Bistability and bifurcation curves for a unimodal piecewise smooth map

This article deals with a two-parameter family of piecewise smooth unimodal maps with one break point, given by a branch of straight line connected at a kink point with a logistic branch. While the dynamic properties of smooth models are well understood, the bifurcation theory of piecewise smooth dynamical systems is much less developed. For such systems we are faced with the so-called border-collision bifurcations. This kind of bifurcation occurs when a trajectory collides with one of the break points of the map. In general, this leads to a discontinuous change in the derivative, and this may cause an abrupt transition in the structure and stability of attracting and repelling invariant sets. The effect of a border-collision bifurcation is not unique. We may have the abrupt transition from an attracting cycle to another attracting cycle of any period, or to cyclical chaotic intervals of any period. The object of the present paper is to describe the border collision bifurcations curves in the two-dimensional parameter plane, connected with the bifurcation curves of the smooth map, and their dynamics effect. This goal is reached by using the superstable cycles and their symbolic representation. Typical for piecewise smooth dynamical systems is the so-called “sausage” structure of the periodicity tongues in the parameter space, first described in the case of a piecewise linear 1D map, and then generalized for higher dimensions. These are evidenced also in the present paper. The peculiar property of this unimodal map is the existence of infinitely many regions of bistability in the parameter plane, corresponding to two coexisting attractors (cycles or chaotic intervals). Clearly their existence is strictly related with the piecewise-smooth nature of the map peculiarity (which is not with negative Schwarzian derivative) and the border collision bifurcations.