Homoclinic shadowing

Brian A Coomes; Huseyin Kocak; Kenneth J. Palmer

doi:10.1007/s10884-005-3146-x

Homoclinic shadowing

Brian A Coomes, Huseyin Kocak, Kenneth J. Palmer

Research output: Contribution to journal › Article

18 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

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Keywords

Chaos
Newton's method
Pseudo orbits
Shadowing
Transversal homoclinic orbits

ASJC Scopus subject areas

Algebra and Number Theory

Cite this

Coomes, B. A., Kocak, H., & Palmer, K. J. (2005). Homoclinic shadowing. Journal of Dynamics and Differential Equations, 17(1), 175-215. https://doi.org/10.1007/s10884-005-3146-x

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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.

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10.1007/s10884-005-3146-x