Heteroclinic tangles in time-periodic equations

Fengjuan Chen, Ali Oksasoglu, Qiu-Dong Wang

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

In this paper we prove that, when a heteroclinic loop is periodically perturbed, three types of heteroclinic tangles are created and they compete in the space of μ where μ is a parameter representing the magnitude of the perturbations. The three types are (a) transient heteroclinic tangles containing no Gibbs measures; (b) heteroclinic tangles dominated by sinks representing stable dynamical behavior; and (c) heteroclinic tangles with strange attractors admitting SRB measures representing chaos. We also prove that, as μ. →. 0, the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections. The theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical Birkhoff-Melnikov-Smale method.

Original languageEnglish (US)
Pages (from-to)1137-1171
Number of pages35
JournalJournal of Differential Equations
Volume254
Issue number3
DOIs
StatePublished - Feb 1 2013

Fingerprint

Tangles
Chaos theory
Differential equations
SRB Measure
Heteroclinic Connection
Strange attractor
Gibbs Measure
Saddlepoint
Dynamical Behavior
Chaos
Unstable
Fixed point
Differential equation
Eigenvalue
Perturbation

Keywords

  • Chaotic dynamics
  • Heteroclinic tangles
  • Separatrix maps
  • Time-periodic differential equations

ASJC Scopus subject areas

  • Analysis

Cite this

Heteroclinic tangles in time-periodic equations. / Chen, Fengjuan; Oksasoglu, Ali; Wang, Qiu-Dong.

In: Journal of Differential Equations, Vol. 254, No. 3, 01.02.2013, p. 1137-1171.

Research output: Contribution to journalArticle

Chen, Fengjuan ; Oksasoglu, Ali ; Wang, Qiu-Dong. / Heteroclinic tangles in time-periodic equations. In: Journal of Differential Equations. 2013 ; Vol. 254, No. 3. pp. 1137-1171.
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AB - In this paper we prove that, when a heteroclinic loop is periodically perturbed, three types of heteroclinic tangles are created and they compete in the space of μ where μ is a parameter representing the magnitude of the perturbations. The three types are (a) transient heteroclinic tangles containing no Gibbs measures; (b) heteroclinic tangles dominated by sinks representing stable dynamical behavior; and (c) heteroclinic tangles with strange attractors admitting SRB measures representing chaos. We also prove that, as μ. →. 0, the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections. The theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical Birkhoff-Melnikov-Smale method.

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