Bernstein Inequalities and Applications to Analytic Geometry and Differential Equations

✍ Scribed by J.-P. Francoise; Y. Yomdin

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 352 KB
• Volume: 146
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3029

✦ Synopsis

We introduce new methods of complex analysis (inequalities of Bernstein type) to study projections of analytic sets. These techniques are applied to problems of bifurcations of periodic orbits of differential equations such as the local Hilbert's 16 th problem.

1997 Academic Press

I. INTRODUCTION

This article relates to two different fields: codimension one analytic hypersurfaces defined on a polydisc and limit cycles of polynomial differential equations of the plane.

Methods of analytic Geometry are quite efficient to get bounds for the number of limit cycles (i.e., periodic orbits which are isolated among periodic orbits). The reader is directed, for instance, to Francoise Pugh [F P], Denkowska [D], and Il'ashenko Yakovenko [I Y]. In [F P], Gabrielov's theorem was used to prove that the number of limit cycles of period less than T (given any T ) is uniformly bounded.

For germs of analytic hypersurface (codimension 1), Gabrielov's theorem can be derived in an elementary way using Rolle's lemma and the property that the ring of germs of analytic functions is Noetherian (cf. Francoise Pugh [F P], Roussarie [R], and Yakovenko [Ya]).