Stability index of linear random dynamical systems
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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 pk with k=0,1,…,n, denote the probabilities that P(X=k). In this paper we obtain either the exact values pk, or their estimations by combining the Monte Carlo method with a least square approach that uses some affine relations among the values pk,k=0,1,…,n. The particular case of n-order homogeneous linear random differential or difference equations is also studied in detail.
CitationCima, A.; Gasull, A.; Mañosa, V. Stability index of linear random dynamical systems. "Electronic journal of qualitative theory of differential equations", 2021, núm. 15, p. 1-27.