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Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle

Zimmermann, Martin and Natiello, Mario LU (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)
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author
organization
publishing date
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
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
2008-12-08 10:31:54
date last changed
2017-03-14 13:42:25
@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},
  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},
  volume       = {8},
  year         = {1998},
}