Probabilistic basin of attraction and its estimation using two Lyapunov functions
type: Item
instance of: scholarly article
title: Probabilistic basin of attraction and its estimation using two Lyapunov functions
zbMATH Open document ID: 1398.93355
DOI: 10.1155/2018/2895658
author: Skuli Gudmundsson
author: Sigurdur Hafstein
published in: Complexity
publication date: 2018-10-04T00:00:00Z
review text: Summary: We study stability for dynamical systems specified by autonomous stochastic differential equations of the form \(d \boldsymbol{X}(t) = \boldsymbol{f}(\boldsymbol{X}(t)) d t + \boldsymbol{g}(\boldsymbol{X}(t)) d \mathbf{W}(t)\), with \((\mathbf{X}(t))_{t \geq 0}\) an \(\mathbb{R}^d\)-valued Itô process and \((\mathbf{W}(t))_{t \geq 0}\) an \(\mathbb{R}^Q\)-valued Wiener process, and the functions \(\mathbf{f} : \mathbb{R}^d \rightarrow \mathbb{R}^d\) and \(\mathbf{g} : \mathbb{R}^d \rightarrow \mathbb{R}^{d \times Q}\) are Lipschitz and vanish at the origin, making it an equilibrium for the system. The concept of asymptotic stability in probability of the null solution is well-known and implies that solutions started arbitrarily close to the origin remain close and converge to it. The concept therefore pertains exclusively to system properties local to the origin. We wish to address the matter in a more practical manner: Allowing for a (small) probability that solutions escape from the origin, how far away can they then be started? To this end we define a probabilistic version of the basin of attraction, the \(\gamma\)-BOA, with the property that any solution started within it stays close and converges to the origin with probability at least \(\gamma\). We then develop a method using a local Lyapunov function and a nonlocal one to obtain rigid lower bounds on \(\gamma\)-BOA.
Mathematics Subject Classification ID: 93E15
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 93D30
Mathematics Subject Classification ID: 93D20
zbMATH DE Number: 6946843
zbMATH Keywords: stability
zbMATH Keywords: dynamical systems
zbMATH Keywords: autonomous stochastic differential equations
zbMATH Keywords: asymptotic stability in probability
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/2895658
OpenAlex ID: W2795235188
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Wikidata QID: Q130042475
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description: scientific article; zbMATH DE number 6946843