Simultaneous Border-Collision and Period-Doubling Bifurcations

TL;DR

This paper analyzes the complex interaction of border-collision and period-doubling bifurcations in piecewise-smooth maps, revealing how these bifurcations unfold and lead to chaos under certain conditions.

Contribution

It provides a detailed unfolding of simultaneous border-collision and period-doubling bifurcations, including classification of local dynamics and chaos conditions.

Findings

A locus of period-doubling bifurcations emanates from border-collision points.

Border-collision bifurcations occur along curves tangent to period-doubling loci.

Conditions for chaos are established in one-dimensional cases.

Abstract

We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that, with sufficient non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics are completely classified; in particular, we give conditions that ensure chaos.