Bifurcations of Invariant Closed Curves in the Piecewise-Linear 3D Border Collision Normal Form

Abstract

Invariant closed curves (ICCs) have received increasing attention in the theory of smooth dynamical systems in recent years, with Neimark–Sacker bifurcations being one of the most prominent mechanisms through which they are created. This thesis investigates whether the theory developed for such invariant curves in the smooth setting can be transferred to piecewise-linear systems, and, where it cannot, identifies the mechanisms responsible for the resulting differences. The investigation is carried out for the piecewise-linear three-dimensional Border Collision Normal Form (BCNF), a piecewise-affine map that serves as a canonical local model for systems exhibiting border collision bifurcations. Within this framework, an attracting resonant ICC is understood as being composed of a saddle cycle, a stable node or focus cycle, and the connecting unstable one-dimensional set of the saddle. Building on this notion, the thesis focuses on the three elementary bifurcations associated with eigenvalues leaving the unit circle under parameter perturbation. The thesis examines for each possibility – leaving the unit circle via "+1", "-1" or as a complex conjugated pair anywhere through the unit circle – how an ICC responds when its constituent cycles cross such boundaries in close succession, comparing the resulting behavior to the corresponding smooth bifurcation scenario in order to identify both parallels and fundamental differences. A key difference is that all (non-border-collision) stability-related bifurcations in the BCNF are inherently degenerate; consequently, the stability of newly created invariant sets is governed by a global, border-collision-induced mechanism rather than by a local, smooth-theory-style criterion. As a byproduct of this investigation, a broad numerical exploration of a slice of the BCNF’s parameter space is conducted to catalogue the wide variety of geometric shapes that ICCs can take on in the nonsmooth setting – ranging from needleand star-shaped curves to tightly wound spirals – and to trace these distinctive shapes back to the underlying eigenvalue structure of the constituent cycles and their proximity to the switching manifold. The results are obtained using the numerical tools PeriodScanner, for the detection and stability classification of periodic orbits, and Manifis, for the high-precision computation of their one-dimensional invariant sets of their saddle cycles.