Tropical linearization and stability analysis of discrete dynamical systems

TL;DR

This paper introduces a tropical linearization method for analyzing the stability of discrete dynamical systems using tropical algebra, providing criteria based on tropical eigenvalues for stability or instability.

Contribution

It develops a novel tropical linearization approach for stability analysis of ultradiscrete systems, linking tropical eigenvalues to fixed point stability.

Findings

Tropical fixed point at the origin is stable if the maximum tropical eigenvalue is negative.

The fixed point is unstable if the maximum tropical eigenvalue is positive.

Results are analogous to classical linearization stability criteria.

Abstract

The tropical semiring is a semiring of extended real numbers, where the operations of `max' and `+' replace the usual addition and multiplication, respectively. Difference equations obtained from the ultradiscrete limit of discrete dynamical systems are described in terms of the tropical semiring. We propose a tropical linearization approach for the stability analysis of difference equations, including those describing ulradiscrete dynamical systems. We show that the fixed point at the tropical origin is asymptotically stable if the maximum eigenvalue of the tropical Jacobian matrix is negative. On the other hand, it is unstable if the maximum eigenvalue of the tropical Jacobian matrix is positive. Since $0$ is the tropical multiplicative identity, these results are analogous to those in the usual linearization process.