Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle
(1998) In International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 8(2). p.359-375- Abstract
- We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codimension-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbit-flip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a... (More)
- We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codimension-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbit-flip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a Smale's horseshoe is found. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1263684
- author
- Zimmermann, Martin and Natiello, Mario LU
- organization
- publishing date
- 1998
- type
- Contribution to journal
- publication status
- published
- subject
- in
- International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
- volume
- 8
- issue
- 2
- pages
- 359 - 375
- publisher
- World Scientific Publishing
- external identifiers
-
- scopus:0032008443
- ISSN
- 0218-1274
- DOI
- 10.1142/S0218127498000218
- language
- English
- LU publication?
- yes
- id
- 339497b8-e8ca-4098-80a2-c57b3458eae5 (old id 1263684)
- alternative location
- http://s0218127498000218.pdf/
- date added to LUP
- 2016-04-01 12:21:46
- date last changed
- 2022-01-27 02:43:07
@article{339497b8-e8ca-4098-80a2-c57b3458eae5,
abstract = {{We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codimension-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbit-flip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a Smale's horseshoe is found.}},
author = {{Zimmermann, Martin and Natiello, Mario}},
issn = {{0218-1274}},
language = {{eng}},
number = {{2}},
pages = {{359--375}},
publisher = {{World Scientific Publishing}},
series = {{International Journal of Bifurcation and Chaos in Applied Sciences and Engineering}},
title = {{Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle}},
url = {{http://dx.doi.org/10.1142/S0218127498000218}},
doi = {{10.1142/S0218127498000218}},
volume = {{8}},
year = {{1998}},
}