Homoclinic shadowing

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

doi:10.1007/s10884-005-3146-x

Homoclinic shadowing

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

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

19 Scopus citations

Abstract

A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in R ⁿ is presented. It is a computer-assisted technique with two main components. First, a global Newton's method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.

Original language	English (US)
Pages (from-to)	175-215
Number of pages	41
Journal	Journal of Dynamics and Differential Equations
Volume	17
Issue number	1
DOIs	https://doi.org/10.1007/s10884-005-3146-x
State	Published - Jan 2005

Keywords

Chaos
Newton's method
Pseudo orbits
Shadowing
Transversal homoclinic orbits

ASJC Scopus subject areas

Analysis

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title = "Homoclinic shadowing",

abstract = "A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in R n is presented. It is a computer-assisted technique with two main components. First, a global Newton's method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.",

keywords = "Chaos, Newton's method, Pseudo orbits, Shadowing, Transversal homoclinic orbits",

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

note = "Funding Information: The Work of B. Coomes and H. Koc¸ak were partially supported by a grant from The National Science Foundation. K. Palmer{\textquoteright}s Work was supported by The National Science Council (Taiwan) grant NSC 92-2115-M-002-022.",

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doi = "10.1007/s10884-005-3146-x",

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AU - Coomes, Brian A.

AU - Koçak, Hüseyin

AU - Palmer, Kenneth J.

N1 - Funding Information: The Work of B. Coomes and H. Koc¸ak were partially supported by a grant from The National Science Foundation. K. Palmer’s Work was supported by The National Science Council (Taiwan) grant NSC 92-2115-M-002-022.

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N2 - A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in R n is presented. It is a computer-assisted technique with two main components. First, a global Newton's method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.

AB - A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in R n is presented. It is a computer-assisted technique with two main components. First, a global Newton's method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.

KW - Chaos

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KW - Pseudo orbits

KW - Shadowing

KW - Transversal homoclinic orbits

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