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Cycles homoclinic to chaotic sets; robustness and resonance

Ashwin, Peter (1997) Cycles homoclinic to chaotic sets; robustness and resonance Chaos, 7. pp. 207-220.


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For dynamical systems possessing invariant subspaces one can have a robust homoclinic cycle to a chaotic set. If such a cycle is stable, it manifests itself as long periods of quiescent chaotic behaviour interrupted by sudden transient `bursts'. The time between the transients increases as the trajectory approaches the cycle. This behavior for a cycle connecting symmetrically related chaotic sets has been called `cycling chaos' by Dellnitz et al. [IEEE Trans. Circ. Sys. I 42, 821–823 (1995)]. We characterise such cycles and their stability by means of normal Lyapunov exponents. We find persistence of states that are not Lyapunov stable but still attracting, and also states that are approximately periodic. For systems possessing a skew-product structure (such as naturally arises in chaotically forced systems) we show that the asymptotic stability and the attractivity of the cycle depends in a crucial way on what we call the footprint of the cycle. This is the spectrum of Lyapunov exponents of the chaotic invariant set in the expanding and contracting directions of the cycle. Numerical simulations and calculations for an example system of a homoclinic cycle parametrically forced by a Rössler attractor are presented; here we observe the creation of nearby chaotic attractors at resonance of transverse Lyapunov exponents.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Ashwin, Peter
Date : 18 March 1997
Additional Information : This is a pre-copy-editing, author-prepared, peer-reviewed PDF of an article published in Chaos, 7, 207-220. Click here to access the publisher's version. © 1997 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:41
Last Modified : 31 Oct 2017 14:02

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