponding value for the denominator. These should b e chosen s o that m+n equals the degree of the truncated Taylor series. In this example we take m = n = 3 and obtain: (C4) sqrt ( l+e ) ; (D4)

SQRT(E + 1)

(C5) taylor ( % ,e,0,6 ) ; 2 3 4 5 21 E6 + ... (D5)!T! 1 + � _ � + L _ � + � _ 2 8 1 64 1 2 8 25 6 1024 (C6) pade( % , 3 ,3 ) ; 2 3 (D6 ) [ 7 E + 56 E + 1 1 2 E + 64] E 3 + 24 E 2 + 80 E + 6 4 (C7) [D4,D5,D6] , e=2, numer; (D7) [ 1 . 73205, 0.9375, [ 1 . 73171]] That is, at f = 2 the Pade approximant gives 1.73171, and thus very nearly recovers the exact solution from the divergent perturbation series!

I

i

I I

,

i I

I

i � !

I

152

A VERA GING

Besides offering an improved approximation, there is another important use for Pade approximants. The zeros of the denominator of the Pade approximant often give useful information about the location and nature of the nearest singularity in the perturbation solution. To continue with the above model example, the zeros of the denominator of the Pade approximant in (D6), 10 3 +2410 2 +SO€+64=0, all lie on the negative 10 axis: ( CS) DENOM(PART(D6,1))i

(DS)

E 3 + 24 E2 + SO E + 64

(C9) ALLROOTS(%)j (D9) [E = - 1.23191, E = - 2.57242, E = - 20. 19566] The distance from the point of expansion ( 10=0) to the nearest root is 1.23, and this gives an approximation for the radius of convergence ( = 1) of the series (AlO.1.4). It turns out that in this example the closest zero to the point of expansion ( 10=0) approaches the branch point 10=-1 as the number of terms increases. A10.2 Application to Mathieu ' s equation Although the discussion so far has involved the model perturbation example (A10.1.4), Pade approximants often give excellent results in real perturbation problems. As an example, we take the transition curve in Mathieu ' S equation which goes through the origin in the 8-10 plane: (A10.2.1)

XO +

(8 +

HOS t) x = 0 ,

The first step is to obtain an expression for the transition curve using perturbation methods. As discussed in section 3.S, regular perturbations offers an efficient way of doing this. A MACSYMA program is presented in the section A10.3 which produces the following expansion for the desired transition curve:

A VERA GING

2

4

6

153

f 8 + O ( f 10 ) 294912

o = _ � + � _ � + 68687

(A10.2.2)

32

2

144

In order to evaluate the accuracy of this expression, we can numerically integrate eq.(A10 . 2 . 1 ) for a series of values of 0 and f, noting by inspection which values produce bounded versus unbounded solutions. (Greater accuracy can be achieved by using Floquet theory instead of "inspection" , see Stoker.) E.g. , we may fix 0 and vary f until a transition between stable and unstable behavior is encountered. For o = -0.4, for example, the transition curve in question passes close to f = 1 . 035. Substituting this value of f into the perturbation expansion (A10.2.2) gives 0 = 0.225, a value which is far from the corresponding numerical integration value of 0 = 0.4. -

-

Now we use Pade approximants : (C7) TCj 2 6 4 8 (D7)/T/ _ � + � _ 29 E + 68687 E + . . . 2949 1 2 144 32 2 ( C 8 ) PADE(TC,4,4)j

(D8)

[_

209 2 82 4 E4 + 2244 09 6 E2 882545 E4 + 6 14 9 232 E2 + 4488192

which gives the value value of 0 = 0.4. -

0=

-

0.398 for f

= 1.035, which is very close to the numerical

We may also use the Pade approximant to estimate the radius of convergence of the perturbation power series (A10.2.2):

A VERA GING

154 (C9) DENOM(PART(D8,1));

(D9) 882545 E4 + 6149232 E2 + 4488192 (CI0) ALLROOTS(%); (DI0) [E = 0.91014 %1, E =

-

0.91014 %1, E = 2.47775 %1, E =

-

2.47775 %1]

The root of the denominator (D9) of the Pade approximant (D8) which is closest to the expansion point f=O is a distance 0.91 away. Thus we expect a radius of convergence in f of about 0.9. In fact, taking more terms gives a value for the radius of convergence of about 0.73 (Hunter and Guerrieri) . Thus the value we used above, namely f = 1.035, was outside the radius of convergence, which explains why the perturbation series (A I 0.2.2) did not deliver a meaningful value. However, note how well the Pade approximant improved the convergence of this divergent series. See Fig.3.5.

A VERA GING

1 .0

0

/

\J,

'\

15 5

�

--

\

\ \

\

o

- 0 .4

o

Fig.3.5. Perturbation series (A I0 .2.2) (S) and Pade approximant (D8) (P) displayed in the 0-£ plane. Numerical integration of Mathieu ' s equation (AlO.2. 1 ) gives a stability transition curve which is indistinguishable in this Figure from the Pade curve P . AI0.3 MACSYMA program for regular perturbations on Mathieu ' s equation The following program produces eq.(AlO.2.2) for the transition curve through 6 = " = 0 in the 0-" plane. This particular transition curve is obtained by starting the perturbation method off with 60 = 0 and Xo = 1. See Rand (1984) for a variety of perturbation programs for efficiently computing the transition curves in Mathieu ' s equation.

156

A VERA GING

/* regular perturbations on Mathieu ' s equation * / /* set up expansions * / trunc:read("truncation order")$ depends(x,t ); xx:sum(x[i] *eA i ,i,O,trunc); delta: sum( d[i] *eAi,i,O,trunc); results : [x[O] = I ,d[O]=O] ; / * prepare null function for later * / nUll(t ) : =O ; /* plug into d.e. and collect terms * / del :diff(xx,t ,2)+( delta+e*cos( t ) )*xx$ de2:taylor( del ,e,O,trunc)$ for i:O thru trunc do eq[i] :coeff( de2,e,i)$ /* main loop * / for i: l thru trunc do ( templ :ev( eq[i] ,results), /* apply trig identities * / temp2:expand( trigreduce( expand( tempI» ), /* pick off secular terms * / temp3:subst(null,cos,temp2) , temp4:ev(temp3 ,x[i]=O), /* solve for d[i] * / temp5:solve( temp4,d[i]) , results:append(results,temp5), temp6:ev( temp2,results), temp7:ratsimp( temp6) ,

A VERA GING

/* use o.d.e. package to find x[i] * / temp8:ode2( temp7 ,x[i] ,t) , / * zap arbitrary constants * / temp9:ev( temp8, %k1=O, %k2=O), results:append(results,[temp9]))$ /* end of main loop * / /* output results * / deltafinal:ev( delta, results ) j

157

158

A VERA GING

Appendix 1 1 : The Andronov Bifurcation In section 3.6 we saw how limit cycles were created via Hopf bifurcations as a parameter was tuned. In this appendix we look at another common scenario in which limit cycles are created. We begin with an example, a generalization of van der Pol ' s eq.(3.5.1): (A1 1 . 1 )

x. . + X - f ( p, - x2 ) x. = O

Eq.(A11.1) exhibits a Hopf bifurcation as p, passes through zero (quasistatically) , for fixed f. (It is a supercritical Hopf, and the limit cycle exists for p, > 0 and is stable, cf.Fig.3.2. The amplitude of the limit cycle grows like .fii, . ) On the other hand, if p, is held fixed and f i s permitted t o vary, we have a different situation. E.g. if P, = 1 , we have van der Pol ' s equation, which was investigated in section 3.5. There we found that a limit cycle of amplitude � 2 occurs for small f :f. o. That is, the limit cycle is born with order(l) amplitude, in contrast to the Hopf, where the limit cycle is born with small amplitude. Moreover, the limit cycle here occurs for both positive and negative f, whereas in the supercritical Hopf the limit cycle occurs only for p, > o. What is happening here is that for f = 0 , the x-x phase plane is filled with the circular solutions of the simple harmonic oscillator x ' + x = o . As f is tuned away from 0, one of these circles gives rise to a limit cycle. This situation was studied by Andronov and his colleagues (see References) , and will be referred to as an Andronov bifurcation. (This nomenclature may be objected to on the grounds that the phenomenon is not a bifurcation, Le. , it does not involve the gradual emergence of a new steady state from an already existing one. However, changing f from zero to a non-zero value does produce a change in the topology of the phase space, a property shared by phenomena traditionally called bifurcations. )

�

I

A VERA GING

159

I

I

The condition for such a bifurcation may be given as follows. Suppose the system is a perturbation off of a general two dimensional Hamiltonian system:

I (AI 1.2) y=

-w. +

I f

I

I

g(X,y)

Let C h be a trajectory of the unperturbed f = 0 system corresponding to energy h, and let Rh be its interior (assumed simply connected) . Then in order for a limit cy cle to exist in the perturbed flow, the Andronov condition (valid to first order in f ) is: (AI 1 .3)

I(h) =

JI [� + �]

dx

dy = 0

h

This condition follows from Green ' s theorem, (Al I A)

=

t [�*-*�]

dt = 0

h

The approximate nature of the condition (Al I .3) (which is valid for small f) is due to the substitution of the unperturbed trajectory C h for the actual trajectory (which is unknown). Condition (Al I .3) is equivalent to the results of first order averaging. As an example, consider van der Pol ' s eq. (AI l. I) with p. = 1 , for which H = (x2 +y2 )/2, f = 0 and g = (1 - x2 )y. Condition (A I 1.3) becomes

I

160

A VERA GING

(AI l .S)

I(h) =

I I (1 - x2 ) dx dy = 0

Rh

where Rh is the interior of a circle (of unknown radius a) . In polar coordinates, (AI l .S) becomes 27r

(AI l .6)

I=

a

I dO I (1 - r2 cos 2 0) r dr = 0 o

0

which is easily evaluated as: 2 I = 7r a2 (1 - 4a ) = 0

(AI l. 7)

giving the well known small f result a = 2. In the case that both I(h) and its first derivative dI/dh vanish, the condition (AU.3) predicts a double root, and the limit cycle is degenerate: it is semi-stable. An unfolding of the singularity (by a small additional perturbation of the system) can now produce two limit cycles, one stable and one unstable. As an example, consider the system (AI l .S)

x + x + f x ( I + a x· 2 + /3 x4 ) = 0 .

.

.

Here the Andronov integral (AI l.3) turns out to be (AI l .9) Solving (AU .9) simultaneously with dI/dh = 0 gives (AI l. IO)

.

u

A VERA GING

16 1

These results imply, for example, that in the small f limit, for h = 2, a = -2/3 and f3 = 1/10, a semi-stable limit cycle exists with approximate amplitude a = /lfi. = 2 , in agreement with numerical integration, see Fig.3.6. By changing a and f3 slight ly in an appropriate fashion, it is possible to produce an example with two limit cycles, s ee Exercise 8 . Similarly i f I and its first and second derivatives all were t o simultaneously vanish, an unfolding of the singular vector field would offer an example with three limit cycles, and so on.

L

.

X

Fig. 3.6. Phase plane flow for eq. ( A11.8 ) with a = -2/3, f3 = 0.1 and f = 0 . 1 . The approximate position of the semi-stable limit cycle L is shown dashed.

CHAPTER FOUR METHODS FOR PREDICTING CHAOS

4. 1 Introduction In this Chapter we discuss two perturbation approaches for predicting chaos in systems defined by differential equations. The first approach, Melnikov ' s method, will be applied to conservative one degree of freedom systems which include separatrix loops, and which are perturbed by small forcing and damping. The result of Melnikov ' s method is a bound on the parameters of the problem such that chaos is predicted not to occur. The second approach, Chirikov ' s method, will also be applied to conservative one degree of freedom systems which are perturbed by small forcing, and which exhibit at least two resonances. The result of Chirikov ' s method is a bound on the parameters of the problem such that the chaos associated with each resonance remains locally restricted to a small region of phase space (in contrast to global chaos which involves large regions of phase space. ) In order to understand the treatment of Chirikov ' s method, the reader needs to know how use Lie transforms, see Chapter Two. In the case of Melnikov ' s method, the idea is to show by perturbation expansions that there exists an intersection of the stable and unstable manifolds of an equilibrium point in a two-dimensional map M. This, by the Smale-Birkhoff theorem, implies there is a horseshoe in the map M, the presence of which implies there exist periodic motions of all periods, as well as motions which are not periodic. The horseshoe also exhibits sensitive dependence on initial conditions. A detailed treatment of these topics has been given in Guckenheimer and Holmes. We present here a brief outline

- 162 -

METHODS FOR PREDICTING CHA OS

163

of the main ideas as background for the perturbation method. The Smale-Birkhoff theorem says that if a two dimensional ( differentiable ) map M has a saddle equilibrium a whose stable and unstable manifolds intersect transversely at some point b, then M contains a horseshoe. See Fig.4. l .

u� .

1

g(V) 11

III

Fig.4. l . The Smale-Birkhoff theorem. i ) Point a is an equilibrium point for a two-dimensional map M:x -+ f(x) . S and U are the stable and unstable manifolds of a, respectively. It is assumed that S and U intersect in another point , here called b. ii ) A small region R containing the point a is mapped forward by n iterates of the map M to fn ( R ) . The forward iterates of R hug the unstable manifold U, and n is

chosen large enough so that fn ( R ) contains point b. Similarly, the backward iterates of R hug the stable manifold S , and m is chosen large enough so that rm ( R ) contains point b. iii ) Letting V = rm ( R) , we find fn ( R) = fn+m ( V ) . Defining g (x) = fn+m (x ) , we obtain a region V which is horseshoe-mapped to g ( V ) .

We next turn to horseshoe maps. The reasoning which permits us to leap from the presence of a horseshoe to chaotic dynamics is a one-to-one correspondence between the horseshoe dynamics and a " shift on symbols" . It is shown in Guckenheimer and Holmes that the set of points A which is left invariant by a horseshoe map can be identified with the set � of all doubly-infinite sequences of two symbols, say l ' s and

164

METHODS FOR PREDICTING CHA OS

O ' s, a typical example of which is: (4 . 1 . 1 )

. . . 100101 1100. 111 1001000 . . .

The invariance of the set A means that after one iteration of the map, each point i n A is mapped to some other point in A. Since each point in A can be identified with a sequence of 1 ' s and O ' s in E , the dynamics of the horseshoe on A sets up a corresponding dynamic on E. The amazing thing is that the dynamics on E is extremely simple ( while the dynamics on A seems incomprehensibly complicated. ) The dynamic on E takes a given sequence to a sequence in which the " dot " has been moved to right one place. E.g. , the sequence in ( 4. 1 . 1 ) would be mapped to ( 4.1.2 )

. . . 10010111001.11 1001000 . . .

Without going into details, i t may be said that the reason this works i s that the scheme for identifying a point in A with a sequence in E involves coding the point ' s entire history and future, in terms of which horizontal or vertical strip it is located on after n iterations of the map. See Fig.4.2.

Fig.4.2. Schematic diagram showing iterates of the horseshoe map.

METHODS FOR PREDICTING CHA OS

Once the dynamics of the horseshoe map are identified with a shift on symbol sequences, the dynamics become easy to understand. To begin with, every periodic motion in A corresponds to a periodically repeating symbol sequence. For example, (4. 1.3)

... 1101 10110110. 110110110 1 10 . . .

i n which 1 1 0 i s repeated ad infinitum, has period 3. O n the other hand, there are an uncountable infinity of sequences which are not periodic. These correspond to motions which wander around the horseshoe without ever settling down on a periodic motion. An important property of the correspondence between A and }; is that it is continuous, i.e. , the closer points are to each other in A, the more terms their symbol sequences have in common near the dot, i.e. , away from their tails. Thus although two neighboring points may start near each other and stay near each other for a large number of iterations, they may eventually move far apart . This situation, called sensitive dependence on initial conditions, is often used as a characteristic of chaos. In the horseshoe this results from the stretching and folding of the map. 4.2 Melnikov ' s Method We begin this section with a brief summary: Melnikov ' s method for predicting chaos ( Melnikov, Guckenheimer and Holmes ) is a perturbation method which involves perturbations off of a dynamical system which includes a separatrix loop ( also known as a saddle-loop or a homoclinic orbit ) . The separatrix loop in the f = 0 system will generally be "broken" when the perturbation is applied. The question of whether or not chaos occurs for a particular system depends upon what happens to the broken pieces of the separatrix loop ( the stable and unstable manifolds of the saddle) , i.e. , whether they intersect or not. The method will be applied to a class of systems of the form: (4. 2 . 1 )

x ' - x + N ( x) = f g ( x,x,t )

165

p 166

METHODS FOR PREDICTING CHA OS

where f is a small parameter, where the perturbation g is of the form: g(x,x,t) = A cos wt - B x

(4.2.2)

and where N(x) is a nonlinear function of x such that i) N(O) = 0 , i.e. , the origin x = x = 0 in the f = 0 system is an equilibrium point (a saddle) , and ii) the f = 0 system has a separatrix loop through the origin. See Fig.4.3 . .

X

--�-------+--

x

Fig.4.3. Separatrix loop in the f = 0 version of eq.( 4.2. 1 ) . Although the autonomous f = 0 system can be adequately described b y using the x-x phase plane, the system (4.2 . 1 ) for f > 0 is nonautonomous and requires another dimension in order that its trajectories not intersect. We therefore append to (4.2. 1 ) the dummy equation t = 1 and we imagine the dynamics of (4. 2 . 1 ) as taking place in

!7r

periodic, the vector field in x-x-t space. Since the forcing term f A cos wt is . . . . p h ase space IS 2 In · h s hows t h at t he x-x-t x-x-t t , whIC space repeats every w topologically equivalent to \R2 x S . This permits us to set up a Poincare map T

7r

METHODS FOR PREDICTING CHA OS

corresponding to a surface of section t = 0 mod

!7r, see Fig.4.4.

x

t x

2 11

a "'------

x

0)

Fig.4.4. The Poincare map T takes a point a to a point Ta Now when the E = 0 version of system ( 4.2. 1 ) is viewed with respect to the x-x-t phase space, the integral curves in the x-x phase plane lie on invariant cylinders, see Fig.4.5.

167

>

168

METHODS FOR PREDICTING CHA OS

t

Fig.4.5. The f = 0 separatrix loop of Fig.4.3 lies on a cylinder when embedded in x-x-t space. The saddle point at the origin in the x-x plane becomes a line x = x = 0 in the x-x-t phase space, which is topologically equivalent to a circle ( i.e. to a periodic orbit ) since the planes t = 0 and t =

�1r are identified mod �1r .

Thus the saddle at

the origin in the x-x plane becomes a saddle-like fixed point in the Poincare map T, also at the origin. When f is permitted to vary from zero to a small positive value, the principle of structural stability permits us to conclude that the saddle at the origin in the Poincare map T for f = 0 continues to persist for f > 0 as a saddle in T. The perturbed saddle will lie close to the origin, but not generally at the origin. When viewed in the x-x phase plane, the saddle at the origin in the f = 0 system persists in the f > 0 system, but now corresponds to a periodic motion which lies in the neighborhood of x = x = O. See Fig.4.6.

169

METHODS FOR PREDICTING CHA OS

x

t To =o

o

----++---+--

X

2 11 W

FigA.6. For f > 0, a periodic motion replaces the saddle at the origin in the f = 0 system. The periodic motion is shown as a fixed point a in the Poincare map T. The right-hand figure shows both the projection of the periodic orbit onto the x-x plane, as well as the corresponding point a in the Poincare map. For f small enough, the local topology of FigA.5 in the neighborhood of the separatrlx persists. That is, the two 2-dimensional surfaces which pass through the saddle in FigA.5 for f = 0, have a counterpart in the f > 0 system, see FigA. 7. These surfaces, called the stable and unstable manifolds of the saddle-like periodic motion, contain motions which approach the saddle as t goes to + III (on the stable manifold) or as t goes to - Ill (on the unstable manifold).

170

METHODS FOR PREDICTING CHA OS

x

t

x

x

2 1\ W

Fig.4.7. The stable manifold of the f > 0 saddle-like periodic motion of Fig.4 .6 . Only a small piece of the stable manifold i n the neighborhood o f the saddle i s shown. Note that the flow on the stable manifold of the periodic motion in x-x-t space (left ) generates a map on the stable manifold of the corresponding equilibrium point in the Poincare map ( right ) . Although the local structure of the stable and unstable manifolds persists when f is increased from zero to a small positive value, their global structure (Le., the separatrix loop ) is structurally unstable and will not in general survive the perturbation. For f > 0, we expect the separatrix loop in the Poincare map T to "break" , and to be replaced by one of two possibilities, depending upon whether or not the stable and unstable manifolds intersect. they intersect (transversely ) , a horseshoe is present in the Poincare map, and chaos results. See Fig.4.8.

If

METHODS FOR PREDICTING CHA OS

•

X

171

•

X

Fig.4.8. Upon perturbation, the separatrix loop of Fig.4.3 can break into two different types of Poincare maps. If the stable and unstable manifolds intersect, a horseshoe results and we have chaos (right) . No chaos occurs if the stable and unstable manifolds do not intersect (left). The idea of Melnikov ' s method is to be able to tell whether or not the stable and unstable manifolds intersect by perturbing off of the f = 0 separatrix loop. We begin by considering a perturbation expansion for a solution x + (t) along the perturbed stable manifold: ( 4.2.3) x + (t) approaches the periodic motion corresponding to the saddle in the f = 0 system, see Fig.4.6.

As t

-I W ,

Similarly, we may write a solution x-(t) along the perturbed unstable manifold in the form:

-�-----

p . -..:< .

172

METHODS FOR PREDICTING CHA OS

(4.2.4) As t

-+

-

x ( t) approaches the same periodic motion as does x+ (t). -

00 ,

Note that a plus sign superscript refers to the stable manifold and a minus sign superscript refers to the unstable manifold. Since the f = 0 separatrix loop is common to both stable and unstable manifolds, a solution on it , xO (t), receives no superscript . Rewriting eq. (4.2. l) as a first order system, (4.2.5)

x=y,

y = x - N(x) + f g(x,y,t)

we substitute (4.2.3), (4.2.4) into (4.2.5) and collect terms, giving: (4.2.6) (4.2.7) where (4.2 . 7) holds for both plus and minus superscripts on x l and y l ' respectively. One approach to dealing with the perturbation equations (4.2.6), (4.2.7) would be to solve for x l (t) and Y l (t). Although possible, such a direct course is algebraically complicated. Melnikov ' s method offers an ingenious alternative. It answers the question of whether or not the stable and unstable manifolds intersect, without finding an expression for xl (t) or y 1 (t). With x+ (t) and x-(t) defined as in (4.2.3) and (4.2.4) , the vector

METHODS FOR PREDICTING CHA OS

goes from a point on the perturbed unstable manifold to a point on the perturbed stable manifold, both at time t , and both f-dose to a point (xO (t),yO (t)) on the

+ +)

unperturbed separatrix loop. (For definiteness, (x ,y may be defined to be the point on the stable manifold which lies on the normal to the unperturbed separatrix loop at time t, and similarly for (x-,y-) on the unstable manifold, although this choice never explicitly enters the Melnikov integral.)

+

In order to determine if there is an intersection between the stable and unstable manifolds , Melnikov takes a dot product of (x x-, y +- y ) with a vector which is -

-

normal to the unperturbed separatrix loop at time t . Since (:ic(t ) , y(t)) is tangent to the separatrix loop, (y(t) , -:ic(t )) is a vector normal to the separatrix loop. Moreover, these derivatives are given by eq. ( 4.2.6) when f = O. Thus Melnikov defines the key quantity .6.(t) as: (4.2.8) where the subscript 0 corresponds to the unperturbed f = 0 solution. See Fig.4.9. The function .6.( t ) characterizes the size of the gap between the stable and unstable manifolds . If .6.(t) vanishes for some t, then the stable and unstable manifolds intersect to O ( f2 ) .

173

ji? 174

METHODS FOR PREDICTING CHA OS

(y, - x)

( x ,;)

y

--�-+--.-----�--�-- X

Fig.4.9. A plane t = constant, showing its intersection with the perturbed ( € > 0 ) stable and unstable manifolds (dark lines) and the unperturbed ( € = 0) separatrix loop. A convenient expression for .6.(t) may be derived as follows. First substitute (4.2.3),( 4.2.4) into (4.2.8) to obtain: (4.2.9) Expanding eq. (4.2.9), Melnikov writes: (4.2. 10) where

1 METHODS FOR PREDICTING CHA OS

Differentiating (4.2. 1 1 ) , we find

Substituting (4.2.6),(4.2.7) into (4.2. 13), obtain

(4.2. 14)

Now a minor miracle happens: nearly everything in (4.2. 14) cancels out, and we are left with:

(4.2. 15) Eq. ( 4.2. 15) may be integrated from an arbitrary value of t, call it

r, to

m

to give:

(4.2.16) Note that A + ( m ) = 0 from (4.2. 11) since the vector (xO-N(xO ) ' Y O ) , describing motion along the saddle loop, approaches the origin as t goes to m . Therefore -

(4.2. 1 7)

A + (r) =

Jr yo g(xo ,yo,t) dt m

p 176

METHODS FOR PREDICTING CHA OS

By treating � -(t) in eq. (4.2. 12) in an analogous way, we can show that

� -( r) =

(4.2.18)

J

r -m

yo g(xo ,yo ,t) dt

Thus eq. ( 4.2. 10) becomes

(4.2. 19)

�(r) =

(J

ro

-m

2 Y o g(xo ,y o ,t) dt + 0 ( ( )

The Melnikov integral in eq.(4.2. 19) is to be evaluated on a solution (xO (t ) ,y O (t)) which takes on an arbitrary but definite initial position on the unperturbed separatrix loop at time t = r. Since the = 0 system is autonomous, this means that xo and yO will be functions of t-r.

(

Once the Melnikov integral has been evaluated in (4.2. 19) and an expression for � ( r) has been found, the chaos question reduces to whether or not � ( r) = 0 for some time r. 4.3 Example

As an example, we follow Melnikov and take N(x) = becomes:

2

- r ' whereupon eq.(4. 2 . 1 )

2 . x. . - x - x 2 = ( ( A cos wt - B x )

(4. 3 . 1 )

Our task is to evaluate � ( r) , which from eq.( 4.2. 19) becomes:

(4.3.2)

� ( r) =

(J

ro

xO (t) (A cos wt - B xO (t)) dt + O( f2 ) -m

1 METHODS FOR PREDICTING CHA OS

177

We begin by computing xo . For f = 0 , eq.(4.3.1) becomes: (4.3.3)

x2 O x. . - x - r =

Eq. ( 4.3.3) has equilibria at x = 0 and x = - 2. It has a first integral of the form: 3 x· 2 x2 - x H=r ( 4.3.4 ) - r 6 = constant

I

The integral curve which passes through the origin is:

I,

x3

x· 2 = x2 + 3

(4.3.5)

r

,I \

See Fig.4.10. Note that the separatrix loop (4.3.5) intersects that x-axis at x = -3 .

t «

'I

I

!

X

/

/

/ 'I

x

Fig.4. lO. Phase plane for x ' - x

-3

< x <3.

2

- � = O. Region displayed is -4 < x < 2 ,

,�

178

METHODS FOR PREDICTING CHA OS

Separating variables in (4.3.5) gives the integral:

J

(4.3.6)

j

dx x2 +

= +

f

I dt

.The integral (4.3.6) can be evaluated in closed form:

- 2 arctanh

(4.3.7)

J

1 +

x(r)

�

x(t )

= + (t-r)

-

We choose x( r) = -3 without loss of generality . .Then solving (4.3.7) for x(t) gives: (4.3.8)

[

x(t) = xO (t) = 3 tanh 2

[�] - 1] =

Differentiating (4.3.8) gives an expression for

- 3

[ ]

COSh2 t 2r

xo along the separatrix loop:

[�]

s i nh . xO (t) = 3 ---COS h3

(4.3.9)

[�]

Substituting (4.3.9) into (4.3.2) gives

(4.3.10)

� ( r) = f

I

co

-m

s i nh

[�]

3 --- A cos COS h 3

[�]

Setting z = t-r in (4.3. 10) and writing

wt

s i nh

[�]

CO S h3

[�]

- 3B

dt + O ( f2 )

4

179

METHODS FOR PREDICTING CHA OS

(4. 3 . 1 1 )

cos wt = cos w(z+ r) = cos wz coswr - sin wz sin wr

gives

( 4.3.12)

A ( r) = f

J

00

-w

. S I nh 2"z 3 --cos h3

�

s i nh A (cos wz coswr - sin wz sin wr) - 3B

�

dz + O( f2 )

cos h 3 2"z

�

Note that the term sinh cos wz / cosh 3 function of z. Rearranging (4.3. 12) gives

� integrates to zero since it is an odd

(4.3.13) where (4.3. 14)

11 =

J

00

-w

. S I nh 2"z

z

2"

sin wz dz cos h3 2"z

and

--- dz

It remains to evaluate the integrals 1 1 and 12 , It turns out that 1 2 may be evaluated as an indefinite integral, but for 1 1 we need to use contour integration and the method of residues. We evaluate 1 2 by using MACSYMA (in which %E stands for e, inf for + for - (0 ) :

00

and minf

180

METHODS FOR PREDICTING CHA OS

(C1) integrate(sinh(z/2) ' 2/cosh(z/2) ' 6 ,z);

(D1)

_

2Z Z 3Z 32 ( 1 5 %E 5 %E + 5 % E + 1 ) 3Z 5Z 4Z 2Z Z 60 %E + 300 %E + 6 00 % E + 600 %E + 300 %E + 60

(C2) limit(d1 ,z,inf) ; (D2) (C3) limit (d1 ,z,minf) ;

-

o

(D3) That is, (4.3. 15) In order to evaluate 1 1 ' we write sin wz = 1m ei wz (see Greenberg, for example) , giving

(4.3.16)

Now we choose a contour in the complex z-plane as shown in Fig.4. 1 1 .

METHODS FOR PREDICTING CHA OS

1M

181

Z

-R

R

Re

Fig.4. 1 1 . Contour chosen to evaluate the integral J 1 in the limit as R -i represent singularities.

z

m.

The X ' s

The contribution on the semicircle goes to zero as R -i m , so that (4. 3 . 1 7)

J 1 = 2 7!i

l residues in the upper half plane ;

The integrand of J 1 has singularities where cosh vanishes. Setting z = u + iv, we find (4.3.18)

. U . v U cos 2'v + 1. SIll cos h 2'z = cosh � h 2' SIll �

; to vanish, its real and imaginary parts must vanish. Since cosh � 0 for real u, we must have cos ; = 0, or v = for n odd. Then sin ; '* 0 and sinh � = 0 for the imaginary part of cosh ; to vanish, i .e. , u = o. Thus the

In order for cosh >

ll1r

singularities of the integrand of J 1 occur along the imaginary axis at z = n 7!i for n odd.

182

METHODS FOR PREDICTING CHA OS

We may use MACSYMA to compute the residues in eq.(4.3. 17): (Cl) declare(n,odd)$ (C2) sinh(z/2)*%e � ( %i*w*z)/cosh(z/2 r 3$ (C3) residue(% ,z,%i*n*%pi)j

(D3)

2 - %PI N W 4 W %E

That is, the residue of the integrand of J 1 at z = ll1ri for n odd is 4 w2 e-ll1rw Eq.(4.3. 17) becomes: (4.3. 19)

J1 = 2 m

2

4 w2 e-n1f'W

n = I ,3,5 . . .

=

. 4 1f'W2 1 s i nh 1f'W

-2 1f'W where we have used the geometric series 1 + a + a2 + . . . = � � -a with a = e Using (4.3. 19) in (4.3. 16) we obtain:

( 4.3.20)

2 I 1 - 41f'w s i nh 1f'W _

Substituting eqs . ( 4.3.20),( 4.3.15) for 1 1 and 1 2 into eq.( 4.3. 13), we obtain the following expression for b.( 7) :

METHODS FOR PREDICTING CHA OS

(4.3.2 1 )

2 � ( T) = -3 f [A sin WT 47rw + 3B s i nh 'TrW

�l + O( (2 )

For an intersection of the stable and unstable manifolds, we set � ( T) = 0 and seek a (real value) for T. Eq.(4.3.21) gives: ( 4.3.22)

· WT = sm

2 B s i nh 'TrW S A 'Tr )

For a real solution T, ! sin WT ! � 1 . From (4.3.22) this requires that the ratio of the forcing amplitude A to the damping coefficient B (cf.eq.( 4.3.1)) be larger than the critical ratio: (4.3.23) If AjB is larger than (AjB) cr then Melnikov ' s method predicts that the Poincare map of (4.3. 1 ) , for small f, will contain horseshoes. See Fig.4. 12.

183

p

184

METHODS FOR PREDICTING CHA OS

8 A B

o

w

2

FigA. 12. The result of Melnikov ' s method: Eq.(4.3.23). Horseshoes are predicted to occur for parameters A ,B , and w which lie above the curve. It is important to remember that the existence of a horseshoe does not imply the existence of a chaotic attractor. Although the horseshoe itself is chaotic, its presence may show up as transient chaos if it coexists with a periodic attractor. Numerical and experimental approaches to chaos (see Moon) have shown that the Melnikov criterion generally represents a lower bound on chaos. For example in eq.(4. 3 . 1 ) , for a given value of w, if A/B is smaller than the critical value (4.3.23) , then we see no chaos. For values of A/B which are greater than (A/B) cr ' we see either transient chaos or escape to infinity (Thompson) . See Appendix 12 on Lyapunov exponents .

r METHODS FOR PREDICTING CHA OS

4.4 Vakakis ' Approach In order to better understand the dynamics of Melnikov ' s method, we use a perturbation approach by Vakakis in order to obtain explicit expressions for motions on the stable and unstable manifolds, valid to first order in f. We illustrate the approach by application to eq. ( 4.3.1 ) : ( 4.4.1 )

2 x. . - x - x2 = f ( A cos wt - B x. )

We look for a solution in the form of a regular perturbation series in f: ( 4.4.2 )

Substituting ( 4.4.2 ) into ( 4.4. 1 ) and collecting terms gives ( 4.4.3 ) ( 4.4.4 )

Since we are interested in solutions along the perturbed stable and unstable manifolds, we begin the perturbation method with an expression for Xo on the ( unperturbed ) separatrix. This has been previously derived in eqs. ( 4.3.8 ) , ( 4.3.9 ) : ( 4.4.5 )

( 4.4.6 )

xo ( t ) =

-...,;:3,-_ COSh 2

__

[�]

[�]

s i nh . xO ( t ) = 3 ---COSh3

[�]

185

186

METHODS FOR PREDICTING CHA OS

Before we begin to solve eq. ( 4.4.4 ) for x 1 ( t ) , it is convenient to change independent variable from t to z = t - r. Using the abbreviation ( 4.4.7 )

. s = smh 2"z '

c = cosh

;,

z = t-r

eqs. ( 4.4.5 ) and ( 4.4.6 ) become: ( 4.4.8 )

Eq. ( 4.4.4 ) becomes ( 4.4.9 )

The general solution to ( 4.4.9 ) is of the form: ( 4.4. 10 )

x1 = xcomp + xpart

where xcomp is the complementary solution: ( 4.4. 1 1 )

and where xpart is a particular solution: ( 4.4. 12 )

where XA and XB are particular solutions corresponding to the nonhomogeneities dx gA A cos w{ z+ r) and gB = - B UzO ' respectively.

=

We begin by noting that eq. ( 4.4.9 ) has a complementary solution of the form d x and we therefore take ( cf.eq. ( 4.4.8 )) : xl = az O'

r METHODS FOR PREDICTING CHA OS

187

( 4.4.13) To obtain a second linearly independent complementary solution X2 , we use variation of parameters and set X2 = X l u. Straightforward substitution of the latter into the 1 = 6 This last equation for u may be left-hand side of eq.( 4.4.9) gives = X 2 1 integrated by using MACSYMA. Multiplying the result by X l and rearranging terms gives the following expression for X2 ( where %E stands for e ) :

�.

�

(4.4.14)

X = %E 2

-Z

- 60 Z %E

(%E

2Z

5Z

+ 15

- 144 %E

% E4 Z + 60 Z %E3 Z - 16 %E3 Z

2Z

Z Z 3 + 15 %E + 1)/(8 (%E + 1) )

In order to obtain expressions for the particular solutions XA and XB of eq. ( 4.4. 12), we again use variation of parameters in the form: (4.4. 15) where

(4.4. 16)

and

where the Wronskian W is given by

( 4.4. 1 7) dx In the case of X B , we set g = gB = - B Ozo in eq. ( 4.4. 16), and use MACSYMA to integrate the resulting equations, thereby producing closed form expressions for K 1 and K 2 . Substituting the result into eq. ( 4.4. 15) and rearranging terms gives the

p

188

METHODS FOR PREDICTING CHA OS

following expression for XB (where %E stands for e): 3Z 3Z 2Z -Z + 15 %E X B = B %E (30 Z %E - 30 Z %E

(4.4. 18)

- 95 %E

2Z

Z Z 3 + 15 %E + 1)/(5 (%E + 1) )

In the case of XA , we set g = gA = A cos w(z+ r) in eq. (4.4. 16) , and obtain the following expression for X A : (4.4. 19)

These integrals cannot be evaluated in closed form and must be obtained by numerical quadrature. We now have the general solution for x l ' eqs.(4.4.10)-(4.4. 12). It remains to choose the arbitrary constants C 1 and C 2 in (4.4. 11) so that we obtain solutions on the stable and unstable manifolds, i.e. , so that x remains bounded in the respective limits z -i + (I) and z -i - (I) . ..,...z From eq.(4.4. 13) we see that X l N +4 e + -i 0 as z -i + (I). From eq. (4.4. 14) we see 1 +z that X2 N g e as Z -i + (I). Thus by choosing C 2 appropriately we can balance terms +Z in xpart which grow like e- as Z -i + (I). For convenience, let us write

(4.4.20)

�

�

where C balances the growth of XA and C balances the growth of X B .

189

METHODS FOR PREDICTING CHA OS

�

From eq.( 4.4.18) we see that XB -+ 0 as z -+ + Ill , while XB N e--z as z -+ - Ill . Thus to balance the growth of XB on the stable manifold, we choose C = 0, whereas on , the unstable manifold we choose C = - B.

�

�

i

+z Now consider the growth of XA given by (4.4. 19). Since X 2 (z) N e- as z -+ + z +z the integral X2 ( () cos w( (+r) d( is expected to grow like e- as z -+ + Ill . o -:-Z However, since X l (z) N +4 e+ -+ 0 as z -+ + Ill , the term

�

J

X 1 (z)

J:

Ill ,

X 2 ( () cos w( (+ r) d( in (4.4. 19) is expected to remain bounded as z -+ +

Ill .

In fact , numerical evaluation reveals that it approaches a periodic function of z as z -+ +

Ill .

On the other hand, the integral

approaches limiting values as z -+ +

Ill .

+z in (4.4. 19) blows up like e- as z -+ + constant C

J:

X 1 ( () cos w( (+ r) d ( is convergent and

Thus the term X 2 (z) Ill .

We

J:

X l ( () cos w( (+r) d (

may therefore choose the arbitrary

� to balance the growth of XA by giving it the values:

J

+1Il C =X 1 ( () A cos w( (+ r) d ( 0 S,U

�

(4.4.21)

where the subscripts S and U stand for stable and unstable manifolds and refer respectively to the upper integration limit of +1Il and -w . Expanding the cosine term in (4.4. 2 1 ) , we obtain: (4.4.22) where

c

�S,U = A (- F 1 S,U cos WT + F2 S,U sin wr)

190

METHODS FOR PREDICTING CHA OS

( 4.4.23)

F1

(4.4.24)

F2

S,U

S,U

=

=

+ro J 0

+ro J 0

sinh ( (/2) cosh 3 ( (/2) sinh ( (/2) cosh 3 ( (/2)

cos w( d (

sin w( d(

Since the integrand in (4.4.23) is odd, we obtain:

Jro

sinh ( (/2) cos w( d( F 1 = F1 = O cosh 3 ( (/2) U S

(4.4.25)

Since the integrand in (4.4.24) is even, we obtain:

(4.4.26)

Jro

Jro

sinh ( (/2 ) . s i nh ( (/2) . sm w( d ( sm w( d( = 2"1 F 2 = - F2 = 3 U O cosh 3 ( (/2) S -ro cosh ( (/2 )

But this last integral has been evaluated a s 1 1 i n eqs. ( 4.3. 14),( 4.3.20), giving: ( 4.4.27)

27rw2 F2 = - F2 = S U s i nh 7rW ---

We have now obtained an appropriate choice for the arbitrary constant C 2 in order that the solutions lie on the stable and unstable manifolds. Vakakis has shown that the effect of C 1 , which is undetermined so far, is to time-shift the solution to order f. We take C 1 = O. The results of this perturbation analysis can be summarized as follows: On the stable manifold,

r

METHODS FOR PREDICTING CHA OS

19 1

( 4.4.28) On the unstable manifold, (4.4.29) where Xo is given by (4.4.8), X l by (4.4. 13), X2 by (4.4. 14), X A by (4.4. 19), X B b y (4.4. 18), C A by (4.4.22),(4.4.25) ,(4.4.27), and where C B = - 5"8 B . 2 2U S,U In order to visualize these results, we plot the projection onto the x-x plane of the solutions (4.4.28) and (4.4.29) lying on the stable and unstable manifolds. See Figs.4. 13 and 4.14, which correspond to the common parameter values A = 2, B = 1 , e = 0 . 1 , W = 1 . The difference between these two Figures i s that Fig.4. 13 corresponds to T = 5, while Fig.4 . 14 corresponds to T = 6.

192

METHODS FOR PREDICTING CHA OS

.

X

't = 5 Fig.4. 13. Perturbation solution (4.4.28) ,(4.4.29) projected onto the x-x plane. Region displayed is -4 < x < 2 and -3 < y < 3. S = stable manifold, U = unstable manifold.

r I r METHODS FOR PREDICTING CHA OS

•

X

/

Fig.4. 14. Perturbation solution (4.4.28),(4.4.29) projected onto the x-x plane. Region displayed is -4 < x < 2 and -3 < y < 3. S = stable manifold, U = unstable manifold. Note that the stable manifold lies inside the unstable manifold in Fig.4.13, whereas the opposite is the case in Fig.4. 14. Thus by continuity, the stable and unstable manifolds will intersect for some value of T between 5 and 6. According to Melnikov ' s criterion, eq. ( 4.3.22), the value of T corresponding to this intersection for the parameter values A = 2 , B = 1 , W = 1 satisfies the equation: (4.4.30)

. sinh 7r = 0. 7352 SIn T = -5 7r

:}

T = 5.5479

In order to check the perturbation calculation, we may numerically integrate the equation of motion (4.4. 1 ) . Solutions on the stable and unstable manifolds may be found by seeking an initial condition of the form:

19 3

194 ( 404.31 )

METHODS FOR PREDICTING CHA OS

t = r, x = x(O), X = 0

The value of x(O) is chosen by trial and error so that the resulting motion remains bounded as t -+ + 00 ( in the case of the stable manifold ) and as t -+ - 00 ( in the case of the unstable manifold) . The results are displayed in Figso4.15 and 4.16. Comparison with the perturbation results in Figso4.13 and 4. 14 shows good agreement for the stable manifold as t -+ + 00 and for the unstable manifold as t -+ - 00. However, the perturbation results exhibit divergent behavior for the stable manifold for large negative t ( Figo4.13 ) and for the unstable manifold for large positive t ( Figo4.14 ) . .

X

X

1: = 5 Fig.4. 15. Results of numerical integration of eq. { 404.l ) displayed on the x-x plane. Region displayed is -4 < x < 2 and -3 < y < 3. S = stable manifold, U = unstable manifold. Cf.Figo4. 13.

r

METHODS FOR PREDICTING CHA OS

.

t=6

X

x

s

\

Fig.4.16. Results of numerical integration of eq.{4.4. 1 ) displayed on the x-x plane. Region displayed is -4 < x < 2 and -3 < y < 3. S = stable manifold, U = unstable manifold. Cf.Fig.4. 14. As observed by Vakakis, the perturbation solution can be used to directly compute Melnikov ' s quantity b. ( r). From eq.( 4.2.8), the definition of b. ( r) is: ( 4.4.32)

From eqs. (4.4.5) , (4.4.6 ) , xO { r) = -3 and yO { r) = 0, so that (4.4.33) Substituting eqs.( 4.4.28) and (4.4.29) into (4.4.33), we obtain:

195

jiP

196

METHODS FOR PREDICTING CHA OS

( 4.4.34 )

Evaluating eq. ( 4.4. 14 ) for X2 at z = t-1' = 0 gives X2 ( 1') = -2. Eqs. ( 4.4.22 ) -( 4.4.27 ) give that ( 4.4.35 )

C

� S - C �U = 2A F2 S sin w1' ,

Also, we showed that C

( 4.4.36 )

� U - - � B.

� ( 1') = -

3E [ �

2 F2 = 2 1rw S s i nh 1rW

Substituting all these results into ( 4.4.34 ) gives:

1rW2 A sin w1' + S I nh 1rW

�B

]

+

O ( E2 )

which agrees with the expression ( 4.3.21 ) obtained by evaluating the Melnikov integral. 4.5 Chirikov ' s Method In this section we are interested in integrable Hamitonian systems which are modified by Hamiltonian perturbations of O( E). ( This prohibits damping terms but allows time dependent forcing terms. ) The resulting system, while Hamiltonian, is nonintegrable, but, for small E, is close to an integrable system. For such a system, KAM theory tells us that for small E, most of the phase space is composed of invariant tori, and that local chaos exists in the neighborhood of resonances. As E is increased, the KAM picture of local pockets of chaos separated by invariant tori becomes inapplicable and is replaced by global chaos, often in the form of a sea of chaos surrounding pockets of order ( the latter located near stable periodic motions. ) Chirikov ' s method is a heuristic scheme for estimating the size of the perturbation parameter E at which the transition between local and global chaos occurs in a Hamitonian system. The idea, called Chirikov ' s overlap criterion, is to treat each of the two largest resonances independently, obtaining the location and size in the phase

-.-

--------------------..-

METHODS FOR PREDICTING CHA OS

1 97

space of the associated separatrix as a function of f. Then the critical value for f is given by the value at which the two resonance regions (separatrices) first touch. In order to explain the method, we will illustrate it on an example offered by Chirikov in which a nonlinear oscillator is driven by a forcing function with two distinct frequencies: (4.5. 1 ) which i s derivable from the Hamiltonian H : (4.5.2) The f = 0 problem, being strongly nonlinear, exhibits a greatly enhanced dependency of amplitude on frequency compared with a nearly-linear oscillator, e.g. x · + x + f x3 = o. The choice of the problem (4.5.1) therefore allows the resonance

regions associated with each of the driving frequencies 0 1 and 0 2 to be located relatively distant from one another in phase space for small f. The goal of the perturbation calculation is to obtain approximations for the location and size of the resonance regions, and then to use Chirikov ' s overlap criterion to estimate the critical value of f at which the chaos associated with the individual resonances loses its local character and becomes "global".

We use a Hamiltonian formulation and Lie transforms ( Chapter Two) to treat (4.5. 1 ),(4.5.2). The unperturbed system has an exact solution which involves elliptic functions (see Appendix 4). In order to treat the perturbed problem, however, it is convenient to approximate the elliptic function solution by trig functions. In view of this, there are two ways to proceed. On the one hand we may obtain the exact elliptic function (cn) solution, and then approximate cn by cos; or we may begin with the assumption that the solution of the unperturbed problem can be approximately expressed as a cosine. There are advantages to both schemes, so we will present them both. In particular, the second scheme is applicable to systems in which no elliptic function solution exists to the f = 0 problem (see Exercise 8.)

198

METHODS FOR PREDICTING CHA OS

Scheme 1 The E = 0 version of (4.5.1),(4.5.2) (4.5.3)

x. . + x3 = 0,

H = �1 Px2 + '41 x4

admits the following exact solution: (4.5.4)

x = A cn( wt ,k),

Px = :ic = - A w sn(wt ,k) dn(wt,k)

The veracity of this claim may be seen by substituting (4.5.4) into the first of (4.5.3) and using the identities in Appendix 4, section A4. 1, which gives: (4.5.5)

w=A

From now on we will assume k = l/P in all elliptic functions, and will abbreviate cn(u,l/{l.) by cn(u), and similarly for sn and dn. Substitution of (4.5.4) ,(4.5.5) into the second of (4.5.3) confirms that the Hamiltonian is conserved for this time-independent problem, and gives: (4.5.6) Now we transform to action-angle variables p,q for the E = 0 problem. Since the period of the cn function is 4K, where K = K(k) = K(l/{l.) � 1.854 is an elliptic integral of the first kind, and since we define the angle variable q to change by 21r in the course of a single period, we take: (4.5.7) From (4.5.4) ,(4.5.5), t changes by 4K/ w = 4K/ A in the course of a single period, so that

METHODS FOR PREDICTING CHA OS

199

(4.5.8) We define the action variable p so that it =

� , where H = i A4 from (4.5.6):

Thus we make the canonical transformation from (x,px ) to (q,p): (4.5.10) x =

] [� 3 p] 1 /3 cn [2K 7r q ,

/3 [2K ] [2K ] 2 ] [3 sn 7r q dn 7r q Px = - � P

which produces the simplified € = 0 Hamiltonian:

(4.5. 1 1 )

2

[3 7r] 2 /3

H = "21 Px + 4"1 x4 = a p 4/3 , where a = 4"1 2K

�

0.867

The motivation for replacing cn by cos comes from consideration of the Hamiltonian:

€

f.

0

(4.5. 1 2 )

The O ( € ) term which involves both cn and cos functions will b e difficult to deal with analytically. From Byrd and Friedman, p.304 § 908.02, we find the following Fourier . series expansion of cn: (4.5.13)

[; ]

cn K q = 0.955 cos q + 0.0430 cos 3q + . . .

For convenience in handling the ensuing Lie transforms, we follow Chirikov and make the following bold approximation based on (4.5.13):

200

METHODS FOR PREDICTING CHA OS

[;

]

cn K q � cos q

(4.5. 14)

which, from (4. 5. 12), produces the Hamiltonian: (4.5.15)

[ ]

[ ]

3 11" 1/3 � 1.364. 3 11" 2/3 � 0.867 and (J = 2K where a = 4"1 2K Before treating this system, we present another scheme for obtaining an approximate solution to the unperturbed problem, this time without using elliptic functions. Scheme 2 We seek an approximate solution to the f = 0 problem, (4.5.16)

x. . + x3 = 0,

in the form of a cosine: ( 4.5. 1 7)

x � A cos wt

Note that the parameters A and w in scheme 2 play a parallel role to the A and w in scheme 1 , but they are not identical. The relation between A and w which follows from the approximation (4.5. 17) can be obtained by the method of harmonic balance. We substitute (4.5.17) into the first of (4.5. 16) and neglect all harmonics beyond cos wt . Since (4.5. 18)

cos 3 wt =

i cos wt + � cos 3wt

this means we neglect the cos 3wt term, and equate the coefficient of the cos wt term to zero, giving

201

METHODS FOR PREDICTING CHA OS

2 A=IJ w

(4.5.19)

The approximate nature of the assumption (4.5.17) may again be seen when it is substituted into the second of (4.5. 16): I A2 w2 sm. 2 wt + 1 A4 cos4wt 9 A4 + hIg' her harmomcs. (4 5 20) H where we have set Px x - Aw sin wt and have used (4.5.19). If we neglect the higher harmonics (cos 2wt and cos 4wt) in (4.5.20), a step which is consistent with the approximate nature of (4.5. 17), then the Hamiltonian is approximately conserved: .

= 2"

.

4'

=

= "3"2"

=

(4.5.21)

Next we transform to approximate action-angle variables for the define the angle variable q wt so that

f=0

problem. We

=

�

x A cos q , The action variable p is defined so that q � : ..13 - 8H 8H dA 9 3 dA ... 2 dA 4..13 ... - '2 A - Op" - OA ap - g A ap '" A ap - g- '" P - 4J'J A3 (4 . 5 . 23) q. This leads us to make the approximate canonical transformation from (x,pJ to (q,p): (4.5.22)

=

_

_

_

_

x [IJ4 ] 1/3 p 1/3 cos q , Px - [21J] 1/3 P2/3 sin q Note that the condition for a transformation to be canonical, namely dp dq dp (see Appendix 5 on differential forms), is satisfied if higher harmonicsx (cos 2q) are neglected, consistent with the approximate nature of (4.5. 17). ( 4.5.24)

dx A

=

=

A

=

202

METHODS FOR PREDICTING CHA OS

The transformation (4.5.24) produces the simplified f = 0 Hamiltonian:

(4.5.25)

[]

1/3 � 0.858 H = "2"1 Px2 + 4"1 x4 = a p 4/3 , where a = 4"3 "2"3

The f + 0 Hamiltonian (4.5.2) becomes: (4.5.26)

where a =

! [�]

1/3

�

0.858 and {3 =

[.,t.f ] 4

1/3

�

1 .321.

Note that both scheme 1 and scheme 2 produce the same form of H (cf. eqs . (4.5. 15) and (4.5.26) ) , but with slightly different values for the parameters a and {3. Lie transforms In order to prepare the nonautonomous Hamiltonian for treatment by Lie transforms, we use extended phase space (see Chapter 2, Section 2 . 7) and write the new Hamiltonian H = H + P 2 in the form: (4.5.27) where q2 =t and where p and q have been relabeled P I and q l for convenience. Writing H = H O + f H I ' we find: (4.5.28) (4.5.29)

1

1

I

METHODS FOR PREDICTING CHA OS

Next we use Lie transforms to O( f 2 ) , see eq.(2.2.6):

Eq.(4.5.30) is a near-identity transformation from (qi ' Pi ) to (Qi ' P i ) coordinates. The generating function W I is to be chosen so as to simplify the transformed Kamiltonian K as much as possible, where K K O + f K l + O ( f2 ) , see eq.(2.3.5):

=

(4.5.31) (4.5.32)

In order to simplify K 1 , we choose W I in the form: (4.5.33)

We can make K l vanish by choosing

= 4 1

C (4.5.34)

l

f.I

!

a

P 1 1/3 , C2 P I 1/ 3 + {1 I

j 1 i

11

( 4.5.30)

- 2" JJ

203

j

= 4 1

f.I

- 2" JJ !

a

P 1 1/3 , P 1 1 / 3_ {1 1

204

METHODS FOR PREDICTING CHA OS

1/3 1/3 The vanishing denominators 3"4 a P 1 - 0 1 and 3"4 a P 1 - 0 2 signal the presence of resonance zones near the regions of phase space given by:

(4.5.35) Our goal is to take each resonance separately, find the location and size of the associated resonance zone, and then use Chirikov ' s overlap criterion to predict a value of £ for global chaos. In either case there will be a nonremovable term in K 1 : (4.5.36) The result of our Lie transform analysis in this resonant case is the transformed Kamiltonian: (4.5.37) Next we produce an autonomous one degree of freedom system via a linear canonical transformation from (Qi ' P i ) to (Xi ' Yi ) (cf.eq.(2.8.18) ) : (4.5.38) which gives the Kamiltonian: (4.5.39) Since X 2 is absent from K in (4.5.39), Y2 is a constant of the motion to O( £2 ) . But K is also a constant of the motion since the system (4.5.39) is autonomous. Thus *

K = K -Y2 is a constant of the motion:

J

METHODS FOR PREDICTING CHA OS

205

*

The integral curves of K (X l ' Y l )=constant are displayed in Fig.4.17. The region inside the separatrix is the resonance zone corresponding to the driver cos 0 . t and is 1 the approximate location of the associated local chaos. When transformed back to x-Px-t space, this resonance zone shows up as lying within a circular annulus centered at the origin x = P x = 0 on a Poincare map corresponding to a surface of section t = 0 (mod 27r/O i ) .

c

B

A

A

D

7T

- 7T *

Fig.4.17. A sketch of the integral curves of K (X l ' Y l )=constant. The phase space is * a hal£-cylinder S x IR + since K is 27r-periodic in X l and since Y1 � O . Point A is a saddle equilibrium and point B is a center. Points C and D represent the maximum and minimum values of the action Y1 associated with the 0i resonant zone. We wish to obtain expressions for the location of points C and D in Fig.4.17, as functions of f, assuming f is small. We begin by locating the equilibria A and B from Hamilton ' s equations:

(4.5.41)

•

Xl =

OK

*

71Y1 = 3"4 a Y 1 1/3 - 0i - 61 f (j Y1-2/3 cos X l + O( f2 ) *

(4.5.42)

. OK Y 1 = - 7J'5C = - 2"1 f {j Y 1 1/3 sm X 1 + O( f2 ) 1 •

206

METHODS FOR PREDICTING CHA OS

From (4.5.42) we see that equilibria correspond to sin Xl = 0, i.e. , Xl = 0 or X l = Substituting these values of Xl into (4.5.41), we get the equilibrium conditions:

'Jr.

(4.5.43) To lowest order in E, both A and B are located at the same value of Y 1 :

(4.5.44) The separatrix is the integral curve in (4.5.40) which passes through point B: X l = 3n ' 3 . We may obtain the value of the associated constant in (4.5.40) by y = 1

[4(/

J

*

substituting these values of Xl and Y 1 into K : (4.5.45)

a

[:il

4 -

(l

i

[: � ] i

3 +

�,p

[:il

Points C and D , which measure the extent of the resonant zone, are located at Xl = O. We let the Y values of points C and D be measured relative to the center 1 at point A:

(4.5.46)

Yl =

3n . 3

[rcf]

+

0

Here o is the distance from point B to point C , or from B to D. We substitute (4.5.46) and X l = 0 into (4.5.45) in order to obtain an equation on 0:

'Jr,

METHODS FOR PREDICTING CHA OS

207

(4. 5 . 4 7) a

3 0 .] 4 - 0 [3 0 .] 3 + 1]1 fJ [� 3 0 .] [� i 1 4a

E

(4. 5 . 47) for 8, we set 8 = k Ii + O ( ) Then using MACSYMA to (4. 5 . 47) for small we may solve for k, and so finally obtain the

In order to solve Taylor expand result :

E .

E,

9 I nU 3 /2

i JB 1i + O 8 = + B' 1]3' �

() (4.5.48) Eqs. ( 4. 5 . 4 6) and (4. 5 . 4 8) together give the location of points C and D. a

E

Chirikov ' s Overlap Criterion Then as E is increased, the two resonant zones will first > Let us assume that for the zone ( point D) equals the maximum YI for Y minimum the when overlap the zone ( pOint C ) :

02 0 1 '

01

1

02

(4. 5 .49) (4. 5 . 49)

Solving for E gives the following expression for the critical value of E above which the local chaos associated with each resonant zone is predicted to become global:

(4. 5 . 5 0)

METHODS FOR PREDICTING CHA OS 208 For example, if we take 0 1 = 1, 02 = 2 and use the values of and f3 obtained from scheme 2, eq. (4. 5 . 26 ) , we find fcr = 0. 3 2 ( 4. 5 . 5 1 ) Numerical Integration For small f, the 0 1 and 02 resonance zones emerge from invariant tori of the form l px2 � x4 = C . =constant, where the constant C . is such that the period of the f = oscillations, 4K/ A = 7.416/ A, equals the period of the forcer, 27r/ 0j " Thus A=1.180 0i ' and using the initial condition x(O ) =A, px( O) = O , we find C . = 0. 4 85 0 4. . The resonant tori are thus: ( 4. 5 . 5 2 ) 2"1 Px2 41 x4 - [0.7.476285 forfor 00 1 == 21 2 These curves are displayed in Fig.4.18. a

+

�

°

1

1

�

1

1

+

_

x

-5

5

-5

Fig.4.18. Resonant tori (4. 5 .52) for 0 1 = 1 (inner curve) and 02 = 2 (outer curve) . In order to better understand the significance of Chirikov's overlap criterion, we

209

METHODS FOR PREDICTING CHA OS

numerically integrate (4. 5 .1) for 0 1 1 , 02 2, and for a variety of initial conditions and values of f. The results are displayed on a Poincare map corresponding to a surface of section t 0 (mod 211") in Fig.4.19. The two resonance zones associated with the resonant tori in FigA.18 may be plainly seen in FigA.19, especially for 0. 0 1 and f 0.1. In examining FigA.19, note how the presence of chaotic motions (exemplified by a "sea" of disordered dots) increases in extent as f increases. For f 0. 0 1, the system is still nearly integrable, with invariant tori everywhere. For f 0.1, local chaos can be seen around the origin, at least some of which may be identified with the 0 1 1 resonance zone. For f 0. 2 , the chaotic region around the origin has become more pronounced, but is still "local", i.e. it is separated from the 02 2 resonance zone. In the case of 0. 3 , however, the chaotic region near the origin has "leaked out" into the region around the 02 resonant torus, and may be said to represent "global" ' chaos. Thus the critical value of f (4. 5 . 5 1) as obtained by Chirikov s method, fcr 0. 3 2, roughly corresponds to the numerically-observed change in the local character of the chaos. =

=

=

f=

=

=

=

=

=

=

f=

=

210

METHODS FOR PREDICTING CHA OS

. : . I',' :

.

' " "

.

.

:

.' l J I .' . . . .. I

, . .'

. 1/

:

•

: ' o' ..

•

I

.

•

_ ,: "

. "

' . o'

:

;;

� '\ ::,' .. ..

_ . . . tI " . .. . . . .._ "'_. . . . ..

",

"

.

•

� , " ' \.

"

� .

.

.

.

.

,

I

,

'

,

.

r

:

I

,

:

"

'

.

�::: f/� :

,...1

: ,

.

.

.:

E=

0. 0 1

;

.

'

• •

"

' o t

.

. . ;. ,' .; ..

. . 0 '

'

;

:

, " . ••

.

.

.

.. .

: :: . : . , , ..

.

,

�\ \. " ' . .

,...,. ..-" . .. . . ... .. . . ' _ . . . . ,.. . - . .. . ... . . ., ',"!•

'I I t lo

\\ : '

� ' '.

•

'

::.

'

'

'

.

•

.

\( '. : �. •

I

: ;; ' .

.

E=

0.1

. '

0. 2 0. 3 Fig. 4 .19. Poincare map corresponding to a surface of section t 0 (mod 271') obtained by numerically integrating eq.(4. 5 . 1 ) for four values of Each map is displayed in the plane, for -5 < < 5, -5 < < 5. E=

E= =

E.

x-P x

x

Px

METHODS FOR PREDICTING CHA OS

4. 6 References Byrd,P . and Friedman, M ., " Handbook of Elliptic Integrals for Engineers and Scientists", Springer (1954) Cesari,L ., "Asymptotic Behavior and Stability Problems in Ordinary Differential Equations", Academic Press (1976) Chirikov ,B. V.," A Universal Instability of Many-Dimensional Oscillator Systems", Physics Reports 52:265-376 (1979) Greenberg, M . D ., "Foundations of Applied Mathematics", Prentice-Hall (1978) Guckenheimer, J . and Holmes,P ., "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields", Springer (1983) Jackson,E .A., "Perspectives of Nonlinear Dynamics", 2 volumes, Cambridge (1989) Melnikov,V.K ., "On the Stability of the Center for Time-Periodic Perturbations", Trans. Moscow Math.Soc., 12:1-57 (1963) Moon,F.C., "Chaotic Vibrations", Wiley (1987) Shaw,S.W . and Rand,R.H ., "The Transition to Chaos in a Simple Mechanical System", Int. J . Nonlinear Mechanics 24:41-56 (1989) Thompson, J. M . T ., "Chaotic Escape from a Potential Well", Proc. R.Soc. London A421:195-225 (1989) Vakakis,A. F ., "Exponentially Small Splittings of Manifolds in a Rapidly Forced Duffing System," J.Sound and Vibration (to appear 1994)

211

METHODS FOR PREDICTING CHA OS 212 4. 7 Exercises 1. Use the method of harmonic balance to find an approximation for the periodic motion in eq.(4. 3 .1) which lies close to x :ic 0 for small That is, substitute the following expression for x in eq.(4. 3 .1): x a cos wt + {3 sin wt Trigonometrically reduce the result and set to zero the coefficients of cos wt and sin wt , giving two equations for a and {3. Expand {3 {31 + {32 2 + . . . and substitute into the equations on a and {3 in order to obtain values for aI ' a2 ,{31 and (32 . To this order of approximation, what does the periodic motion look like when projected onto the x-x plane ? (Cf. Fig.4. 6 .) 2. Evaluate the integral J 1 in eq.(4.3.16) by using the following contour (cf. Fig.4.l1): =

=

f.

=

f

=

1M

f

Z

2n i

R

-R

Re

z

Fig. 4 . 2 0. Alternate contour for evaluating the integral J 1 in Melnikov's method. The X represents a singularity at z rio =

METHODS FOR PREDICTING CHA OS

3. Use Melnikov's method to find the critical ratio �cr ( cf.eq. (4. 3 . 23)) for the system: x. . -x + x3 (A cos wt - B x. ) = f

Answer: �� cosh r 4. Use Melnikov's method to find the critical ratio �cr ( cf.eq. (4. 3 . 23)) for the system: x'

- sin x ( A cos wt - B x) = f

Answer: � cosh r 5. This exercise concerns the transition to chaos in a system consisting of an inverted pendulum with rigid barriers on either side of the unstable equilibrium position ( Shaw and Rand ) . Collisions of the pendulum with the barrier are assumed to be perfectly elastic, i. e ., x -x. See FigA. 2 1. -+

<

A

/

cos

>

cu t

Fig. 4 . 2 1. An inverted pendulum is driven horizontally with sinusoidal forcing.

21 3

...

METHODS FOR PREDICTING CHA OS 214 We assume the barriers are close enough to permit the following linearized model to be used: d2x -x (A cos wt - B (IT)' I x l < 1. dt -:-:2

dx

= f

The phase plane for the 0 equation consists of a linear saddle with reflections at x + 1 . See Fig. 4 . 2 2, in which the dotted lines show instantaneous jumps corresponding to collisions with the barrier (which are located at x + 1 ) . Note that the straight-line trajectories are part of a homoclinic orbit . f=

=

=

y=x .

/

/ / I A

\ '"\

"" "---I---""",*,---+--�

""

X

/

-1

o

\

'V

/

1

Fig. 4 . 2 2. The unperturbed phase portrait. a) In preparation for Melnikov's method, we write the problem in the form: dx

(IT = Y

METHODS FOR PREDICTING CHA OS

215

and set where Xo and yO represent the f = 0 solution, i.e., the unforced, undamped problem. Find a solution xO (t ) on the homoclinic orbit (for f = 0) which satisfies the initial condition t = r , x = l. Note that the form of the solution for xO( t ) will involve two distinct expressions, one valid for t < r ( corresponding to the upper half of the phase space, y = x > 0, and x 0 as t ) and the other for t > r (corresponding to y = x < 0, and x 0 as t +(0). b) Use Melnikov' s method to compute a critical value of the ratio AlB ( as a function of ) above which there exists an intersection of stable and unstable manifolds. Hint: Use eq. (4. 2 .19) . 6. Obtain a closed form expression for the stable and unstable manifolds in the system: �

� --,n

�

�

w,

1 Work to O( f) and follow the procedure given in section 4.4. In particular, for the linear differential equation on Xl : a) Obtain complementary solution Xl by differentiating xO . b) Obtain complementary solution X2 by setting X2 = X l u. c) Obtain particular solutions XB and XA by variation of parameters. , f«

jP'

METHODS FOR PREDICTING CHA OS 216 d) Choose the arbitrary constants C 1 and C2 to keep x bounded as t + e) Plot your approximations for the stable and unstable manifolds on the x-x plane for 1, .1, B 1, A 2, and for two values of 5 and '6. This entails numerically evaluating the associated integrals (e.g., by Simpson s Rule). As a shortcut, you may compute x by finite differencing neighboring values of x. f) Check your results by numerically integrating the original differential equation for the same parameters to obtain the stable and unstable manifolds. This entails some trial and error to find the approximate initial conditions (of the form t x 0, x ?) which take a motion to the hyperbolic periodic motion close to the origin. Make two plots on the x-x plane for the same parameters as above, and compare your results. 7. This exercise involves numerically generating the Lyapunov exponent >. for the forced, damped pendulum: (4. 7 .1) x · sin x (A cos wt - B x) , < < 1 For given values of the parameters A and B, choose random initial conditions and numerically integrate eq.(4.7.1) with step size no larger than 0. 0 1 time units. Wait for some time for transients to die out (not less than 500 time units), then begin to integrate the linear variational equation on e(t) associated with (1), choosing its initial conditions randomly. (You may, without loss of generality, choose e(0)2 + e(0)2 1 ) Continue integrating both equations for at least another 1000 time units. Then the Lyapunov exponent >. will be approximately given by -+

w=

f=

=

=

Ill .

r=

= r,

=

=

-

= f

f

f, w,

=

.

where T is the total time for which you have integrated the e equation. As discussed

METHODS FOR PREDICTING CHA OS

in Appendix 1 2 , to avoid an overflow error due to the exponential growth, you shoul d renormalize e every, say, 10 steps. Do the above for the parameter choices f = 0 . 1 , B = 1, w = 0 . 1 ,0.2,0.3, . . . ,1

and A = 1,2,3, . . . ,20

Display your results on the w-A parameter plane by marking those points for which ).

>

°

differently than ). � 0. Draw the curve of

�cr as a function of w derived by

Melnikov ' s method on the same plot. (See exercise 4.) Comment on the relati onship between these two different approaches to chaos. 8. Use Chirikov ' s method to determine fcr for the system: (4. 7.2)

x. . + x5 = f ( cos 0 1 t + cos 0 2 t )

Follow the procedure in section 4.5. Note that no elliptic function solution is available for the f = ° problem, so use scheme 2. Check your analytic result by numerically integrating eq.{4. 7.2) and displaying the results in a Poincare map as in Fig.4. 19.

217

218

METHODS FOR PREDICTING CHA OS

Appendix 12: Lyapunov Exponents A12 . 1 Introduction The stretching of phase space plays an important role in chaos. For example, stretching effects in a horseshoe map produce sensitive dependence on initial conditions. Lyapunov exponents (Moon, Jackson) are a popular measure of chaos based on quantifying the degree of stretching which occurs in a dynamical system. The computation of Lyapunov exponents is a numerical or experimental method based on waiting for transients to die out and then characterizing the steady state motion of the system. For example, the steady state may be a " strange attractor" (behavior described by the words chaotic, random, unstable) , or it may be a periodic motion (regular, stable). The idea of a Lyapunov exponent is to characterize the local stretching in the neighborhood of a steady state by a single number .>., which, by its sign, indicates whether the system is chaotic or not . Let the system (A I 2 . 1 . 1 )

x = f(x,t) ,

x f lRn

have a solution (AI2 . 1 .2)

x = u(t)

In order to measure the stretching in the neighborhood of x = u(t) , we consider another motion, x = v(t), which is close to u(t) at t = 0, and we compute the distance between the points x = u(t) and x = v(t) for large t. To accomplish this, we set (A I 2 . 1 .3)

x = u(t) + e(t)

where e(t) = v(t ) - u(t) represents a vector from u(t) to v(t) . Substituting (AI2 . 1 .3) into (AI2 . 1 . 1 ) , we obtain

,

I

METHODS FOR PREDICTING CHA OS

(A I 2 . 1 .4)

u

+ e = f(u(t )+ e,t)

=>

219

e = f(u(t)+ e,t) - f(u(t),t)

where we have used u = f(u(t),t). Next we linearize (AI2 . 1 .4) around x = u(t). The heuristic reason for this move is that we are interested in the local behavior in the neighborhood of x = u(t), i.e. , that the length I e(t) I of the e(t)-vector is small. The practical reason for the linearization is that linear differential equations are better understood and thus easier to work with than nonlinear differential equations. The result is:

(AI2 . 1 .5)

e = Df(u(t),t) e = A(t) e

where A = Df is the matrix of partial derivatives aij =

a v aej .

A basic result from the theory of linear differential equations (Cesari) is that if the coefficients aij (t) in (A I 2. 1.5) are continuous bounded functions of t as t -+ m , then every solution e(t) grows no faster than e At , for some A. This result , known to Lyapunov, led to his concept of "type numbers" . A12.2 Lyapunov ' s type numbers Given a scalar function g(t) , its Lyapunov type number A is defined to be: (AI2 . 2 . 1 )

A = lim sup in l g(t) 1 t -+ m t

[Before proceeding, we offer the following reminder of what "lim sup" means: A sequence a 1 ,a2 , . . . ,an , . . . is said to have an accumulation point A if there are an infinite number of terms ai in any neighborhood of A. E.g. , the sequence (AI2.2.2)

1 1 , '3"1 1 , 4"1 1 , 1 ' . . . ' ' ' D

'2"

has two accumulation points, 1 and O . The lim sup of a sequence i s its largest

p 220

METHODS FOR PREDICTING CHA OS

accumulation point . In the case of a function g(t), consider a sequence g(t 1 ),g(t 2 ) , . . · ,g(t n ) , . . · , where t l
(A12.2.3)

where we have used L ' Hospital ' s Rule. For g(t) = t e-3t we obtain

(A12.2.4)

-3t -3t I = l i m in t + l i m in e >. = lim sup in I t e =-3 � ro t � ro t t -+ ro t

For g( t) = e3t sin t we obtain

(A12.2.5)

3t 3t >. = lim sup in I e sin t l = l i m in e + lim sup in 1 sin t l t -+ ro t t-+ ro t t -+ ro t

Now l i m in l si n t l does not exist . Nevertheless lim s up in l sin t l = 0, giving the t-+ ro t t -+ ro t result >. = 3. It is easy to invent a function which does not have a finite type number. E.g. for 2 g(t) = et we obtain

METHODS FOR PREDICTING CHA OS

2 ln l et� 1 t2 = + � = lim sup A = lim sup t � (I) t t � (I) t

(A I2.2.6)

221

(I)

2 Moreover, it is easy to invent a linear differential equation which has e = et as a 2 solution. Since e- = 2t et , the equation e- = 2t e possesses this solution, and hence has a solution which does not have a finite type number. However, the coefficient 2t (cf. (AI2 . 1 .5» is not bounded. Lyapunov ' s theorem is that if the coefficients of a linear differential equation are continuous bounded functions of t , then all solutions have finite type numbers. Note that the definition (A I 2.2.1) may be extended to a vector function g(t) if I g(t) I is interpreted as a norm of g(t) . E.g. if g = [g l ' g 2 , . . . ,gn] ' we may take 1/2 I g(t) 1 = g l 2 +g 2 2 + - - - +gn2

[

]

A12.3 Lyapunov exponents Lyapunov exponents are a numerical/experimental version of Lyapunov type numbers. We return to the first variational equation (AI2.1.5):

(A I 2.3. 1) where A(t) = Df(u(t),t). For e f IRn , the general solution to (A I 2.3.1) will be of the form: (A I 2.3.2) where the ei (t) ' s are linearly independent and thus may each have a distinct type number Ai . For a "general" initial condition, i.e., one for which none of the ci are zero, the solution (AI2.3.2) will possess a type number A = max(A i ) .

1

11

1 1

......

...

222

METHODS FOR PREDICTING CHA OS

We may obtain a numerical approximation for A by computing the solution �(t) to (A12.3.1) by finite differences, and then calculating (A12.3.3) One is hopeful that as larger values of t are taken in (A12.3.3), a more accurate value of A is obtained. The quantity A which results from this calculation is called a Lyapunov exponent. The significance of this computation is that if A > 0, then solutions of the original differential equation (A12. 1 . 1 ) are being stretched apart in the neighborhood of x = u(t). Steady states !!ill. for which A > 0 are identified as being chaotic. If A < 0 , then neighboring solutions are contracting and the steady state is nonchaotic. There is a practical problem associated with the computation of A in (A12.3. 3 ) , namely that i f �(t) is growing exponentially, i t may cause an ove�flow o n the digital computer executing the numerical integration of (A12.3. 1). Here is a computational strategy to avoid such an overflow: Begin the numerical integration with a "general" initial condition �O for which 1 �O 1 = 1. (In practice we may choose the "direction" of eO at random. Since (A12.3. 1) is a linear differential equation, we may take 1 �O 1 = 1 without loss of generality.) After a certain number of steps, which must be small enough so that no overflow occurs, say 10 steps, � has the value � 1O' Define a1 = 1 � 10 I · Now we stop the numerical integration and reset the initial condition by scaling � 10 -I � 10 1 a1 ' i.e. so that the new value of � 10 has unit norm. We use this initial condition to compute � 11 and so on, until 10 more steps have been completed. Then we set a2 = 1 �20 I , etc. We continue in this manner until we are done, say after 1000 steps, at which point a100 = 1 � 1000 I · By the linearity of the first variational equation (A12.3.1) these scalings may be superimposed, so that if no scalings had been used we would have found that (A12.3.4) From (A12.3.3) we obtain the following approximation for the Lyapunov exponent :

METHODS FOR PREDICTING CHA OS

(A12.3.5)

A�

1 0 00

in l e 1000 1 = � l � 1000 1000

in

i =1

a·

1

In order to apply this process to a steady state x = u(t) of eq.(A12. 1 . 1 ) , we must have already numerically integrated (A12. 1.1) until the transients have died out . In practice we wait an arbitrarily chosen length of time, say 500 time steps, for steady state to be achieved. Then we begin the foregoing process of computing the Lyapunov exponent. This involves simultaneously numerically integrating both (A12 . 1 . 1 ) to obtain u(t) and (A12.3.1) to obtain e(t) (since the coefficients in (A12.3.1) depend on u(t)). Note that the results of this computation are statistical in the sense that they depend upon the randomly chosen initial conditions, both for the x equation (A12. 1 . 1 ) and the e eq.(A12.3. 1 ) . In particular, if the phase flow of (A12. 1 . 1 ) contains more than one attractor, each with its own basin of attraction, then the computed Lyapunov exponent will depend upon which steady state is excited. A12A Examples As an example of the numerical computation of Lyapunov exponents , we take eq.(4.3.1) which has been treated by Melnikov ' s method:

(A12A. 1 )

2 x" " - x - x2 = f ( A cos wt - B x" )

As might be expected from the nature of the f = 0 phase portrait , FigA. 10, this system exhibits motions which may escape to infinity. Computation of Lyapunov exponents for motions which start at t = 0 inside the unperturbed separatrix loop (specifically at x = -2, x = 0, the unperturbed center) gives A < 0 for all motions which do not escape to infinity. See FigA.23.

223

224

METHODS FOR PREDICTING CHA OS

10

�------���

A B

o

w 2 Fig.4.23. Results of numerical integration of eq. (A12.4.1) for I: = 0 . 1 and B = 1 , for a grid of values of w and A covering 0 � w � 2 and 0 � A � 10. Points in the shaded area correspond to motions which do not escape to infinity and which turn out to have negative Lyapunov exponents. Unshaded region represents motions which escape to infinity. The U-shaped curve is Melnikov ' s critical ratio of A/B , eq.( 4.3.23) , cf. Fig.4.12.

Systems which lie above the Melnikov curve, but which do not escape to infinity, exhibit transient chaos due to the presence of horseshoes, but do not involve a strange attractor, since the Lyapunov exponent is negative. See Thompson for a discussion of this system. As another example we take the system of Exercise 3 (Chapter 4): (A12.4.2)

. x. . - x + x3 = 1: ( A cos wt - B x)

r

METHODS FOR PREDICTING CHA OS

22 5

which has a figure--eight separatrix in its f = 0 phase portrait . See Fig .4.2 4 . .

X

X

Fig.4.24. Phase plane for x ' - x + x3 = O. Region displayed is -2 < x < 2,

-2 < x < 2 .

The Melnikov calculation for eq. ( A12.4.2 ) gives the result: ( A12.4.3 )

A

If

cr

2 .j1I 7rW = 37rW cosh 2

Eq. ( A12.4.3 ) is displayed in Fig.4.25 together with the results of the numeri cal computation of Lyapunov exponents for eq. ( A12.4.2 ) . Note that all parameters which were found to correspond to positive Lyapunov exponents ( displayed as dots) were located in the region above the Melnikov curve ( A12.4.3 ) . These d otted points correspond to systems with chaotic steady states ( strange attract ors ) , and thus are expected not to lie below the Melnikov curve, where the stable and unst able manifolds do not intersect . Systems with negative Lyapunov exponents whi ch are

226

METHODS FOR PREDICTING CHA OS

located in the region above the Melnikov curve (undotted pOints) exhibit transient chaos but have a nonchaotic steady state (e.g. a periodic motion.)

10 A B o

�. . . .

co

3

Fig.4.25. Results of numerical integration of eq.(A12.4.2) for f = 0. 1 and B = 1 , for a grid of values of w and A: 0 � w � 3 in steps of 0. 1 and 0 � A � 10 in steps of O . l . Dotted points correspond t o motions which have positive Lyapunov exponents. Undotted points represent motions with negative Lyapunov exponents. The curve is Melnikov ' s critical ratio of A/B , eq.(A12.4.3).

It

r f

I

INDEX

Andronov bifurcation 122,128,158-161 averaging 91-149 Chirikov ' S method 196-210,217 contour integration 180-182,212 differential forms 83-90 Duffing eq. 44-47,50-53, 73,78-81, 122,125,139, 146-149, 197,211,213,215,224-22 6 elliptic functions 52,70,72,78-82,122-127,139-149,197-200,211,217 exterior derivative 84,87,90 generalized speed 1 , 2,5-7, 1 1,12,18 generating function averaging 95,98,105,113, 132, 134,136 canonical transformations 73,89-90 Lie transforms 42,50,65,203 harmonic balance 200,211 Henon-Heiles system 62-69, 75 homoclinic orbit 165,214,215 Hopf bifurcation 108-112,122, 123,125,158 horseshoe 162-165 , 1 70,171, 183,184,218,224 invariant tori 66-68,196,208,209 Kane ' s eqs. (see method of generalized speeds) Lagrange multiplier 2 , 1 1 , 13,16,21-23,29 Lie transforms 40-77 limit cycle 103-112,122,123, 125-128,139-142,144-149,158-161 Lyapunov exponent 184,216-226

- 227 -

•

228

INDEX

MACSYMA averaging second order 129-131 third order 132-133 nth order 136-138 nonautonomous 134-138 using elliptic functions 142-149 Chirikov ' s method 207 elliptic functions in 79 Lagrange ' s eqs. 38-39 Lie transforms 76-77 Melnikov ' s integral 179-180,182 method of generalized speeds 36-38 Pade approximants 150-154 period of nonlinear oscillator 53 Poincare map in 2 dof system 66 Poisson bracket 50 regular perturbations 1 56-157 Vakakis ' perturbation calculation 187-188 Mathieu ' S eq. 53-62,73,1 14-121, 126,128,134-138,152-157 Melnikov ' s method 165-184,195,196,212-217,223-226 method of generalized speeds 2,3,15 nonholonomic constraint 9-13, 16-18,22,29,30 Pade approximants 122,128,150-157 partial velocity 2,6,8,12,19,21 partial angular velocity 14, 15,19 pendulum 1 ,27,28,213-2 1 7 Poincare map 60-62,66-70,75 , 166-171, 183,205,209,210,21 7 Poincare ' s lemma 87,90 quadratic nonlinearity 62-69,72,81-82,108-1 10,122,126,127,139-146, 176-196, 223-224

INDEX

separatrix 162, 165-178,185 , 197,205,206,223,225 Smale-Birkhoff theorem 162-163 stable/unstable manifold 162-165,169-174,183-185,188-195 , 215-216,225 strange (chaotic) attractor 184 subharmonic 56, 1 1 6 , 123 transient chaos 184 Vakakis' approach 185-196,215,216 van der Pol ' s eq. 1 03-109, 123, 125-126,132-133, 158_160

22 9