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The Center and Cyclicity Problems Valery Romanovski

The Center and Cyclicity Problems By Valery Romanovski

The Center and Cyclicity Problems by Valery Romanovski


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Summary

Using a computational algebra approach, this work addresses the center and cyclicity problems as behaviors of dynamical systems and families of polynomial systems. The self-contained text contains exercises as well as historical notes and algorithms.

The Center and Cyclicity Problems Summary

The Center and Cyclicity Problems: A Computational Algebra Approach by Valery Romanovski

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 Poincare 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 isfollowed by a discussionregarding 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 suitablemainly as a textbook for a graduate course in the subject but also as a reference for researchers.

The Center and Cyclicity Problems Reviews

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)

Table of Contents

Polynomial Ideals and Their Varieties.- Stability and Normal Forms.- The Center Problem.- The Isochronicity and Linearizability Problems.- Invariants of the Rotation Group.- Bifurcations of Limit Cycles and Critical Periods.

Additional information

NPB9780817647261
9780817647261
0817647260
The Center and Cyclicity Problems: A Computational Algebra Approach by Valery Romanovski
New
Paperback
Birkhauser Boston Inc
2009-04-16
330
N/A
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