Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes

TL;DR

This paper develops an analytical method to approximate the probability distribution of responses in systems experiencing rare, heavy-tailed instabilities due to correlated stochastic excitation, validated through two prototype systems.

Contribution

It introduces a novel approach that separates stable and unstable response regimes to analytically capture heavy-tail behavior in intermittently unstable systems.

Findings

Analytical approximations align well with Monte Carlo simulations.

Method effectively captures heavy-tail statistics in response distributions.

Applicable to mechanical oscillators and turbulent systems with stochastic excitation.

Abstract

In this work, we consider systems that are subjected to intermittent instabilities due to external stochastic excitation. These intermittent instabilities, though rare, have a large impact on the probabilistic response of the system and give rise to heavy-tailed probability distributions. By making appropriate assumptions on the form of these instabilities, which are valid for a broad range of systems, we formulate a method for the analytical approximation of the probability distribution function (pdf) of the system response (both the main probability mass and the heavy-tail structure). In particular, this method relies on conditioning the probability density of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable state. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods for random dynamical systems. The unstable regime is inherently transient and in order to analyze this regime we characterize the statistics under the assumption of an exponential growth phase and a subsequent decay phase until the system is brought back to the stable attractor. The method we present allows us to capture the statistics associated with the dynamics that give rise to heavy-tails in the system response and the analytical approximations compare favorably with direct Monte Carlo simulations, which we illustrate for two prototype intermittent systems: an intermittently unstable mechanical oscillator excited by correlated multiplicative noise and a complex mode in a turbulent signal with fixed frequency, where multiplicative stochastic damping and additive noise model interactions between various modes.