Limit Cycles of Differential Equations

by Christopher, Colin; Li, Chengzhi

• ISBN13: 9783764384098
• ISBN10: 3764384093
• Format: Paperback
• Copyright: 2007-07-30
• Publisher: Birkhauser

Summary

This textbook contains the lecture series originally delivered at the "Advanced Course on Limit Cycles of Differential Equations" in the Centre de Rechercha Mathematica Barcelona in 2006. The topics covered are the center-focus problem for polynomial vector fields, and the application of abelian integrals to limit cycle bifurcations. Both topics are related to the authors' interests in Hilbert's sixteenth problem, but would also be of interest to those working more generally in the qualitative theory of dynamical systems.

Table of Contents

Around the Center-Focus Problem	p. 1
Preface	p. 3
Centers and Limit Cycles	p. 5
Outline of the Center-Focus Problem	p. 5
Calculating the Conditions for a Center	p. 9
Bifurcation of Limit Cycles from Centers	p. 10
Darboux Integrability	p. 17
Invariant Algebraic Curves	p. 17
The Darboux Method	p. 18
Multiple Curves and Exponential Factors	p. 21
Liouvillian Integrability	p. 25
Differential Fields and Liouvillian Extensions	p. 25
Proof of Singer's Theorem	p. 26
Riccati equations	p. 29
Symmetry	p. 33
Algebraic Symmetries	p. 33
Centers for analytic Lienard equations	p. 34
Centers for polynomial Lienard equations	p. 37
Cherkas' Systems	p. 41
Monodromy	p. 49
Some Basic Examples	p. 49
The Model Problem	p. 50
Applying Monodromy to the Model Problem	p. 51
The Tangential Center-Focus Problem	p. 55
Preliminaries	p. 56
Generic Hamiltonians	p. 57
Relative exactness	p. 59
Monodromy of Hyperelliptic Abelian Integrals	p. 63
Some Group Theory	p. 64
Monodromy groups of polynomials	p. 65
Proof of the theorem	p. 67
Holonomy and the Lotka-Volterra System	p. 71
The monodromy group of a separatrix	p. 72
Integrable points in Lokta-Volterra systems	p. 73
Other Approaches	p. 79
Finding components of the center variety	p. 79
Extending Centers	p. 80
An Experimental Approach	p. 82
Bibliography	p. 85
Abelian Integrals and Applications to the Weak Hilbert's 16th Problem	p. 91
Preface	p. 93
Hilbert's 16th Problem and Its Weak Form	p. 95
Hilbert's 16th Problem	p. 95
Weak Hilbert's 16th Problem	p. 99
Abelian Integrals and Limit Cycles	p. 111
Poincare-Pontryagin Theorem	p. 111
Higher Order Approximations	p. 116
The Integrable and Non-Hamiltonian Case	p. 120
The Study of the Period Function	p. 122
Estimate of the Number of Zeros of Abelian Integrals	p. 127
The Method Based on the Picard-Fuchs Equation	p. 127
A Direct Method	p. 130
The Method Based on the Argument Principle	p. 133
The Averaging Method	p. 138
A Unified Proof of the Weak Hilbert's 16th Problem for n=2	p. 143
Preliminaries and the Centroid Curve	p. 143
Basic Lemmas and the Geometric Proof of the Result	p. 145
The Picard-Fuchs Equation and the Riccati Equation	p. 149
Outline of the Proofs of the Basic Lemmas	p. 155
Proof of Theorem 4.6	p. 156
Bibliography	p. 159
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