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

TL;DR

This paper investigates how stability and bifurcation properties of one-dimensional continuous dynamical systems are preserved through tropical discretization and ultradiscretization, highlighting conditions for bifurcation emergence.

Contribution

It establishes a connection between continuous, discretized, and ultradiscrete systems, showing bifurcation preservation through tropical discretization with positivity constraints.

Findings

Ultradiscrete max-plus systems retain bifurcation structures of continuous systems.

Introduction of discretized time as a bifurcation parameter reveals flip bifurcations.

Discretization with positivity enables correspondence between continuous and ultradiscrete dynamics.

Abstract

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 emergence condition of an additional bifurcation, flip bifurcation, is identified. 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.