Relation of stability and bifurcation properties between continuous and ultradiscrete dynamical systems via discretization with positivity: One dimensional cases

Abstract

The stability and bifurcation properties of one-dimensional discrete dynamical systems with positivity, which are derived from continuous ones by tropical discretization, are studied. The discretized time interval is introduced as a bifurcation parameter in the discrete dynamical systems, and the emergence condition of an additional bifurcation, flip bifurcation, is identified. The correspondence between the discrete dynamical systems with positivity and the ultradiscrete ones derived from them is discussed. It is found that the derived ultradiscrete max-plus dynamical systems can retain the bifurcations of the original continuous ones via tropical discretization and ultradiscretization.

Types and stability of fixed points for positivity-preserving discretized dynamical systems in two dimensions

“…With the above problem in mind, we have recently studied dynamical properties of the local bifurcations (transcritical, saddle-node, pitchfork) in one-dimensional dynamical systems [8,9]. In these previous studies, we argued stabilities of fixed points in continuous differential equations and their tropically discretized ones.…”