Periodic Cycles and Bifurcation Curves for One-Dimensional Maps with Two Discontinuities

Abstract

Abstract Starting from a family of discontinuous piece-wise linear one-dimensional maps, recently introduced as a dynamic model in social sciences, we propose a geometric method for finding the analytic expression of the bifurcation curves, in the space of the parameters, that bound the regions characterized by the existence of stable periodic cycles of any period. The conditions for the creation and the destruction of periodic cycles, as well as the analytic expressions of the bifurcation conditions, are obtained by studying the occurrence of border-collision bifurcations. In this paper we consider the case of maps formed by three linear portions separated by two discontinuity points. After summarizing the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point. Finally we show how the considered map can be obtained as the limit case of a family of continuous maps as a parameter is increased without bounds, and we show how the low period cycles, which are typical of the discontinuous map we consider, emerge from the more complex (i.e. chaotic) behaviors observed in the continuous maps when a parameter value is large enough. From the point of view of the social application the increasing values of the parameter can be interpreted as higher degrees of impulsivity of the agents involved in binary decisions.