Periodic perturbation of quadratic systems with two infinite heteroclinic cycles

Marcelo Messias

doi:10.3934/dcds.2012.32.1881

Article Contents

2012, Volume 32, Issue 5: 1881-1899. Doi: 10.3934/dcds.2012.32.1881

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Periodic perturbation of quadratic systems with two infinite heteroclinic cycles

Marcelo Messias^1,

1.
Departamento de Matemática, Estatística e Computação, Faculdade de Ciências e Tecnologia, Univ Estadual Paulista - UNESP, Cx.Postal 266, 19060-900, Presidente Prudente, SP

Received: October 31, 2010

Revised: February 28, 2011

Published: April 2012

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Abstract

We study periodic perturbations of planar quadratic vector fields having infinite heteroclinic cycles, consisting of an invariant straight line joining two saddle points at infinity and an arc of orbit also at infinity. The global study concerning the infinity of the perturbed system is performed by means of the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in $\mathbb{R}^3$, whose boundary plays the role of the infinity. It is shown that for certain type of periodic perturbation, there exist two differentiable curves in the parameter space for which the perturbed system presents heteroclinic tangencies and transversal intersections between the stable and unstable manifolds of two normally hyperbolic lines of singularities at infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the perturbed system solutions in a finite part of the phase space. Numerical simulations are performed for a particular example in order to illustrate this behavior, which could be called "the chaos arising from infinity", because it depends on the global structure of the quadratic system, including the points at infinity.

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Mathematics Subject Classification: Primary: 37C29, 37D45; Secondary: 37B55.

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