The Center and Cyclicity Problems

In the last three decades, advances in methods for investigating polynomial ideals and their varieties have provided new possibilities for approaching two long-standing problems in the theory of differential equations: the Poincaré center problem and the cyclicity problem (the problem of bifurcation of limit cycles from singular trajectories).

Using a computational algebra approach, this work addresses the center and cyclicity problems as behaviors of dynamical systems and families of polynomial systems. The text first lays the groundwork for computational algebra and gives the main properties of ideals in polynomial rings and their affine varieties; this is followed by a discussion regarding the theory of normal forms and stability of differential equations. The center and cyclicity problems are then explored in detail.

The book contains numerous examples, pseudocode displays of all the computational algorithms, historical notes, nearly two hundred exercises, and an extensive bibliography. Completely self-contained, it is thus suitable mainly as a textbook for a graduate course in the subject but also as a reference for researchers.

• Groebner bases
• Lyapunov's method
• algebra
• bifurcations
• computational algebra
• limit cycles
• normal forms
• stability
• ordinary differential equations
• partial differential equations

From the reviews:

“This book treats in detail the problem of computing the center conditions and the cyclicity … . The book is written very clearly and … entirely self-contained. This makes it quite approachable for beginning researchers as well as a reference for those who know the area well. … furnished with a large number of exercises and examples, together with pseudo-code for the computations. An appendix gives some of these routines in Mathematica. Altogether an ideal book for those wishing to start research in this area.” (Colin J. Christopher, Mathematical Reviews, Issue 2010 h)

“The primary object in this interesting book is to study small-amplitude periodic solutions of planar polynomial autonomous systems which concerns the Poincare center (center-focus) problem and the cyclicity problem (the local 16th Hilbert problem). … the book is designed for the use as a primary textbook in an advanced graduate course … . The book is of interest not only for those working in the field of differential equations, but also for researches from the field of computational algebra and symbolic computation … .” (Alexander Grin, Zentralblatt MATH, Vol. 1192, 2010)

Policies and ethics