TOPICS IN NONLINEAR DYNAMICS WITH COMPUTER ALGEBRA
Computation in Education: Mathematics, Science and Engineering A series edited by Robert Gilbert, University of Delaware, Newark
Volume 1 Topics in Nonlinear Dynamics with Computer Algebra Richard H. Rand
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TOPICS IN NONLINEAR DYNAMICS WITH COMPUTER ALGEBRA
Richard H. Rand
Cornell University Ithaca, New York
Gordon and Breach Science Publishers
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Library of Congress Cataloging-in-Publication Data Rand, R.
H.
(Richa rd
H.)
Topics in nonlinea r dynamics with computer a l g ebra / Richa rd p.
cm. - (Computation in education; v . 1)
H.
Rand.
Incl udes index. ISBN 2-88449-113-9. - ISBN 2-88449-114-7 (pbk.) 1.
Dyn amics-Data processing . 2. Nonlin e a r theori es-Data
processin g . 3. Alg ebra -Data processin g . 4. Pert u rbation (Mathematics)-Data processing. I. T itle.
5. MA CSYMA
(Computer system)
II. S er i es.
QC133.R36 1994 531.1'1 '078-dc20
94-2869 CIP
No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording , or by any information storage or retrieval system, without permission in writing from the publisher. Printed in the United States of America.
CONTENT S Introduction to the Series
ix
Preface ONE 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11
THE METHOD OF GENERALIZED SPEEDS Introduction The Method for a System of Particles Example Nonholonomic Constraints Example Rigid Bodies Example Computer Algebra Comments References Exercises
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11
1 1 2 3 9 12 14 15 24 24 26 27
Derivations MACSYMA Programs for Deriving Equations
31
of Motion
36
LIE TRANSFORMS
40
Appendix 1: Appendix 2:
TWO
vii
Introduction The Near-Identity Transformation The Kamiltonian Example Higher Order Approximation Example Nonautonomous Hamiltonians and Extended Phase Space Example A Two Degree of Freedom System References Exercises
Appendix 3:
MACSYMA Program for Deriving the Kamiltunian
Appendix 4:
Elliptic Function Solutions
Appendix 5:
Differential Forms
40 42
43 44 47 50
53 54
62 70 72 76 78
83
CONTENTS
vi
THREE 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11
THE METHOD OF AVERGING
Introduction The Method of Averaging Example Second Order Averaging Limit Cycles and van der Pol's Equation Hopf Bifurcations Nonautonomous Systems Example Related Topics References Exercises
Appendix 6:
Appendix 7:
MACSYMA Program for Second Order Averaging MACSYMA Program for Third Order Averaging
Appendix 8:
MACSYMA Program for Averaging
Appendix 9:
in a Nonautonomous System Averaging with Elliptic Functions
Appendix 10: Pade Approximants Appendix 11: The Andronov Bifurcation
FOUR 4.1 4.2 4.3 4.4 4.5 4.6 4.7
METHODS FOR PREDICTING CHAOS
Introduction Melnikov's Method Example Vakakis' Approach Chirikov's Method References Exercises
91 91 93 96 98 103 108 112 114 122 123 125 129 132 134 139 150 158
162 162 165 176 185 196 211 212
Appendix 12: Lyapunov Exponents
218
Index
227
INTRODUCTION TO THE SERIES Computation in Education: Mathematics, Science and Engineering comprises a series of textbooks and monographs covering traditional topics with an innovative use of computers. Topics include modeling and simulation, optimal control, neural networks, computer graphics and books containing numerical recipes. Software specific books for a particular discipline will also be featured. This series is aimed at graduate and undergraduate students in the mathematical sciences and engi neering, and university graduates wishing to stay current with new software implementation in their fields. As developments in one field impact on other fields, we anticipate that a series devoted to the use of computers in engineering and the mathematical sciences will further this cross-fertilization.
Robert Gilbert
PRE:FACE This book consists of material which I have used in a number of graduate courses in the department of theoretical and applied mechanics at Cornell University. These have included courses entitled Intermediate Dynamics, Advanced Dynamics, Nonlinear Vibrations, and Perturbation Methods. All these courses include com puter algebra. I frequently use MACSYMA in real time in the classroom in order to perform a derivation or complete an algebraically complicated step in doing a problem. I regularly assign homework which is to be completed using computer algebra. I offer the uninitiated student a single introductory lecture on MACSYMA, but I accept any computer algebra system on the homework. Maple and Mathemat ica are the most frequently used alternatives to MACSYMA. I find that replacing a long calculation by hand on the blackboard with a computer algebra computation saves valuable class time and demonstrates to the student an increase in efficiency, accuracy, and convenience. Much of the material can be viewed as a supplement to the many established textbooks on advanced dynamics or nonlinear vibrations. When using such texts, I found it convenient to provide the students with handouts on the method of generalized speeds, Lie transforms, Melnikov's and Chirikov's criteria for chaos, differential forms, elliptic functions, and so on. The supplementary notes which I prepared for my classes are the basis for the present book. All four chapters are relatively independent and can be read without having read the others, except for section 4.5 of chapter four on Chirikov's method, which requires knowledge of chapter two on Lie transforms. The role played by computer algebra in this book may be contrasted with that in a previous book with Dieter Armbruster, Perturbation Methods, Bifurcation Theory
and Computer Algebra (Springer, 1987). The latter work presented MACSYMA programs which automated the solution of certain classes of frequently encountered problems.
This served to demonstrate to users who already knew perturbation
methods how computer algebra could be used to replace hand computation. In using that book as a supplementary text in a variety of courses, however, I found that the student-who was generally initially unfamiliar with the perturbation methods-was not learning so much about the methods themselves, as about how to operate the programs. In the present work no such general purpose programs are supplied. Some sample MACSYMA programs are given, mainly to serve as examples for the reader to use as a pattern. Computer algebra is mainly utilized in an interactive mode in order to complete a derivation or to work out the details in an example. The goal here is for the student to understand the method as completely as if it were being worked out by hand, while still having the computational advantages of a commercial computer algebra package. l owe thanks to many people for the help they gave me in developing these notes. I can't name them all, but I do wish to thank specifically A. von Flowtow, D. Rosenthal,
F. Moon, P. Holmes, J. Guckenheimer, and V. Coppola.
My current e-mail address is
[email protected].
CHAPTER ONE THE METHOD OF GENERALIZED SPEEDS (KANE ' S EQUATIONS)
1.1 Introduction The method of generaliz ed speeds is a scheme for obtaining the equations of motion of a mechanical system (Kane and Levinson). Like Lagrange ' s equations (which we assume the reader is familiar with) , the method of generaliz ed speeds automatically eliminates constraint forces from the governing equations of motion. (A constraint is a given restriction on the motion of the system, e.g. the length of a pendulum is 1. A constraint force is a force which is required, by Newtons ' equations, to satisfy a constraint , e.g. the tension along the length of the pendulum. S ince constraint forces are not explicitly specified, but rather are given in terms of their effects on the motion, their presence in the equations of motion is usually undesirable. ) The method of generaliz ed speeds has two features which offer an improvement over Lagrange ' s equations. In Lagrange ' s equations, once generalized coordinates qi are chosen, all references to speeds are made in terms of the generaliz ed vel ocities q i . The method of generaliz ed speeds, however, permits the user to define generaliz ed speeds ui which are an affine function of the q i . If the ui are chosen appropriately, the equations of motion may be significantly simplified. The second advantage which the method of generaliz ed speeds offers over Lagrange ' s equations involves the manner in which nonholonomic constraints are treated. (A nonholonomic, or nonintegrable constraint is a given restriction on the motion of a
-1-
..
2
METHOD OF GENERALIZED SPEEDS
system which involves velocities, e. g. a wheel rolling on a plane without slipping. ) Treatment of nonholonomic constraints by Lagrange ' s equations involves the use of Lagrange multipliers, which increases the number of unknowns in a problem and thereby nullifies to some extent the advantage of eliminating constraint forces. In the method of generaliz ed speeds, however, nonholonomic constraints can be treated in a natural and convenient manner, without using Lagrange multipliers. A disadvantage of the method of generaliz ed speeds is that it requires the computation of acce lerations, whereas Lagrange ' s equations only require that velocities be computed. However, Lagrange ' s equations require that the Lagrangian be differentiated in order to get the equations of motion, which requires about as much work as computing the accelerations. We begin by offering an outline of the method of generaliz ed speeds for a system of particles , followed by an example. A derivation is presented in Appendix 1 . 1.2 The Method for a System of Particles Let a system consist of N particles with n degrees of freedom. Thus we have N vectors I j from the origin of some inertial frame to each particle, as well as N velocities V j and N accelerations aj , these respectively being the first and second time-- derivatives of the I j ' s, taken in the inertial frame. Let mj be the mass of the jth particle. We also have n generaliz ed coordinates qi. We will use the convention that the subscript j tags particles and goes from 1 to N, while the subscript i tags generaliz ed coordinates and goes from 1 to n. 1 . Choose the ui ' s as an affine functi on of the
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)
LIE TRANSFORMS
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
LIE TRANSFORMS
( 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 (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
+
1
•
W
1 +
H 0
1 +
•
W
1
•
1
2
W 2
---------
6
+
W 2
-------
•
I
W 2
•
2
H
2
1
-------
---------
H
+
1
H 0
1
2
----------
3
W
W
•
---------
H K 3
•
H 0
=
+
H , 2 H o
+
H )] 3
•
W
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 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 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. -+
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 . = 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
......
...
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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
