Polynomial first integrals via the Poincare series

[摘要] Many models of the nature use differential equation systems in the plane and applying the qualitative theory of differential equations, introduced by Poincare, the behavior of these systems in the majority of the cases can be known. One of the problems that persists in order to control the behavior of this type of systems, is to distinguish between a focus or a center (the center problem). The resolution of this problem goes through the computation of the so called Poincare-Liapunov constants. On the other hand, differential systems that have centers at certain singular points are very interesting due to the fact that perturbations of these systems give rise to rich bifurcations of limit cycles. Another open problem consist on determining what differential systems are integrable, i.e., differential systems that have a first integral, see for instance [5]. The search of first integrals is a classical tool in the classification of all trajectories of a differential system.

[发布日期] 2005-12-15

[关键词] center-focus problem;polynomial first integral;nonlinear differential equations