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Abstract:
We study the stability of deterministic systems, given sequences of large, jump-like perturbations. Our main result is the derivation of 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.
A central problem in the study of dynamical systems is quantifying the stability of an attractor. Looking at small perturbations leads to linear stability and Lyapunov exponents. Many applications instead require us to consider the chance that a large perturbation will not kick the system out of the basin of attraction called basin stability.1 If we add noise to the system, we can further ask about the expected time it takes for the system to first exit the basin.
In this work, we show how to use ideas from basin stability to study first exit times, if the noise is a sequence of large, but sufficiently rare jumps. We quantify precisely what sufficiently rare means and derive a bound for the first exit time distribution. Crucially, this bound is expressed in terms of quantities that can be evaluated efficiently for high-dimensional systems.
Thus, the results of this paper will have wide applications to high-dimensional systems with noise that can be approximated in the above way, e.g. power grids subjected to intermittent wind and solar power, neuronal networks, or ecosystems.
