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

• A boundary equilibrium bifurcation involving stable and saddle foci is considered.
• A two-dimensional return map is constructed and approximated by a one-dimensional map.
• A trapping region and Smale horseshoe are identified for a Rössler-like attractor.
• Bifurcation diagrams reveal period-doubling cascades and windows of periodicity.

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 the bifurcation structure. The robustness of the attractor is also investigated by studying the influence of nonlinear terms.