Border Collision Bifurcations in n-Dimensional Piecewise Linear Discontinuous Maps

TL;DR

This paper analyzes border collision bifurcations in n-dimensional discontinuous maps using piecewise linear approximation, extending previous work to derive conditions for fixed points and illustrating with a 2D example.

Contribution

It provides a general framework for analyzing border collision bifurcations in n-dimensional discontinuous maps, expanding on prior work by Feigin.

Findings

Derived conditions for existence of fixed points before and after bifurcation

Extended analysis to n-dimensional maps using piecewise linear approximation

Illustrated method with a specific 2D discontinuous map example

Abstract

In this paper we report some important results that help in analizing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. Proceeding along similar lines, we obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific example of a two-dimensional discontinuous map.