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arXiv · 1911.04578

Chaotic attractors from border-collision bifurcations: stable border fixed points and determinant-based Lyapunov exponent bounds

D. J. W. Simpson
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Abstract

The collision of a fixed point with a switching manifold (or border) in a piecewise-smooth map can create many different types of invariant sets. This paper explores two techniques that, combined, establish a chaotic attractor is created in a border-collision bifurcation in $\mathbb{R}^d$ $(d \ge 1)$. First, asymptotic stability of the fixed point at the bifurcation is characterised and shown to imply a local attractor is created. Second, a lower bound on the maximal Lyapunov exponent is obtained from the determinants of the one-sided Jacobian matrices associated with the fixed point. Special care is taken to accommodate points whose forward orbits intersect the switching manifold as such intersections can have a stabilising effect. The results are applied to the two-dimensional border-collision normal form focusing on parameter values for which the map is piecewise area-expanding.

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D. J. W. Simpson. 2019-11-11. Chaotic attractors from border-collision bifurcations: stable border fixed points and determinant-based Lyapunov exponent bounds. https://arxiv.org/abs/1911.04578

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