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 | |
| 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
Homoclinic shadowing. / Coomes, Brian A; Kocak, Huseyin; Palmer, Kenneth J.
In: Journal of Dynamics and Differential Equations, Vol. 17, No. 1, 01.2005, p. 175-215.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Homoclinic shadowing
AU - Coomes, Brian A
AU - Kocak, Huseyin
AU - Palmer, Kenneth J.
PY - 2005/1
Y1 - 2005/1
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
KW - Newton's method
KW - Pseudo orbits
KW - Shadowing
KW - Transversal homoclinic orbits
UR - http://www.scopus.com/inward/record.url?scp=23044457657&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=23044457657&partnerID=8YFLogxK
U2 - 10.1007/s10884-005-3146-x
DO - 10.1007/s10884-005-3146-x
M3 - Article
AN - SCOPUS:23044457657
VL - 17
SP - 175
EP - 215
JO - Journal of Dynamics and Differential Equations
JF - Journal of Dynamics and Differential Equations
SN - 1040-7294
IS - 1
ER -