Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form
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.
Keywords
Cite
@article{arxiv.2211.05917,
title = {Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form},
author = {P. A. Glendinning and D. J. W. Simpson},
journal= {arXiv preprint arXiv:2211.05917},
year = {2023}
}