Dynamics of homoclinic tangles in periodically perturbed second-order equations

Qiudong Wang, Ali Oksasoglu

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

We obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second-order ordinary differential equations with dissipation. Let Μ be the size of perturbation and ΛΜ be the entire homoclinic tangle. We prove in particular that (i) for infinitely many disjoint open sets of Μ, ΛΜ contains nothing else but a horseshoe of infinitely many branches; (ii) for infinitely many disjoint open sets of Μ, ΛΜ contains nothing else but one sink and one horseshoe of infinitely many branches; and (iii) there are positive measure sets of Μ so that ΛΜ admits strange attractors with Sinai-Ruelle-Bowen measure. We also use the equation. d2q/dt2+(Λ-γq2)dq/dt-q+q2=Μq2sinωt to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation.

Original languageEnglish (US)
Pages (from-to)710-751
Number of pages42
JournalJournal of Differential Equations
Volume250
Issue number2
DOIs
StatePublished - Jan 15 2011

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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