Limit Cycles of Differential Equations (Advanced Courses in Mathematics - CRM Barcelona)

Foreword
Contents
Part I Around the Center-Focus Problem
	Preface
	Chapter 1 Centers and Limit Cycles
		1.1 Outline of the Center-Focus Problem
		1.2 Calculating the Conditions for a Center
		1.3 Bifurcation of Limit Cycles from Centers
		Notes
	Chapter 2 Darboux Integrability
		2.1 Invariant Algebraic Curves
		2.2 The Darboux Method
		2.3 Multiple Curves and Exponential Factors
		Notes
	Chapter 3 Liouvillian Integrability
		3.1 Differential Fields and Liouvillian Extensions
		3.2 Proof of Singer’s Theorem
		3.3 Riccati equations
		Notes
	Chapter 4 Symmetry
		4.1 Algebraic Symmetries
		4.2 Centers for analytic Li´enard equations
		4.3 Centers for polynomial Li´enard equations
		Notes
	Chapter 5 Cherkas’ Systems
		Notes
	Chapter 6 Monodromy
		6.1 Some Basic Examples
		6.2 The Model Problem
		6.3 Applying Monodromy to the Model Problem
		Notes
	Chapter 7 The Tangential Center-Focus Problem
		7.1 Preliminaries
		7.2 Generic Hamiltonians
		7.3 Relative exactness
		Notes
	Chapter 8 Monodromy of Hyperelliptic Abelian Integrals
		8.1 Some Group Theory
		8.2 Monodromy groups of polynomials
		8.3 Proof of the theorem
		8.4 The symmety of the differential
		Notes
	Chapter 9 Holonomy and the Lotka–Volterra System
		9.1 The monodromy group of a separatrix
		9.2 Integrable points in Lokta–Volterra systems
		9.3 Holonomy on more general curves
		Notes
	Chapter 10 Other Approaches
		10.1 Finding components of the center variety
		10.2 Extending Centers
		10.3 An Experimental Approach
		Notes
		Bibliography
Part II Abelian Integrals and Applications to the Weak Hilbert’s 16th Problem
	Preface of the first edition
	Preface of the second edition
	Chapter 1 Hilbert’s 16th Problem and Its Weak Form
		1.1 Hilbert’s 16th Problem
			1.1.1 The finiteness problem
			1.1.2 Configuration of limit cycles
			1.1.3 Some results on quadratic systems
			1.1.4 Some results on cubic and higher degree systems
			1.1.5 Some results on Liénard equations
		1.2 Weak Hilbert’s 16th Problem
			1.2.1 The study of Z(2)
			1.2.2 Perturbations of elliptic and hyperelliptic Hamiltonians
	Chapter 2 Abelian Integrals and Limit Cycles
		2.1 Poincaré–Pontryagin Theorem
		2.2 Higher Order Approximations
		2.3 The Integrable and Non-Hamiltonian Case
		2.4 The Study of the Period Function
	Chapter 3 Estimate of the Number of Zeros of Abelian Integrals
		3.1 The Method Based on the Picard–Fuchs Equation
		3.2 A Direct Method
		3.3 The Method Based on the Argument Principle
		3.4 The Averaging Method
		3.5 The Averaging Method in Piecewise Systems
		3.6 Other Methods and Related Works
	Chapter 4 A Unified Proof of the Weak Hilbert’s 16th Problem for n=2
		4.1 Preliminaries and the Centroid Curve
		4.2 Basic Lemmas and the Geometric Proof of the Result
		4.3 The Picard–Fuchs Equation and the Riccati Equation
		4.4 Outline of the Proofs of the Basic Lemmas
		4.5 Proof of Theorem 4.6
	Bibliography