Poincare Map Method for Limit Cycles in a Max-Plus Dynamical System

TL;DR

This paper introduces a Poincare map approach to analyze the stability and basin structures of limit cycles in a two-dimensional max-plus dynamical system, simplifying the system to a one-dimensional piecewise linear map.

Contribution

The paper applies the Poincare map method to max-plus systems, revealing hierarchical basin structures and connecting it with existing piecewise linear mapping techniques.

Findings

Basins exhibit hierarchical structures.

Poincare map reduces system to one-dimensional form.

Relationship with integrable system theory discussed.

Abstract

Dynamical properties of limit cycles in a two-dimensional max-plus dynamical system are discussed. We apply a Poincare map method to the limit cycles in order to reveal their stabilities. This method reduces the two dimensional system to a one-dimensional piecewise linear discrete dynamical system composed of the Poincare map and its cross section. Basins for one of the limit cycles are derived by considering the inverse system of the original model. It is found that the obtained basins show a hierarchic structure. Relationship between the Poincare map method and the method of piecewise linear mapping studied in integrable system theory for the limit cycles is discussed.