Bounding the first exit from the basin: Independence Times and Finite-Time Basin Stability

TL;DR

This paper introduces a probabilistic framework for assessing the stability of deterministic systems under large, jump-like perturbations by defining independence time and finite-time basin stability, providing practical bounds for system resilience.

Contribution

It develops the concept of independence time and finite-time basin stability, offering new tools to evaluate system stability under rare, large perturbations.

Findings

Derived a lower bound for basin stability under rare perturbations

Introduced the concept of independence time for perturbed systems

Finite-time basin stability is a new probabilistic stability measure

Abstract

We study the stability of deterministic systems given sequences of large, jump-like perturbations. Our main result is to dervie a lower bound for the probability of the system to remain in the basin, given that perturbations are rare enough. This bound is efficient to evaluate numerically. To quantify rare enough, we define the notion of the independence time of such a system. This is the time after which a perturbed state has probably returned close to the attractor, meaning that subsequent perturbations can be considered separately. The effect of jump-like perturbations that occur at least the independence time apart is thus well described by a fixed probability to exit the basin at each jump, allowing us to obtain the bound. To determine the independence time, we introduce the concept of finite-time basin stability, which corresponds to the probability that a perturbed trajectory returns to an attractor within a given time. The independence time can then be determined as the time scale at which the finite-time basin stability reaches its asymptotic value. Besides that, finite-time basin stability is a novel probabilistic stability measure on its own, with potential broad applications in complex systems.