Border-collision bifurcations in a generalized piecewise linear-power map

Abstract

In this paper a class of generalized piecewise smooth maps is studied, which is linear at one side and nonlinear with power dependence at the other side. According to the value of the power in the term x^z the bifurcations occurring in this map are classified into five types: z >, z = 1,0 < z < 1,z = 0 and z < 0. We derive the occurrence conditions of border collision bifurcation and smooth fold and flip bifurcation, especially the codimension-2 bifurcation points describing the interaction between border collision bifurcation and smooth bifurcation. The general results are then applied to the specific cases of the power z, and different bifurcation scenarios are shown for individual cases, from which the period-adding scenario is found to be general for any power.