Ultradiscrete bifurcations for one dimensional dynamical systems

Abstract

Bifurcations of one dimensional dynamical systems are discussed based on some ultradiscretized equations.The ultradiscrete equations are derived from the normal forms of one-dimensional nonlinear differential equations,each of which has saddle-node,transcritical,or pitchfork bifurcations. An additional bifurcation, which is similar to flip bifurcation,is also discussed. 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 FitzHugh-Nagumo model, and discuss its dynamical properties.