A one-dimensional piecewise-smooth map with some economic applications

We study the dynamics of a one-dimensional piecewise-smooth map consisting of two branches topologically related to the logistic function. This map is obtained by simplifying a recently proposed economic banking-duopoly framework; at the same time, it is mathematically interesting for its own right. The bifurcation structure of the economically meaningful regions of the parameter space of this dynamical map is analytically studied and detailed, especially focusing on border-collision bifurcations of its fixed points, and on the specific case where the fixed points are virtual. Numerical simulations are provided to support our analytic results.