# Existence of Noise Induced Order, a Computer Aided Proof

## TL;DR

This paper provides a rigorous computer-aided proof of Noise Induced Order in a chaotic chemical reaction model, demonstrating how increased noise amplitude can induce a transition from chaos to order.

## Contribution

It introduces a novel computer-assisted method to rigorously prove the existence of Noise Induced Order in a specific dynamical system, with potential applicability to similar systems.

## Key findings

- Proves the transition from chaos to order as noise increases.
- Establishes Lipschitz continuity of the stationary measure under perturbations.
- Shows Lyapunov exponent varies H"older continuously with noise amplitude.

## Abstract

We prove the existence of Noise Induced Order in the Matsumoto-Tsuda model, where it was originally discovered in 1983 by numerical simulations. This is a model of the famous Belosouv-Zabotinsky reaction, a chaotic chemical reaction, and consists of a one dimensional random dynamical system with additive noise. The simulations showed that an increase in amplitude of the noise causes the Lyapunov exponent to decrease from positive to negative; we give a mathematical proof of the existence of this transition. The method we use relies on some computer aided estimates providing a certified approximation of the stationary measure in the $L^{1}$ norm. This is realized by explicit functional analytic estimates working together with an efficient algorithm. The method is general enough to be adapted to any piecewise differentiable dynamical system on the unit interval with additive noise. We also prove that the stationary measure of the system varies in a Lipschitz way if the system is perturbed and that the Lyapunov exponent of the system varies in a H\"older way when the noise amplitude increases.