The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations

TL;DR

This paper demonstrates that in piecewise-smooth systems, a stable equilibrium can instantaneously bifurcate into a chaotic attractor at a critical parameter value, supported by numerical simulations and a simplified map model.

Contribution

It provides numerical evidence of instantaneous chaos onset during a boundary equilibrium bifurcation in three-dimensional systems, linking it to a unimodal map approximation.

Findings

Chaotic attractors can emerge instantaneously at bifurcation points.

A period-doubling cascade to chaos is observed.

The dynamics are well approximated by a one-dimensional unimodal map.

Abstract

An attractor of a piecewise-smooth continuous system of differential equations can bifurcate from a stable equilibrium to a more complicated invariant set when it collides with a switching manifold under parameter variation. Here numerical evidence is provided to show that this invariant set can be chaotic. The transition occurs locally (in a neighbourhood of a point) and instantaneously (for a single critical parameter value). This phenomenon is illustrated for the normal form of a boundary equilibrium bifurcation in three dimensions using parameter values adapted from of a piecewise-linear model of a chaotic electrical circuit. The variation of a secondary parameter reveals a period-doubling cascade to chaos with windows of periodicity. The dynamics is well approximated by a one-dimensional unimodal map which explains this bifurcation structure. The robustness of the attractor is also investigated by studying the influence of nonlinear terms.