Border-collision bifurcations in a generalizedpiecewise linear-power map

Abstract

In this paper a class of generalized piecewise smooth maps isstudied, which is linear at one side and nonlinear with powerdependence at the other side. According to the value of the powerin the term $x^z$, the bifurcations occurring in this map areclassified into five types: $z>1$, $z=1$, $0&#60;z<1$, $z=0$, and$z<0$. We derive the occurrence conditions ofborder collision bifurcation and smooth fold and flipbifurcation, especially the codimension-2bifurcation points describing the interaction between bordercollision bifurcation and smooth bifurcation. The general results are thenapplied to the specific cases of the power $z$, and differentbifurcation scenarios are shown for individual cases, from which theperiod-adding scenario is found to be general for any power.