Incomplete approach to homoclinicity in a model with bent-slow manifold geometry

Rajesh, S and Ananthakrishna, G (2000) Incomplete approach to homoclinicity in a model with bent-slow manifold geometry. In: Physica D: Nonlinear Phenomena, 140 (3-4). pp. 193-212.

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Abstract

The dynamics of a model, originally proposed for a type of instability in plastic flow, has been investigated in detail. The bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical behavior, including period bubbling and period adding or Farey sequences. The complex bifurcation sequences, characterized by mixed mode oscillations, exhibit partial features of Shilnikov and Gavrilov–Shilnikov scenario. Utilizing the fact that the model has disparate time scales of dynamics, we explain the origin of the relaxation oscillations using the geometrical structure of the bent-slow manifold. Based on a local analysis, we calculate the maximum number of small amplitude oscillations, s, in the periodic orbit of $L^s$ type, for a given value of the control parameter. This further leads to a scaling relation for the small amplitude oscillations. The incomplete approach to homoclinicity is shown to be a result of the finite rate of ‘softening’ of the eigenvalues of the saddle focus fixed point. The latter is a consequence of the physically relevant constraint of the system which translates into the occurrence of back-to-back Hopf bifurcation.

Item Type:	Journal Article
Publication:	Physica D: Nonlinear Phenomena
Publisher:	Elsevier
Additional Information:	Copyright of this article belongs to Elsevier.
Keywords:	Chaos;Stick-slip dynamics;Slow manifold;Mixed mode oscillations.
Department/Centre:	Division of Chemical Sciences > Materials Research Centre
Date Deposited:	28 Jul 2008
Last Modified:	19 Sep 2010 04:48
URI:	http://eprints.iisc.ac.in/id/eprint/15313

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