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arXiv · 1606.00073

The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations

D. J. W. Simpson
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Abstract

An attractor of a piecewise-smooth continuous system of differential equations can bifurcate from a stable equilibrium to a more complicated invariant set when it collides with a switching manifold under parameter variation. Here numerical evidence is provided to show that this invariant set can be chaotic. The transition occurs locally (in a neighbourhood of a point) and instantaneously (for a single critical parameter value). This phenomenon is illustrated for the normal form of a boundary equilibrium bifurcation in three dimensions using parameter values adapted from of a piecewise-linear model of a chaotic electrical circuit. The variation of a secondary parameter reveals a period-doubling cascade to chaos with windows of periodicity. The dynamics is well approximated by a one-dimensional unimodal map which explains this bifurcation structure. The robustness of the attractor is also investigated by studying the influence of nonlinear terms.

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D. J. W. Simpson. 2016-05-31. The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations. https://doi.org/10.1016/j.physleta.2016.07.033

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