A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics

Brian A. Coomes; Hüseyin Koçak; Kenneth J. Palmer

doi:10.1007/s10884-015-9437-y

A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics

Brian A. Coomes, Hüseyin Koçak, Kenneth J. Palmer

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

A general theorem that guarantees the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous differential equation in Rⁿnear a suitable approximate connecting orbit given the invertibility of a certain explicitly given matrix is proved. Numerical implementation of the theorem is described using five examples including two Sil’nikov saddle-focus homoclinic orbits and a Sil’nikov saddle-focus heteroclinic cycle.

Original language	English (US)
Pages (from-to)	1081-1114
Number of pages	34
Journal	Journal of Dynamics and Differential Equations
Volume	28
Issue number	3-4
DOIs	https://doi.org/10.1007/s10884-015-9437-y
State	Published - Sep 1 2016

Keywords

Approximate orbits
Chaos
Connecting orbits
Heteroclinic cycle
Homoclinic orbits

ASJC Scopus subject areas

Analysis

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10.1007/s10884-015-9437-y

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title = "A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics",

abstract = "A general theorem that guarantees the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous differential equation in Rnnear a suitable approximate connecting orbit given the invertibility of a certain explicitly given matrix is proved. Numerical implementation of the theorem is described using five examples including two Sil{\textquoteright}nikov saddle-focus homoclinic orbits and a Sil{\textquoteright}nikov saddle-focus heteroclinic cycle.",

keywords = "Approximate orbits, Chaos, Connecting orbits, Heteroclinic cycle, Homoclinic orbits",

author = "Coomes, {Brian A.} and H{\"u}seyin Ko{\c c}ak and Palmer, {Kenneth J.}",

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N2 - A general theorem that guarantees the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous differential equation in Rnnear a suitable approximate connecting orbit given the invertibility of a certain explicitly given matrix is proved. Numerical implementation of the theorem is described using five examples including two Sil’nikov saddle-focus homoclinic orbits and a Sil’nikov saddle-focus heteroclinic cycle.

AB - A general theorem that guarantees the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous differential equation in Rnnear a suitable approximate connecting orbit given the invertibility of a certain explicitly given matrix is proved. Numerical implementation of the theorem is described using five examples including two Sil’nikov saddle-focus homoclinic orbits and a Sil’nikov saddle-focus heteroclinic cycle.

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KW - Connecting orbits

KW - Heteroclinic cycle

KW - Homoclinic orbits

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A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics

Abstract

Keywords

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