Limit Cycles and Differential Equations Colin Christopher and Chengzhi Li Series: Advanced Courses in Mathematics - CRM Barcelona Birkhäuser (2007), 171 pp., Price: US$39.95 ISBN 978-3-7643-8409-8.
Reviewer: Douglas S. Shafer Mathematics Department, University of North Carolina at Charlotte, Charlotte, North Carolina, U.S.A.

This book is a set of polished lecture notes based on lectures given by the authors in the Advanced Course on Limit Cycles and Differential Equations at the Centre de Recerca Matematica at the Universitat Autonoma de Barcelona in June of 2006 as part of its year-long research program on Hilbert's Sixteenth Problem. As stated in the forward, "The common goal of the two sets of notes is to help young mathematicians enter a very active area of research lying on the borderline between dynamical systems, analysis and applications." The book is divided into two parts of roughly equal length, each one written by one of the authors. The first, by Colin Christopher of the University of Plymouth, is entitled "Around the Center-Focus Problem." The second, by Chengzhi Li of Peking University, is entitled "Abelian Integrals and Applications to the Weak Hilbert's Sixteenth Problem." Both parts of the book are written at a high level and, in keeping with the purpose both of the book and the lecture series, are appropriate really only for those whose research interests center on the theory of limit cycles in polynomial systems. Each one has an extensive bibliography; even allowing for overlap there are probably well over 200 entries.

To describe the specific content of the book we must first recall that the second half of Hilbert's Sixteenth Problem is to obtain a uniform bound \(H(n)\) on the number of limit cycles that can appear in the phase portrait of any planar system of ordinary differential equations whose right hand sides are polynomials of degree at most \(n\). Let \(H(x,y)\) be a polynomial of degree \(m \geq 2\) that vanishes at the origin and whose level curves near zero include a family of ovals about the origin, let \(P(x,y)\) and \(Q(x,y)\) be polynomials for which \(\max\{ \deg P, \deg Q \} = n \geq 2\), and consider the system

a polynomial perturbation of a Hamiltonian system with a center at the origin. The abelian integral (by which is meant the integral of a rational 1-form about an algebraic oval) \(I(h) = \int _{\gamma_h} P \, dy - Q \, dx\), where \(\gamma_h\) is the small oval \(H = h\) about the origin, is the first order term when the displacement along a transversal in the period annulus, in one turn about the origin, is expanded in powers of \(\epsilon\). The weak (or tangential or infinitesimal) Hilbert 16th Problem is to find the maximum number \(Z(m,n)\) of isolated zeros of the abelian integral \(I\). The reason for the terminology is that this number is an upper bound for the number of limit cycles that bifurcate from the period annulus for small \(\epsilon\), hence \(\widetilde Z(n) = Z(n+1,n) \leq H(n)\).

In his first chapter Chengzhi Li reviews this theory and similar matters and gives a survey of known results. His second chapter is a careful and thorough treatment of the relation of abelian integrals to the creation of limit cycles from the period annuli of Hamiltonian centers (with a section on more general centers). The third chapter discusses methods for estimating the number of zeros of \(Z(m,n)\), including methods based on Picard-Fuchs differential equations, the argument principle, and averaging. Finally, his last chapter presents the unified proof, by him and his collaborators, that \(\widetilde Z(2) = 2\).

In his first chapter Colin Christopher outlines the center-focus problem and explains its relevance to the problem of counting limit cycles in polynomial systems. Centers of polynomial systems typically arise through two mechanisms, presence of sufficiently many invariant algebraic curves (the Darboux theory) and presence of symmetry, which are discussed in the next four chapters. This comprises the first half of this part of the book. The remainder of this part of the book is devoted to approaches to centers based on the concept of monodromy, the study of how objects that depend on a parameter, but are in some regard locally constant, change as the parameter is varied along a non-trivial path. The first situation treated gives the flavor of the rest. Let us consider the system (1). The origin is a tangential center if \(I(h) \equiv 0\) for \(h\) sufficiently near zero. It is proved that for a generic Hamiltonian a tangential center occurs when the 1-form \(P \, dy - Q \, dx\) is ``relatively exact," meaning that it has the form \(dA + BdH\) for some polynomials \(A\) and \(B\). The method of proof is to complexify and show that in moving around a non-trivial loop in \(\mathbb{C}\) minus the critical values of \(H\) one obtains the whole homology group \(H_1(\varphi_h, \mathbb{Z})\) of the Riemann surface \(\varphi_h = \{(x,y) : H(x,y) = h \}\). This result (first proved by Yu. Ilyashenko) can then be used to count the dimension of the space of non-trivial perturbations of the original Hamiltonian system and thus provide a lower bound on the maximum number of limit cycles that can be produced by such perturbations. A particularly interesting aspect of this part of the book is the inclusion of many suggestions for promising areas for future research.

In summary, this work is a high level treatment of several specialized topics in the qualitative theory of differential equations by two highly respected leaders in the field. To my mind it exactly fulfills its stated objective. Birkhäuser is to be commended for making these valuable lecture notes economically available.