Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert's 16th problem? Section 2: The first part of Hilbert's 16th problem. Section 3: The second part of Hilbert's 16th problem: introduction. Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert's 16th problem. Section 7: Global and local bifurcations of Zq–equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.

Hilbert's 16th problem and bifurcations of planar polynomial vector fields

The second part of Hilbert’s 16th problem deals with polynomial differential equations in the plane. It remains unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the problem inspired significant progress in the geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry. The dramatic history of the problem, as well as related developments, are presented below. §1. The problem and its counterparts What may be said about the number and location of limit cycles of a planar polynomial vector field of degree n? (The limit cycle is an isolated closed orbit of a vector field.) This second part of Hilbert’s 16th problem appears to be one of the most persistent in the famous Hilbert list [H], second only to the Riemann ζ-function conjecture. Traditionally, Hilbert’s question is split into three, each one requiring a stronger answer. Problem 1. Is it true that a planar polynomial vector field has but a finite number of limit cycles? Problem 2. Is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only? The bound on the number of limit cycles in Problem 2 is denoted by H(n) and known as the Hilbert number. Linear vector fields have no limit cycles; hence H(1) = 0. It is still unknown whether or not H(2) exists. Problem 3. Give an upper bound for H(n). A solution to any of these problems implies a solution for the previous ones. Only the first problem is solved now. The positive answer was established in [E92], [I91]. There are analytic counterparts of Problems 1 and 2. Received by the editors December 2001. 2000 Mathematics Subject Classification. Primary 34Cxx, 34Mxx, 37F75.

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Centennial History of Hilbert’s 16th Problem

Normal forms and desingularization Singular points of planar analytic vector fields Local and global theory of linear systems Functional moduli of analytic classification of resonant germs and their applications Global properties of complex polynomial foliations Appendix. First aid Bibliography Index.

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Lectures on analytic differential equations

In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion of the tangent vector to an arc in a manifold is used in §8.2 and §8.3, and the notion of a vector field is introduced in an auxiliary capacity in §8.3. The Campbell-Hausdorff formula is introduced only to establish the First and Second Principles of §8.1 below; if you are willing to take those on faith the formula (and exercises dealing with it) can be skimmed. Exercises 8.27–8.29 give alternative descriptions of the Lie algebra associated to a Lie group, but can be skipped for now.

Lie Algebras and Lie Groups

Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let

 ${X_H} = {H_y}\frac{\partial }{{\partial x}} - {H_x}\frac{\partial }{{\partial y}}$
 be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of X

 H
 in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles.

The infinitesimal 16th Hilbert problem in the quadratic case

We prove that the Petrov module P f associated to an arbitrary semiweighted homogeneous polynomial f ϵ C[ x , y ] is free and finitely generated. We compute its generators and use this to obtain a lower bound for the maximal number of zeros of complete Abelian integrals.

Petrov modules and zeros of Abelian integrals

Nous montrons que toute fonction de Poincare-Pontryagin d'ordre superieur, associee au feuilletage polynomial perturbe defini par df - e (Pdx + Qdy) = 0, verifie une equation fuchsienne.

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Higher order Poincaré-Pontryagin functions and iterated path integrals

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The period function of a Hamiltonian quadratic system

We propose a program for finding the cyclicity of period annuli of quadratic
systems with centers of genus one. As a first step, we classify all such
systems and determine the essential one-parameter quadratic perturbations
which produce the maximal number of limit cycles. We compute the associated
Poincare-Pontryagin-Melnikov functions whose zeros control the number of
limit cycles. To illustrate our approach, we determine the cyclicity of the
annuli of two particular reversible systems.

Lubomir Gavrilov

Papers

The infinitesimal 16th Hilbert problem in the quadratic case

Petrov modules and zeros of Abelian integrals

Higher order Poincaré-Pontryagin functions and iterated path integrals

The period function of a Hamiltonian quadratic system

Perturbations of quadratic centers of genus one