Ultradiscrete Bifurcations for One Dimensional Dynamical Systems

Bifurcations of one dimensional dynamical systems are discussed based on some ultradiscrete equations. The ultradiscrete equations are derived from normal forms of one-dimensional nonlinear differential equations, each of which has saddle-node, transcritical, or supercritical pitchfork bifurcations. An additional bifurcation, which is similar to the flip bifurcation, is found in ultradiscrete equations for supercritical pitchfork bifurcations. Dynamical properties of these ultradiscrete bifurcations can be characterized with graphical analysis. As an example of application of our treatment, we focus on an ultradiscrete equation of the FitzHugh–Nagumo model and discuss its dynamical properties.