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

5 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

Access to Document

10.1007/s10884-015-9437-y

Cite this

@article{82e91929646c4b54aec71f222aa3eb09,

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.}",

note = "Funding Information: The work of H.K. was supported by NSF Grant CMG-0417425 and by the National Research Council of Taiwan during several visits; the work of K.P. was supported by the National Research Council of Taiwan and by the Department of Computer Science at the University of Miami during several visits. Publisher Copyright: {\textcopyright} 2015, Springer Science+Business Media New York.",

year = "2016",

month = sep,

day = "1",

doi = "10.1007/s10884-015-9437-y",

language = "English (US)",

volume = "28",

pages = "1081--1114",

journal = "Journal of Dynamics and Differential Equations",

issn = "1040-7294",

publisher = "Springer New York",

number = "3-4",

}

TY - JOUR

T1 - A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics

AU - Coomes, Brian A.

AU - Koçak, Hüseyin

AU - Palmer, Kenneth J.

N1 - Funding Information: The work of H.K. was supported by NSF Grant CMG-0417425 and by the National Research Council of Taiwan during several visits; the work of K.P. was supported by the National Research Council of Taiwan and by the Department of Computer Science at the University of Miami during several visits. Publisher Copyright: © 2015, Springer Science+Business Media New York.

PY - 2016/9/1

Y1 - 2016/9/1

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.

KW - Approximate orbits

KW - Chaos

KW - Connecting orbits

KW - Heteroclinic cycle

KW - Homoclinic orbits

UR - http://www.scopus.com/inward/record.url?scp=84924193035&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84924193035&partnerID=8YFLogxK

U2 - 10.1007/s10884-015-9437-y

DO - 10.1007/s10884-015-9437-y

M3 - Article

AN - SCOPUS:84924193035

VL - 28

SP - 1081

EP - 1114

JO - Journal of Dynamics and Differential Equations

JF - Journal of Dynamics and Differential Equations

SN - 1040-7294

IS - 3-4

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this