Small-scale instabilities in dynamical systems with sliding

Sieber, J and Kowalczyk, P (2010) Small-scale instabilities in dynamical systems with sliding. Physica D, 239 (1-2). pp. 44-57. ISSN 0167- 2789

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We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit remains stable, and its local return map becomes piecewise linear. However, when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.

Item Type: Article
Uncontrolled Keywords: Vector fields with sliding; Singular perturbation; Discontinuity induced bifurcation, CICADA
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
Depositing User: Dr Piotr Kowalczyk
Date Deposited: 08 Jan 2010
Last Modified: 20 Oct 2017 14:12

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