Small-scale instabilities in dynamical systems with sliding

Details Small-scale instabilities in dynamical systems with sliding Small-scale instabilities in dynamical systems with sliding

2008-11-12

arxiv.org/abs/0811.1944v3

J. Sieber, P. Kowalczyk: Small-scale instabilities in dynamical systems with sliding. Physica D 239 pp.44-57, 2010

Nonlinear Sciences

Chaotic Dynamics

26 pages, 9 figures (corrected figure caption in Fig C2)

Scientific paper

10.1016/j.physd.2009.10.003

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.

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