Stochastic Stability of Monotone Dynamical Systems. I. The Irreducible Cooperative Systems

TL;DR

This paper investigates the stochastic stability of irreducible cooperative monotone dynamical systems, showing that under noise, the system's long-term behavior concentrates on stable equilibria using large deviation theory.

Contribution

It establishes the stochastic stability of cooperative irreducible systems and characterizes the zero-noise limit as a convex combination of stable equilibria.

Findings

Zero-noise limit concentrates on stable equilibria

Stochastic stability characterized using large deviation theory

Applicable to classical positive feedback systems

Abstract

The current series of papers is concerned with stochastic stability of monotone dynamical systems by identifying the basic dynamical units that can survive in the presence of noise interference. In the first of the series, for the cooperative and irreducible systems, we will establish the stochastic stability of a dynamical order, that is, the zero-noise limit of stochastic perturbations will be concentrated on a simply ordered set consisting of Lyapunov stable equilibria. In particular, we utilize the Freidlin--Wentzell large deviation theory to gauge the rare probability in the vicinity of unordered chain-transitive invariant set on a nonmonotone manifold. We further apply our theoretic results to the stochastic stability of classical positive feedback systems by showing that the zero-noise limit is a convex combination of the Dirac measures on a finite number of asymptotically stable equilibria although such system may possess nontrivial periodic orbits.