Three forms of dimension reduction for border-collision bifurcations

TL;DR

This paper investigates how dimension reduction can be applied to analyze border-collision bifurcations in switching dynamical systems, especially when eigenvalues change discontinuously but some remain continuous, simplifying complex chaotic dynamics.

Contribution

It introduces a novel approach to reduce the dimensionality of systems undergoing border-collision bifurcations by leveraging eigenvalue continuity, facilitating analysis of complex dynamics.

Findings

Dimension reduction is possible when one eigenvalue remains continuous during bifurcation.

The method simplifies the analysis of chaotic dynamics in switching systems.

Comparison with other codimension-two scenarios demonstrates the effectiveness of the approach.

Abstract

For dynamical systems that switch between different modes of operation, parameter variation can cause periodic solutions to lose or acquire new switching events. When this causes the eigenvalues (stability multipliers) associated with the solution to change discontinuously, we show that if one eigenvalue remains continuous then all local invariant sets of the leading-order approximation to the system occur on a lower dimensional manifold. This allows us to analyse the dynamics with fewer variables, which is particularly helpful when the dynamics is chaotic. We compare this to two other codimension-two scenarios for which dimension reduction can be achieved.