Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems

TL;DR

This paper investigates how non-hyperbolic cycles in nonsmooth systems undergo bifurcations when interacting with discontinuity boundaries, revealing a common geometric feature of bifurcation curves with rigorous proof and practical examples.

Contribution

It identifies a universal geometric pattern in bifurcations involving non-hyperbolic cycles and discontinuities, supported by rigorous proof and diverse scientific examples.

Findings

A third bifurcation curve emanates tangentially at intersections.

The phenomenon is proven under very general conditions.

Examples from science and engineering illustrate the results.

Abstract

We analyse three codimension-two bifurcations occurring in nonsmooth systems, when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in continuous- and discrete-time) interacts with one of the discontinuity boundaries characterising the system's dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of nonsmooth system and the geometry of the involved boundary, we concentrate on the geometric features that are common to all scenarios. We show that, at a generic intersection between the smooth and discontinuity induced bifurcation curves, a third curve generically emanates tangentially to the former. This is the discontinuity induced bifurcation curve of the secondary invariant set (the other cycle, the double-period cycle, or the torus, respectively) involved in the smooth bifurcation. The result can be explained intuitively, but its validity is proven here rigorously under very general conditions. Three examples from different fields of science and engineering are also reported.