Center conditions for a class of planar rigid polynomial differential systems

Jaume Llibre; Roland Rabanal

doi:10.3934/dcds.2015.35.1075

Article Contents

2015, Volume 35, Issue 3: 1075-1090. Doi: 10.3934/dcds.2015.35.1075

This issue Previous Article Recurrence properties and disjointness on the induced spaces Next Article On the limit cycles bifurcating from an ellipse of a quadratic center

Center conditions for a class of planar rigid polynomial differential systems

Jaume Llibre^1, and
Roland Rabanal^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.
Departamento de Ciencias, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, San Miguel, Lima 32

Received: February 28, 2014

Revised: July 31, 2014

Published: February 2015

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Abstract

In general the center--focus problem cannot be solved, but in the case that the singularity has purely imaginary eigenvalues there are algorithms to solving it. The present paper implements one of these algorithms for the polynomial differential systems of the form \[ \dot x= -y + x f(x) g(y),\quad \dot y= x+y f(x) g(y), \] where $f(x)$ and $g(y)$ are arbitrary polynomials. These differential systems have constant angular speed and are also called rigid systems. More precisely, in this paper we give the center conditions for these systems, i.e. the necessary and sufficient conditions in order that they have an uniform isochronous center. In particular, the existence of a focus with the highest order is also studied.

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Mathematics Subject Classification: Primary: 34C07, 34C05; Secondary: 34C25, 37G15.

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