Lectures on Dynamical Systems, Structural Stability, and their Applications

471 Pages·1992·27.499 MB·English

by  Kotik K. Lee

Lectures on
Dynamical Systems,
Structural Stability
and their Applications
KOTIKK. LEE
Dept. of Electrical and Computer Engineering
University of Colorado
h
World Scientific
f"
' Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P 0 Box 128, Farrer Road, Singapore 9128
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
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LECTURES ON DYNAMICAL SYSTEMS, STRUCTURAL
STABILITY AND THEIR APPLICATIONS
Copyright© 1992 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof. may not be reproduced in any form
or by any means, electronic or mechanical, including photocopying, recording or any
information storage and retrieval system now known or to be invented, without
wrillen permission from the Publisher.
ISBN 9971-50-965-2
Printed in Singapore by JBW Printers and Binders Pte. Ltd.
Dedicated to
Peter G. Bergmann and Heinz Helfenstein,
who taught me physics and mathematics,
and to Lydia,
together they taught me humanity.
v

PREFACE
In this past decade, we have witnessed the enormous
growth of an interdisciplinary field of study, namely,
dynamical systems. Yet, dynamical systems, as a subfield of
mathematics, has been established since late last century by
Poincare [1881], and reinforced by Liapunov (1892]. Parts of
such a surge are due to many major advances in differential
topology, the geometric theory of differential equations,
algebraic geometry, nonlinear functional analysis, and
nonlinear global analysis, just to name a few. This is
partly due to computers being readily available. Nowadays,
any second-year college student can use a personal computer
to program the evolution of a nonlinear difference equation
and finq all kinds of chaotic behavior. With the new level
of sophistication of graphics, it can evolve into
"mathematical entertainment" or "arts". Each popular science
or engineering magazine has at least one article on
nonlinear dynamical systems a year. It not only caught the
fascination of the scientists, but also attracted the
attention of the enlightened public. Ten to fifteen years
ago, there were but a dozen papers on dynamical systems in
each major mathematics or physics journal per year. At
present, the majority of these journals contain sections on
nonlinear sciences, and indeed there are several new
journals solely devoted to this subject. Nonetheless, the
communication and infusion of knowledge between the
mathematicians working on the analytic approach and the
scientists and engineers working mostly on the applications
and numerical simulations have been less than ideal.
Part of the reason is cultural. Mathematicians tend to
approach the problem in a more generic sense, that is, they
tend to ask questions and look for answers for more general
properties and the underlying structures of the systems.
Books written by mathematicians usually treat the subject
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with mathematical rigor, but lack of some motivation for
wanting to study the underlying mathematical structures of
the system. The treatment by scientists and engineers
usually encompasses too many details and misses the
underlying structures which may transcend the usual
boundaries of various disciplines. Thus, scientists and
engineers may not be aware of the advances of the same type
of problems in other disciplines. We shall give examples of
such underlying mathematical structures for diverse
disciplines. This volume intends to bridge the gap between
these two categories of books treating nonlinear dynamical
systems. Thus, we would like to bridge the gap and foster
communication between scientists and mathematicians. In the
following, proofs of theorems are usually not given.
Instead, examples are provided so that the readers can get
the meaning of the theorems and definitions. In other words,
we would like the readers to get some sense of the concepts
and techniques of the mathematics as well as its "culture".
This volume is based on the lecture notes of a graduate
course I gave in 1983-4 while I was with TRW.
Chapter 1 introduces the concept of dynamical systems
and stability with examples from physics, biology, and
economics. We want to point out that even though the
differential or difference equations governing those
phenomena are different, nonetheless, the mathematical
procedures in analyzing them are the same. We also try to
motivate the reader about the concept and the need to study
the structural stability.
In Chapter 2, we assemble most of the definitions and
theorems about basic properties of algebra, points set (also
called general) topology, algebraic and differential
topology, differentiable manifolds and differential
geometry, which are needed in the course of the lectures.
Our main purpose is to establish notations, terminology,
concepts and structures. We provide definitions, examples,
and essential results without giving the arguments of proof.
The material presented is slightly more than absolutely
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necessary for the course of this lecture series. For
instance, we certainly can get by without explicitly talking
about algebraic topology, such as homotopy and homology.
Nonetheless, we do utilize the concepts of orientable
manifolds or spaces so that an everywhere non-zero volume
element can be found. We also discuss the connected
components and connected sums of a state space. Nor do we
have to discuss tubular neighborhoods, even though when
dealing with return maps, the stability of orbits and
periodic orbits, etc. we implicitly use the concept of
tubular neighborhoods. Nonetheless, these and many other
concepts and terminology are frequently used in research
literature. It is also our intent to introduce the reader to
a more sophisticated mathematical framework so that when the
reader ventures to research literature or further reading,
he will not feel totally lost due to different "culture" or
terminology. Furthermore, in global theory, the state spaces
may be differentiable manifolds with nontrivial topology. In
such cases, concepts from algebraic topology are at times
essential to the understanding.
In light of the above remarks, for the first reading the
reader may want to skip Section 2.3, part of Section 2.4, de
Rham cohomology part of Section 2.8 and Section 2.9.
The subjects discussed in Chapter 3 are not extensively
utilized in the subsequent chapters, at least not
explicitly. Nonetheless, some of the concepts and even
terminology do find their way to our later discussions. This
chapter is included, and indeed is lectured, to prepare the
readers with some concepts and understanding about global
analysis in general, and some techniques important to the
global theory of dynamical systems. In particular, the
reader may find it useful when they venture to theoretically
oriented research literature.
In the next chapter, we shall discuss the general theory
of dynamical systems. Most of the machinary developed in the
last chapter is not used immediately. Only in the last
section of Chapter 4, the idea of linearization of nonlinear
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differential operators will be utilized for the discussion
of linearization of dynamical systems. Nor does Chapter 5
depend on the material of Chapter 3. As a consequence, one
may want to proceed from Chapter 2 to Chapters 4 and 5,
except Section 6 of Chapter 4. Then come back to Chapter 3
and continue to Chapter 6.
In Section 4.7 we briefly discuss the linearization
process based on some results from Chapter 3. For most
scientists and engineers, the linearization process is
"trivial". Everyone has done this since their freshman year
many times. Unfortunately, the linearization process is the
most misunderstood and frequently mistaken procedure for
scientists and engineers in dealing with nonlinear phenomena
and nonlinear dynamical systems at large. This is
particularly true for highly nonlinear systems. Two
illustrations from our earlier training in mathematics can
make the point clear. First, in our freshman calculus
course, we learned about the condition of continuity of a
function at a point, and its derivative at that point. For
the derivative to have a meaning, not only the change of the
independent variable, say, ~x, has to be very small (i.e.,
local), but also the limits lim x ~ O+ ~YI ~x and lim x ~ 0_
~y;~x agree at x = x0• In other words, the usual
linearization makes sense, only when it is done locally, and
any "displacement" from the point has to be consistent. The
consistency can be illustrated by the following example. As
we shall discuss in Chapter 2, a differentiable manifold M
is a topological manifold (a topological space with certain
nice properties) endowed with a differentiable structure.
For any two points p and q in M, A, B are sufficiently small
neighborhoods of p and q in M respectively. At p and q in A
and B, a rectangular coordinate system can be attached to
each of p and q, and every variable looks linear in A and B.
What makes the coordinate system or the differentiable
structure go beyond the confine of A or B is that if the
intersection of A and B is non-empty, then the coordinate
values in A and B have to agree at all points in the
X
intersection of A and B. That is, the differentiable
structure has to be consistently and continuously agreed
between A to B. Consequently, for a highly nonlinear system
when one linearizes such a system, one has to be certain
that the linearization procedure is "consistent" in the
above sense. Otherwise, except at or very near the
equilibrium point(s) in the phase space, one has no
assurance as to the correctness of the results of
linearization.
Related to the above discussion of the linearization
procedure, we would also like to caution the reader about
numerical simulation of nonlinear dynamical systems, even
when only dealing with a local situation. First and foremost,
do not just code the differential equations and let the
computer do the rest. One has to analyze the characteristics
of the system, namely, how many and what kind of fixed
points, periodic orbits, attractors, etc. Then, let the
computer do the dirty work, and compare with the analysis to
~ee whether or not the numerical simulation agrees with the
analysis on the number and type of characteristics.
Otherwise, there is no way one can be sure the simulation is
correct and meaningful.
Another important point sometimes scientists and
engineers have overlooked is whether or not the system has a
Cauchy data set. If the system does not admit a Cauchy data
set, it is tantamount to say that the system does not admit
a unique time. Thus, the time evolution of the system loses
its meaning. One may argue that who cares about t ~ ~. But
even for a finite t, the system may become unpredictable,
with or without chaos, etc.
It is also appropriate to point out a practical point
which relates to the numerical schemes for studying the
nonlinear dynamical system. The implicit method is a very
popular and efficient scheme to study the linear or weakly
nonlinear differential equations. But for highly nonlinear
systems, this scheme may leads to erroneous results. This is
because the interpolation definitely introduces errors,
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