Border Collision Bifurcations and Coexisting Attractors in an Economic Bimodal Map

In this paper, we enhance the study of the tâtonnement model with cautious price adjustment introduced by Weddepohl in 1995. Specifically, we investigate the discrete time dynamics of the model when the original assumption of equal values for the maximum rate of increase and decrease of the price is relaxed. As a result, its analytic definition is expressed by a bimodal one-dimensional continuous piecewise smooth map which depends on three parameters. As it happens in general for bimodal maps, it is possible to describe the bifurcation structure in some regions of the parameter space of the model using the skew tent map scenario as a normal form. Nonetheless, we show that some border collision bifurcations which play a fundamental role for the asymptotic behavior of the map essentially pertain to its bimodal shape. Among them we highlight the ones that lead, for some specific parameter values, to the coexistence of two chaotic attractors. Moreover, we identify degenerate border collision bifurcations responsible for the peculiar shapes of the chaotic attractors that distinguish the model.