DEGENERATE PERIOD ADDING BIFURCATION STRUCTURE OF ONE-DIMENSIONAL BIMODAL PIECEWISE LINEAR MAPS

Abstract

Motivated by a problem in the management of ecological populations, we study the bifurcation structure known as period adding structure for a family of one-dimensional bimodal piecewise linear maps. This structure is rather degenerate compared to the general case usually addressed in the literature. The degeneracy affects both the type of border collision bifurcations constituting the bifurcation structure and the number and location of the bifurcation points in the parameter space. We provide rigorous theoretical results that yield a complete description of the degenerate border collision bifurcations and a full determination of the bifurcation structure. This allows us to extend partial results previously reported about a similar problem. From an ecological point of view, we provide numerical simulations showing potential risks and opportunities associated with the bifurcation structure studied here. Moreover, we provide examples of applications of our results to some well-known population models, showing that the period adding structure ranges from very simple to very intricate.