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.
Author(s): Colin Christopher, Chengzhi Li
Edition: 1
Year: 2007
Language: English
Pages: 171
Cover......Page 1
Advanced Courses in Mathematics CRM Barcelona......Page 3
Limit Cycles of Differential Equations......Page 4
9783764384098......Page 5
Foreword......Page 6
Contents......Page 8
I: Around the Center-Focus Problem - Colin Christopher ......Page 10
Preface ......Page 12
1.1 Outline of the Center-Focus Problem ......Page 14
1.2 Calculating the Conditions for a Center ......Page 18
1.3 Bifurcation of Limit Cycles from Centers ......Page 19
2.1 Invariant Algebraic Curves ......Page 26
2.2 The Darboux Method ......Page 27
2.3 Multiple Curves and Exponential Factors ......Page 30
3.1 Differential Fields and Liouvillian Extensions ......Page 34
3.2 Proof of Singer’s Theorem ......Page 35
3.3 Riccati equations ......Page 38
4.1 Algebraic Symmetries ......Page 42
4.2 Centers for analytic Liénard equations......Page 43
4.3 Centers for polynomial Liénard equations......Page 46
5 Cherkas’ Systems......Page 50
6.1 Some Basic Examples ......Page 58
6.2 The Model Problem......Page 59
6.3 Applying Monodromy to the Model Problem ......Page 60
7 The Tangential Center-Focus Problem ......Page 64
7.1 Preliminaries ......Page 65
7.2 Generic Hamiltonians ......Page 66
7.3 Relative exactness ......Page 68
8 Monodromy of Hyperelliptic Abelian Integrals ......Page 72
8.1 Some Group Theory ......Page 73
8.2 Monodromy groups of polynomials ......Page 74
8.3 Proof of the theorem ......Page 76
9 Holonomy and the Lotka–Volterra System ......Page 80
9.1 The monodromy group of a separatrix ......Page 81
9.2 Integrable points in Lokta–Volterra systems ......Page 82
10.1 Finding components of the center variety ......Page 88
10.2 Extending Centers ......Page 89
10.3 An Experimental Approach ......Page 91
Bibliography (missing) ......Page 94
II: Abelian Integrals and Applications to the Weak Hilbert’s 16th Problem - Chengzhi Li......Page 100
Preface ......Page 102
1.1 Hilbert’s 16th Problem ......Page 104
1.2 Weak Hilbert’s 16th Problem ......Page 108
2.1 Poincaré–Pontryagin Theorem......Page 120
2.2 Higher Order Approximations ......Page 125
2.3 The Integrable and Non-Hamiltonian Case ......Page 129
2.4 The Study of the Period Function ......Page 131
3.1 The Method Based on the Picard–Fuchs Equation ......Page 136
3.2 A Direct Method......Page 139
3.3 The Method Based on the Argument Principle ......Page 142
3.4 The Averaging Method......Page 147
4.1 Preliminaries and the Centroid Curve ......Page 152
4.2 Basic Lemmas and the Geometric Proof of the Result ......Page 154
4.3 The Picard–Fuchs Equation and the Riccati Equation ......Page 158
4.4 Outline of the Proofs of the Basic Lemmas ......Page 164
4.5 Proof of Theorem 4.6 ......Page 165
Bibliography ......Page 168
