On the relative coexistence of fixed points and period-two solutions near border-collision bifurcations

TL;DR

This paper classifies the possible scenarios for the coexistence of fixed points and period-two solutions near border-collision bifurcations in piecewise-smooth maps, showing that one previously hypothesized scenario cannot occur, leaving four feasible cases.

Contribution

It demonstrates that one of the five previously proposed coexistence scenarios is impossible, refining the understanding of bifurcation behavior in piecewise-smooth maps.

Findings

One scenario of coexistence is impossible.

Four feasible coexistence scenarios remain.

Provides a complete classification of scenarios near border-collision bifurcations.

Abstract

At a border-collision bifurcation a fixed point of a piecewise-smooth map intersects a surface where the functional form of the map changes. Near a generic border-collision bifurcation there are two fixed points, each of which exists on one side of the bifurcation. A simple eigenvalue condition indicates whether the fixed points exist on different sides of the bifurcation (this case can be interpreted as the persistence of a single fixed point), or on the same side of the bifurcation (in which case the bifurcation is akin to a saddle-node bifurcation). A similar eigenvalue condition indicates whether or not there exists a period-two solution on one side of the bifurcation. Previously these conditions have been combined to obtain five distinct scenarios for the existence and relative coexistence of fixed points and period-two solutions near border-collision bifurcations. In this Letter, it is shown that one of these scenarios, namely that two fixed points exist on one side of the bifurcation and a period-two solution exists on the other side of the bifurcation, cannot occur. The remaining four scenarios are feasible. Therefore there are exactly four distinct scenarios for fixed points and period-two solutions near border-collision bifurcations.