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Tropical linearization and stability analysis of discrete dynamical systems

Yuki Nishida, Sennosuke Watanabe, Yoshihide Watanabe

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.

Tropical linearization and stability analysis of discrete dynamical systems

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 is the tropical multiplicative identity, these results are analogous to those in the usual linearization process.
Paper Structure (5 sections, 7 theorems, 66 equations)

This paper contains 5 sections, 7 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Tropical semiring
  3. Tropical linear approximation
  4. Stability analysis
  5. Concluding remarks

Key Result

Theorem 2.1

The maximum eigenvalue of $A \in \mathbb{R}_{\max}^{n \times n}$ is identical to the maximum average weight of all circuits in $\mathcal{G}(A)$.

Theorems & Definitions (12)

  • Theorem 2.1: Green1979
  • Theorem 2.2: Cyclicity Theorem, Cohen1985
  • Proposition 3.1
  • Example 3.2
  • Remark 3.3
  • Lemma 3.4
  • Theorem 4.1
  • Lemma 4.2: Sergeev2009
  • Theorem 4.3
  • Example 4.4
  • ...and 2 more