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arXiv · 1904.05725

Stability index of linear random dynamical systems

Anna Cima·Armengol Gasull·Víctor Mañosa
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Abstract

Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of the stable manifold of the zero solution. In particular, for the $n$ dimensional case, the zero solution is globally asymptotically stable if and only if this stability index is $n.$ Fixed $n,$ let $X$ be the random variable that assigns to each linear random dynamical system its stability index, and let $p_k$ with $k=0,1,\ldots,n,$ denote the probabilities that $P(X=k)$. In this paper we obtain either the exact values $p_k,$ or their estimations by combining the Monte Carlo method with a least square approach that uses some affine relations among the values $p_k,k=0,1,\ldots,n.$ The particular case of $n$-order homogeneous linear random differential or difference equations is also studied in detail.

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Anna Cima, Armengol Gasull, Víctor Mañosa. 2019-04-11. Stability index of linear random dynamical systems. https://doi.org/10.14232/ejqtde.2021.1.15

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