Stochastic stability of the classical Lorenz flow under impulsive type forcing

TL;DR

This paper introduces a new impulsive random perturbation model for the Lorenz flow to analyze its stochastic stability, relevant for modeling slowly varying phenomena like climate change effects.

Contribution

It presents a novel impulsive perturbation approach for the Lorenz system and proves its stochastic stability using random dynamical systems and semi-Markov process frameworks.

Findings

Proves stochastic stability of the Lorenz flow under impulsive perturbations.

Models slowly varying phenomena such as climate forcing.

Uses piecewise deterministic Markov process framework.

Abstract

We introduce a novel type of random perturbation for the classical Lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow. The perturbation acts on the system in an impulsive way, hence is not of diffusive type as those already discussed in \cite{Ki}, \cite{Ke}, \cite{Me}. Namely, given a cross-section $\mathcal{M}$ for the unperturbed flow, each time the trajectory of the system crosses $\mathcal{M}$ the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the umperturbed flow is then carryed on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes.