A discrete-time dynamical system with four types of codimension-one bifurcations

Abstract

Usually, several discrete-time difference equations are shown in introductory courses on dynamical systems theory, in order to illustrate the occurrence of the most common bifurcations, which are saddle-node, transcritical, pitchfork, and flip. For instance, transcritical and flip bifurcations are found in the well-known logistic map. Here, a first-order difference equation undergoing these four types of bifurcations is presented. The bifurcation diagram is analytically derived and the rationale behind the construction of this equation is explained. The main goal of this didactic work is to give tips on how to write difference equations exhibiting various types of bifurcations, which can be associated with real-world scenarios.