Unstable dimension variability and heterodimensional cycles 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 emphasise the existence of 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. We explicitly identify parameter values for heterodimensional cycles and argue that the existence of heterodimensional cycles between two given saddles can be dense in parameter space. We numerically identify key
bifurcations associated with unstable dimension variability by studying a one-parameter subfamily that transitions continuously from where periodic solutions are all saddles to where they are all repellers. This is facilitated by fast and accurate computations of periodic solutions; indeed the piecewise-linear form should provide a useful test-bed for further study.
Original languageEnglish
Article numberL022202
JournalPhysical Review E: covering statistical, nonlinear, biological, and soft matter physics
Volume108
DOIs
Publication statusPublished - 3 Aug 2023

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