CENTERS FOR THE KUKLES HOMOGENEOUS SYSTEMS WITH EVEN DEGREE

,Jaume Llibre Jaume Llibre,Claudia Valls Claudia Valls

doi:10.11948/2017093

Abstract

For the polynomial differential system Ẋ=-y, ẏ=x + Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n ≥ 2 has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all n even. More precisely, we prove both conjectures in the case n=4 and for n ≥ 6 even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of n odd was studied in[8].

Highlights

Is a center were found in [11]
In [4] the center problem for the class of system (1.1) with a7 = 0 was solved, and it was proved that at most five limit cycles bifurcate from the origin
In [12] it was solved the center problem for system (1.1) when a2 = 0 and it was proved that at most six limit cycles bifurcate from the origin

Summary

Introduction

Is a center were found in [11]. During may years it had been thought that these conditions were necessary and sufficient conditions, but some new centers have been found later on, see [2,10]. In the theorem we prove both conjectures in the case n = 4 and for n ≥ 6 even under the assumption that if the system has a center or an isochronous center at the origin, it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space.

Results

Conclusion