Dynamical Systems: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 19–27, 1978

302 Pages·1980·18.98 MB·English

by  GUCKENHEIMER,  MOSER,  NEWHOUSE

C. Marchioro (Ed.)

Dynamical Systems

Lectures given at aSummerSchoolof the
Centro Internazionale Matematico Estivo (C.I.M.E.),
held in Bressanone (Bolzano), Italy,
June 19-27, 1978
C.I.M.E. Foundation
c/o Dipartimento di Matematica “U. Dini”
Viale Morgagni n. 67/a
50134 Firenze
Italy
[email protected]

ISBN 978-0-8176-3024-9 ISBN 978-1-4899-3743-8 (eBook)
DOI 10.1007/978-1-4899-3743-8

©Springer-Verlag Berlin Heidelberg 1980
Originally published by Springer-Verlag Berlin Heidelberg New York in 1980.
Reprint of the 1ste d. C.I.M.E., Ed. Liguori, Napoli & Birkhäuser 1980
W ith kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com
CONTENTS
J. GUCKENHEIMER: Bifurcations of Dynamical Systems Pag. 5
M. MISIUREWICZ.: Horseshoes for Continuous mappings Pag. 125
of an Interval
J. MOSER Various aspects of integrable
Hamiltonian systems Pag. 137
A. CHENCINER Hopf Bifurcation for Invariant Tori Pag. 197
S.E. NEWHOUSE Lectures on Dynamical Systems Pag. 209
CENTRO INTERNAZIONALE MATEMATICO ESTIYO
(C.I.M.E.)
BIFURCATIONS OF DYNAMICAL SYSTEMS
JOHN GUCKENHEIMER
Bifurcations of Dynamical Systems
John Guckenheimer
University of California, Santa Cruz
§1 Introduction
The subject of these lectures is the bifurcation theory of dynamical
systems. They are not comprehensive, as we take up some facets of bifurcation
theory and largely ignore others. In particular, we focus our attention on
finite dimensional systems of difference and differential equations and say
almost nothing about infinite dimensional systems. The reader interested in
the infinite dimensional theory and its applications should consult the re-
cent survey of Marsden [66) and the conference proceedings edited by
Rabinowitz [89) . We also neglect much of the multidimensional bifurcation
theory of singular points of differential equations. The systematic ex-
position of this theory is much more algebraic than the more geometric ques-
tions considered here, and Arnold [7,9) provides a good survey of work in this
area. We confine our interest to questions which involve the geometric orbit
structure of dynamical systems. We do make an effort to consider applications
of the mathematical phenomena illustrated. For general background about the
theory of dynamical systems consult [102). Our style is informal and our
intent is pedagogic. The current state of bifurcation theory is a mixture of
mathematical fact and conjecture. The demarcation between the proved and un-
proved is small [11). Rather than attempting to sort out this confused state
of affairs for the reader, we hope to provide the geometric insight which will
8
allow him to explore further.
The problems we deal with concern the asymptotic behavior of a dynamical
system as time tends to There are three kinds of systems we examine on a
finite dimensional CQ ) manifold M:
(1) smooth, continuous time flows ~: Mxm ~ M obtained from in-
tegrating a vector field or solving a system of ordinary differential equa-
tions on M,
(2) smooth, discrete time flows obtained by iterating a diffeomorphism
f: M ~ M, and
(3) non-invertible, discrete time semiflows obtained by iterating a
smooth map f: M ~ M.
Similar problems arise in each of these situations, and we shall pass rather
freely from one to the other as we select the most convenient setting for each
problem we study. In some applications, we shall find examples of all three,
A common procedure will be to pass from (1) to (2) when examining periodic
phenomena and from (2) to (3) as a singular limit or projection. At each of
these steps, the dimension of the state space M is reduced which makes our
analysis easier and our geometric intuition keener.
Dynamical systems theory has placed a major emphasis upon the elu-
cidation of the typical, or generic behavior of dynamical systems. we shall
say that a phenomenon is robust if it persists under perturbations of the
system. Bifurcation theory goes one step beyond the study of robust pro-
perties to examine those which are "almost" robust. There are several re-
lated ways of doing this. One way is to examine those properties of para-
metrized families of dynamical systems which persist under perturbation of
the family. Another outlook is to search for the typical way a robust pro-
perty disappears or changes into another robust property. This corre-
sponds to looking at the boundaries of regions in the space of dynamical
9
systems which enjoy similar properties. A third viewpoint is to search for
low codimension hypersurfaces in the space of systems with qualitatively
similar properties. We shall rely more upon the first approach than the
others, but all of them will appear at times during the discussion.
Attempts to provide a truly coherent approach to bifurcation theory have
been singularly unsuccessful. In contrast to the singularity theory for
smooth maps, viewing the problems as one of describing a stratification of a
space of dynamical systems quickly.leads to technical considerations that
draw primary attention from the geometric phenomena which need description.
This is not to say that the theory is incoherent but that it is a labyrinth
which can be better organized in terms of examples and techniques than in
terms of a formal· mathematical structure. Throughout its history, examples
suggested by applications have been a motivating force for bifurcation theory.
Hoping to convey this spirit we shall try to avoid as much abstraction as
possible. The penalty is that we shall leave the generality of many results
unspecified. The sophisticated reader desiring a precise accounting of the
state of the art will not find it here.
Let us lay out our pl~n of action. We begin with a description of the
elementary bifurcations. These are bifurcations which involve the mildest
lack of robustness in the local behavior of a dynamical system. We illus
trate both the continuous and discrete cases. All of the examples for con
tinuous flows can be described in the plane. In the plane or on the two
sptere, complicated recurrent behavior does not occur. Thus the theory
proceeds much more smoothly than in other situations. Sotomayor [104] has
given a systematic account of the codimension one phenomena encountered on
the sphere, and our exposition of the elementary bifurcations is largely
drawn from his work.
The two dimensional theory has been carried further to the analysis of
higher codimension bifurcations of singular points. The most complete
10
results are those of Dumortier [24]. Takens' concept of nonnal forms [111]
plays a central role, so we describe it in some detail and illustrate with
an example. The example we use is a codimension two bifurcation studied by
Takens [~14] and Bogdanov [16]. They independently computed the unfolding of
this bifurcation. Holmes and RanQ [SO] have used this example in a problem
involving non-linear oscillations, and we shall exhibit it in the differential
equations of a continuous, stirred tank chemical reactor.
After describing these local results, much of our attention will be de
voted to bifurcations which involve homoclinic behavior of dynamical systems.
Topological constraints prevent homoclinic phenomena from occurring in two
dimensional continuous flows, but it can be found with three dimensional
vector fields, two dimensional diffeomorphisms, and one dimensional maps.
Two examples, the forced van der Pol equation and the Lorenz attractor pro
vide the setting for us to describe the relationship between these three con
texts. The interest in homoclinic behavior comes from its role in dynamical
systems which have complicated asymptotic behavior (called "chaos" in some of
the applied literature) and sensitivity to initial conditions [91]. Math
ematical models with these properties are appearing in a rapidly growing list
of disciplines which now includes chemical kinetics, geodynamics, fluid
mechanics, electric~! circuit theory, ecology, and physiology.
The bifurcation behavior of one dimensional maps has been an area of
substantial advances in the past few years. We shall describe these results
in the language of kneading sequences introduced by Milnor and Thurston.
The order properties of the line restrict the order in which various bi
furcations can occur. This is the only situation for which it is possible
to give explicit relations between large sets of bifurcations. We shall
formulate topological results about rotation numbers for homeomorphisms
of the circle in these terms.
11
Armed with the kneading theory., we reconsider the three dimensional
vector fields introduced earlier. The bifurcation theory of the Lorenz
attractor gives us an example of a vector field which has moduli for topo-
logical equivalence. The Lorenz vector field is structurally unstable and
cannot be perturbed to a structurally stable vector field. Nonetheless, the
topological equivalence classes of all of the nearby vector fields are
c~aracterized by two parameters. In the case of the van der Pol equation,
o~ analysis suggests sensitive phenomena which have not been observed
numerically.
The next section is a brief introduction to bifurcations of population
models. We describe several population models which have complicated dy-
namics and bifurcation theory. These models raise a number of practical
questions that we mention in passing. The final section introduces a number
of other bifurcation phenomena which involve global bifurcations of dynamical
systems more than the material of the preceding sections. We touch upon
loops of saddle separatrices for plane vector fields, the wild hyperbolic
sets of plane diffeomorphisms studied by Newhouse [74], and the moduli of
saddle connections discovered by Palis [85].
Before embarking upon our exposition, it i~ necessary to set the frame-
work for our discussions. This requires that we provide a minimal back-
ground from the theory of dynamical systems and that we outline the general
strategy of bifurcation theory. After establishing this context, we give a
list of basic examples in the next section.
Let ~: MX~ + M be a continuous time, smooth flow on a smooth manifold
M. Recall that this means that ~t = ~(·,t) is a one parameter group of
diffeomorphisms of M. Associated with each flow is its vector field de-
fined by X(x) = dd t(~(x,t)). The map from flows to vector fields is
a bijection when M is a compact manifold. This is a consequence of the