Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations

Abstract

We consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation–dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting, but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural networks and robustness of their estimates.

Asymptotic behaviour of a non‐autonomous multispecies Holling type II model with a complex type of noises

“…Nevertheless, these autonomous systems, including stochastic prey–predator model with Holling type II, (1) and (2), often struggle to accurately represent fluctuating ecosystems influenced by seasonal variations, temperature and humidity, as these factors introduce dynamic parameter variations. Thus, some authors have introduced stochastic non‐autonomous models with simple linear diffusion components to capture this phenomenon (Branicki & Uda, 2021; Du, 2014; Ji & Jiang, 2017; Jiang et al, 2008; Jiang et al, 2017; Li & Mao, 2009; Li & Shuai, 2010; Liu, 2015; Mehdaoui et al, 2023; Sengupta & Das, 2019; Zhang et al, 2017). However, most of these models concentrate on introducing stochasticity through linear environmental noise affecting the populations or the functional response, failing to account for the diverse impact of different noises on the dynamical system.…”