Dynamical Systems Lectures given at a Summer School of the Centro In
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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
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 1st ed. C.I.M.E., Ed. Liguori, Napoli & Birkhäuser 1980 With kind permission of C.I.M.E.
Printed on acid-free paper
Springer.com
CONTENTS
J. GUCKENHEIMER: Bifurcations of Dynamical Systems M. MISIUREWICZ.: Horseshoes for Continuous mappings of an Interval Various aspects of integrable J. MOSER Hamiltonian systems Hopf Bifurcation for Invariant Tori A. CHENCINER Lectures on Dynamical Systems S.E. NEWHOUSE
Pag. 5 Pag. 125
Pag. 137 Pag. 197 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 recent 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 questions 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. proved is small [11).
The demarcation between the proved and un-
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 finite
dimensional
(1)
Q)
C
There are three kinds of systems we examine on a manifold
M:
smooth, continuous time flows
~:
Mxm
~
M obtained from in-
tegrating a vector field or solving a system of ordinary differential equations on
M,
(2)
smooth, discrete time flows obtained by iterati
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