Periodically forced double homoclinic loops to a dissipative saddle

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Abstract

In this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point. We prove the existence of many complicated dynamical objects for a large class of non-autonomous second order equations, ranging from attractive quasi-periodic torus to Newhouse sinks and Hénon-like attractors, and to rank one attractors with SRB measures and full stochastic behavior. This theory enables us to apply rigorously many profound dynamics theories on non-uniformly hyperbolic maps developed in the last forty years, including the Newhouse theory, the theory of SRB measures, the theory of Hénon-like attractors and the theory of rank one attractors, to the analysis of the strange attractors in a periodically perturbed Duffing equation.

Original languageEnglish (US)
JournalJournal of Differential Equations
DOIs
StateAccepted/In press - Jul 25 2015

ASJC Scopus subject areas

  • Analysis

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