A simple model for ultradiscrete Hopf bifurcation

TL;DR

This paper introduces a simple ultradiscrete model based on max-plus algebra that captures the key features of Hopf bifurcation, including limit cycles and excitability, serving as a potential normal form for ultradiscrete bifurcations.

Contribution

It proposes a novel ultradiscrete model derived from classical Hopf bifurcation models, providing a normal form for ultradiscrete Hopf bifurcation phenomena.

Findings

Limit cycles depend on bifurcation parameter values.

Limit cycles consist of finite discrete states.

The model exhibits excitability.

Abstract

Dynamical properties of ultradiscrete Hopf bifurcation, similar to those of the standard Hopf bifurcation, are discussed by proposing a simple model of ultradiscrete equations with max-plus algebra. In ultradiscrete Hopf bifurcation, limit cycles emerge depending on the value of a bifurcation parameter in the model. The limit cycles are composed of a finite number of discrete states. Furthermore, the model exhibits excitability. The model is derived from two different dynamical models with Hopf bifurcation by means of ultradiscretization; it is a candidate for a normal form for ultradiscrete Hopf bifurcation.