## 2006.413: Homoclinic Snaking near a Heteroclinic Cycle in Reversible Systems

2006.413:
J. Knobloch and T. Wagenknecht
(2005)
*Homoclinic Snaking near a Heteroclinic Cycle in Reversible Systems.*
Physica D, 206 (1-2).
pp. 82-93.
ISSN 0167-2789

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DOI: 10.1016/j.physd.2005.04.018

## Abstract

Snaking curves of homoclinic orbits have been found numerically in a number of ODE models from water wave theory and structural mechanics. Along such a curve infinitely many fold bifurcation of homoclinic orbits occur. Thereby the corresponding solutions spread out and develop more and more bumps (oscillations) about their own centre. A common feature of the examples is that the systems under consideration are reversible. In this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria, one of which is a bi-focus. Using Lin’s method a snaking of 1-homoclinic orbits is proved to occur in an unfolding of such a cycle. Further dynamical consequences are discussed. As an application a system of Boussinesq equations is considered, where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi-focus equilibrium.

Item Type: | Article |
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Uncontrolled Keywords: | Bifurcation; Heteroclinic cycle; Homoclinic snaking; Lin’s method; Boussinesq system |

Subjects: | PACS 2003 > 02 Mathematical methods in physics PACS 2003 > 05 Statistical physics, thermodynamics, and nonlinear dynamical systems |

MIMS number: | 2006.413 |

Deposited By: | Thomas Wagenknecht |

Deposited On: | 07 December 2006 |

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