Subharmonic bifurcations near infinity

Abstract

In this paper are considered periodic perturbations, depending on two parameters, of planar polynomial vector fields having an annulus of large amplitude periodic orbits, which accumulate on a symmetric infinite heteroclinic cycle. Such periodic orbits and the heteroclinic trajectory can be seen only by the global consideration of the polynomial vector fields on the whole plane, and not by their restrictions to any compact region. The global study envolving infinity is performed via the Poincaré Compactification. It is shown that, for certain types of periodic perturbations, one can seek, in a neighborhood of the origin in the parameter plane, curves Cm of subharmonic bifurcations, to which the periodically perturbed system has subharmonics of order m, for sufficiently large integer m. Also, in the quadratic case, it is shown that, as m tends to infinity, the tangent lines of the curves Cm, at the origin, approach the curve C of bifurcation to heteroclinic tangencies, related to the periodic perturbation of the infinite heteroclinic cycle. The results are similar to those stated by Chow, Hale and Mallet-Paret in [4], although the type of systems and perturbations considered there are quite different, since they are restricted to compact regions of the plane.