Limit Cycles Coming from Some Uniform Isochronous Centers

Joan Torregrosa,Haihua Liang,Jaume Llibre

doi:10.1515/ans-2015-5010

Abstract

Abstract This article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers x ˙ = - y + x 2 ⁢ y ⁢ ( x 2 + y 2 ) n , y ˙ = x + x ⁢ y 2 ⁢ ( x 2 + y 2 ) n , $\dot{x}=-y+x^{2}y(x^{2}+y^{2})^{n},\quad\dot{y}=x+xy^{2}(x^{2}+y^{2})^{n},$ of degree 2 ⁢ n + 3 ${2n+3}$ and we perturb them inside the class of all polynomial differential systems of degree 2 ⁢ n + 3 ${2n+3}$ . For n = 0 , 1 ${n=0,1}$ we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n = 2 ${n=2}$ we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center.

Highlights

This article is about the weak 16–th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems
A weaker problem than the 16th Hilbert’s problem, known as the weak 16th Hilbert’s problem was proposed by Arnold [2], who asked for the maximum number Z(m, n) of isolated zeros of Abelian integrals of all polynomial 1–form of degree n over algebraic ovals of degree m, for more details on the weak 16th Hilbert’s problem see [4, 9, 19], and the hundreds of references quoted in these articles
The weak 16th Hilbert’s problem is a particular case of the problem of studying the maximum number of limit cycles that can bifurcate from the periodic orbits of a center of a polynomial differential system of degree m − 1 when it is perturbed inside the class of all polynomial differential systems of degree n

Summary

Introduction

This article is about the weak 16–th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. Here the main work is to study the maximum number of simple zeros of the obtained averaged functions, because not always the standard study of Extended Chebyshev systems (ET-systems) can be applied (see Appendix 2).

Results

Conclusion