Basin of Attraction Analysis in Piecewise-Linear Systems with Big-Bang Bifurcation for the Period-Increment Phenomenon

This paper investigates the basins of attraction of periodic orbits arising in one-dimensional piecewise-linear discrete dynamical systems as the system parameters vary in a neighborhood of a Big-Bang bifurcation point associated with the period-increment phenomenon. In this setting, the Big-Bang point corresponds to a parameter value through which infinitely many bifurcation curves pass, leading to the successive emergence of periodic orbits whose periods increase incrementally. The analysis is carried out using a fully analytical approach, exploiting the one-dimensional nature of the system and the occurrence of border-collision bifurcations. Within this framework, we construct analytical sequences that characterize the convergence of any initial condition on the real line toward a periodic point belonging to a periodic orbit, either isolated or coexisting with another periodic orbit. As the main results, we explicitly characterize the basins of attraction of periodic orbits generated in the period-increment Big-Bang scenario and provide explicit analytical conditions on the system parameters for the existence of these periodic orbits. Moreover, we show that, in certain regions of the parameter plane, at most two periodic orbits can coexist, and we describe explicitly the structure of their corresponding basins of attraction. This work provides a new analytical perspective on basin organization in piecewise-linear systems exhibiting the period-increment phenomenon.