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

TL;DR

This paper investigates complex dynamical phenomena such as heterodimensional cycles and blenders within the border-collision normal form, a piecewise-linear map relevant to various applications, providing explicit parameter analysis and bifurcation insights.

Contribution

It explicitly identifies parameter values for heterodimensional cycles and blenders in the border-collision normal form, advancing understanding of unstable dimension variability in these systems.

Findings

Explicit parameter values for heterodimensional cycles

Identification of bifurcations in unstable dimension variability

Exact analysis facilitated by piecewise-linear structure

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.