On a Maximal Number of Period Annuli

Yelena Kozmina,Felix Sadyrbaev

doi:10.1155/2011/393875

Abstract

We consider equationx′′+g(x)=0, whereg(x)is a polynomial, allowing the equation to have multiple period annuli. We detect the maximal number of possible period annuli for polynomials of odd degree and show how the respective optimal polynomials can be constructed.

Highlights

Consider equation x g x 0, 1.1 where g x is an odd degree polynomial with simple zeros
Recall that a critical point O of 1.2 is a center if it has a punctured neighborhood covered with nontrivial cycles
A period annulus is every connected region covered with nontrivial concentric cycles

Summary

Introduction

Consider equation x g x 0, 1.1 where g x is an odd degree polynomial with simple zeros. Recall that a critical point O of 1.2 is a center if it has a punctured neighborhood covered with nontrivial cycles. A period annulus is every connected region covered with nontrivial concentric cycles. A period annulus enclosing several more than one critical points will be called a nontrivial period annulus. Our task in this article is to define the maximal number of nontrivial period annuli for 1.1.

Nontrivial Period Annuli

Polynomials

Existence of Polynomials with Optimal Distribution