Ultradiscrete equations for bifurcations in low-dimensional dynamical systems (Recent Developments in Dynamical Systems and their Applications)

Abstract

We discuss ultradiscrete equations in low-dimensional, especially, one-dimensional dynamical systems. The ultradiscrete equations are derived from the normal forms of the local bifurcations i.e., saddle-node, transcretical and pitchfork bifurcations, of one-dimensional continuous dynamical systems. With the aid of the graphical analysis, the dynamical properties of the obtained ultradiscrete equations are revealed. In particular, we show that these ultradiscrete equations exhibit the bifurcations characterized by piecewise linearity, say ultradiscrete bifurcations.