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

Jifa Jiang, Xi Sheng, Yi Wang

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.

Stochastic Stability of Monotone Dynamical Systems. I. The Irreducible Cooperative 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.
Paper Structure (6 sections, 10 theorems, 110 equations, 1 figure, 1 table)

This paper contains 6 sections, 10 theorems, 110 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Main Results
  4. Quasipotential for Monotone Flows
  5. The Proof of Main Results
  6. Applications to Stochastic Biochemical Control Circuit

Key Result

Lemma 2.1

For each compact set $K\subset \mathbb{R}^r$, there is a $L>0$ such that

Figures (1)

  • Figure 1: The graph of $h(z)$ for $z\geq0$.

Theorems & Definitions (19)

  • Lemma 2.1
  • proof
  • Definition 2.2: Freidlin--Wentzell uniform large deviations principle over $\mathcal{K}$
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 4.1
  • Proposition 4.2: Quasipotential from unordered chain-transitive sets
  • proof
  • ...and 9 more