Subsumed homoclinic connections and infinitely many coexisting attractors in piecewise-linear maps

TL;DR

This paper demonstrates a deep connection between stable periodic solutions and homoclinic connections in piecewise-linear maps, revealing a codimension-three phenomenon with implications for understanding complex dynamics.

Contribution

It establishes an equivalence between stable periodic solutions and subsumed homoclinic connections in N-dimensional piecewise-linear maps, a novel codimension-three phenomenon.

Findings

Existence of infinitely many stable periodic solutions linked to subsumed homoclinic connections.

Exact expressions for solutions and manifolds in piecewise-linear maps enable analysis.

Practical method for identifying this phenomenon in parameter space demonstrated.

Abstract

We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for $N$-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solutions and components of the stable and unstable manifolds which are available because the maps are piecewise-linear. We also describe a practical approach for finding this phenomenon in the parameter space of a map and illustrate the results with the three-dimensional border-collision normal form.