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Turing Instability Suppressed and Induced by Multiplicative Noise in Brusselator System

Qasim Khan, Anthony Suen, Bao Quoc Tang

Abstract

The effect of multiplicative noise to the Turing instability of the Brusselator system is investigated. We show that when the noise acts on both of the concentrations with the same intensities, then the Turing instability is suppressed provided that the intensities are sufficiently large. This aligns with the stabilizing effect of multiplicative noise in partial differential equations. Utilizing the linearized system, we can quantify the magnitude of noise which stabilizes the system. On the other hand, when the noise is involving only one concentration, then the Turing instability can be triggered with suitable intensities. These are confirmed by numerical simulations.

Turing Instability Suppressed and Induced by Multiplicative Noise in Brusselator System

Abstract

The effect of multiplicative noise to the Turing instability of the Brusselator system is investigated. We show that when the noise acts on both of the concentrations with the same intensities, then the Turing instability is suppressed provided that the intensities are sufficiently large. This aligns with the stabilizing effect of multiplicative noise in partial differential equations. Utilizing the linearized system, we can quantify the magnitude of noise which stabilizes the system. On the other hand, when the noise is involving only one concentration, then the Turing instability can be triggered with suitable intensities. These are confirmed by numerical simulations.

Paper Structure

This paper contains 11 sections, 2 theorems, 36 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Stochastic Brusselator system
  3. Linear stability analysis for the stochastic system
  4. The case of the same noise intensities
  5. Proof:
  6. Proof:
  7. The case of different noise intensities
  8. Numerical Simulations for Nonlinear Systems
  9. Turing instability prevented by noise
  10. Turing instability induced by noise
  11. Conclusion and Future Work

Key Result

Lemma 3.1

There exists $\sigma_0\in \mathbb R$ and $\omega > 0$ such that for all $\sigma^2 \ge \sigma_0^2$ where $\mathfrak{S}(M)$ denotes the spectrum of a matrix $M$.

Figures (10)

  • Figure 1: Turing instability without noise of the linearized system \ref{['LR_sys']}.
  • Figure 2: Suppression by noise with the same noise intensities.
  • Figure 3: Global stability of system \ref{['LR_sys']} with high noise intensities $\sigma_u = \sigma_v = 2$.
  • Figure 4: Destabilization by noise with different noise intensities.
  • Figure 5: The final profiles of solutions and Lyapunov exponents for different eigenmodes.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Lemma 3.1
  • Theorem 3.2