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Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle

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Author

Summary, in English

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.

Department/s

  • Dynamical systems

Publishing year

1998

Language

English

Pages

359-375

Publication/Series

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volume

8

Issue

2

Document type

Journal article

Publisher

World Scientific Publishing

Topic

  • Mathematics

Status

Published

Research group

  • Analysis and Dynamics
  • Dynamical systems

ISBN/ISSN/Other

  • ISSN: 0218-1274