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arXiv · 2211.05917

Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form

P. A. Glendinning·D. J. W. Simpson
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Abstract

Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to demonstrate these phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise-linear they are relatively amenable to an exact analysis and we are able to explicitly identify parameter values for heterodimensional cycles and blenders. For a one-parameter subfamily we identify bifurcations involved in a transition through unstable dimension variability. This is facilitated by being able to compute periodic solutions quickly and accurately, and the piecewise-linear form should provide a useful test-bed for further study.

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P. A. Glendinning, D. J. W. Simpson. 2022-11-10. Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form. https://doi.org/10.1103/physreve.108.l022202

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