Periodic perturbation of quadratic systems with two infinite heteroclinic cycles

Abstract

We study periodic perturbations of planar quadratic vector fieldshaving infinite heteroclinic cycles, consisting of an invariantstraight line joining two saddle points at infinity and an arc oforbit also at infinity. The global study concerning the infinity ofthe perturbed system is performed by means of the Poincarécompactification in polar coordinates, from which we obtain a systemdefined on a set equivalent to a solid torus in $\mathbb{R}^3$, whoseboundary plays the role of the infinity. It is shown that forcertain type of periodic perturbation, there exist twodifferentiable curves in the parameter space for which the perturbedsystem presents heteroclinic tangencies and transversalintersections between the stable and unstable manifolds of twonormally hyperbolic lines of singularities at infinity. Thetransversality of the manifolds is proved using the Melnikov methodand implies, via the Birkhoff-Smale Theorem, in a complex dynamicalbehavior of the perturbed system solutions in a finite part of thephase space. Numerical simulations are performed for a particularexample in order to illustrate this behavior, which could be called'the chaos arising from infinity', because it depends on theglobal structure of the quadratic system, including the points atinfinity.