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Perturbation Theory of Dynamical Systems

Perturbation Theory of Dynamical Systems - Perturbation...

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arXiv:math.HO/0111178 v1 15 Nov 2001 Perturbation Theory of Dynamical Systems Nils Berglund Department of Mathematics ETH Z¨urich 8092 Z¨urich Switzerland Lecture Notes Summer Semester 2001 Version: November 14, 2001
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Preface This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Summer term 2001, to undergraduate Mathematics and Physics students. It covers a few selected topics from perturbation theory at an introductory level. Only certain results are proved, and for some of the most important theorems, sketches of the proofs are provided. Chapter 2 presents a topological approach to perturbations of planar vector fields. It is based on the idea that the qualitative dynamics of most vector fields does not change under small perturbations, and indeed, the set of all these structurally stable systems can be identified. The most common exceptional vector fields can be classified as well. In Chapter 3, we use the problem of stability of elliptic periodic orbits to develop perturbation theory for a class of dynamical systems of dimension 3 and larger, including (but not limited to) integrable Hamiltonian systems. This will bring us, via averaging and Lie-Deprit series, all the way to KAM-theory. Finally, Chapter 4 contains an introduction to singular perturbation theory, which is concerned with systems that do not admit a well-defined limit when the perturbation parameter goes to zero. After stating a fundamental result on existence of invariant manifolds, we discuss a few examples of dynamic bifurcations. An embarrassing number of typos from the first version has been corrected, thanks to my students’ attentiveness. Files available at http://www.math.ethz.ch/ berglund Please send any comments to [email protected] Z¨urich, November 2001 3
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Contents 1 Introduction and Examples 1 1.1 One-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Forced Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Singular Perturbations: The Van der Pol Oscillator . . . . . . . . . . . . . . 10 2 Bifurcations and Unfolding 13 2.1 Invariant Sets of Planar Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 The Poincar´ e–Bendixson Theorem . . . . . . . . . . . . . . . . . . . 21 2.2 Structurally Stable Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Definition of Structural Stability . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Peixoto’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Singularities of Codimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.1 Saddle–Node Bifurcation of an Equilibrium . . . . . . . . . . . . . . 29 2.3.2 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.3 Saddle–Node Bifurcation of a Periodic Orbit . . . . . . . . . . . . . 34 2.3.4 Global Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 Local Bifurcations of Codimension 2 . . . . . . . . . . . . . . . . . . . . . . 39 2.4.1 Pitchfork Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.2 Takens–Bogdanov Bifurcation . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Remarks on Dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Regular Perturbation Theory 49 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.1 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.2 Basic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Averaging and Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.1 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.2 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.3 Lie–Deprit Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Kolmogorov–Arnol’d–Moser Theory . . . . . . . . . . . . . . . . . . . . . . 67 3.3.1 Normal Forms and Small Denominators . . . . . . . . . . . . . . . . 68 3.3.2 Diophantine Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.3 Moser’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.4 Invariant Curves and Periodic Orbits . . . . . . . . . . . . . . . . . . 77 3.3.5 Perturbed Integrable Hamiltonian Systems . . . . . . . . . . . . . . 81 5
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0 CONTENTS 4 Singular Perturbation Theory 85 4.1 Slow Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1.1 Tihonov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.1.2
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