# Stability index of linear random dynamical systems

## TL;DR

This paper investigates the stability index of linear random dynamical systems, providing methods to compute or estimate the probabilities of different stability indices using Monte Carlo and least squares techniques.

## Contribution

It introduces a novel approach combining Monte Carlo simulations with least squares to estimate stability index probabilities for random dynamical systems.

## Key findings

- Derived exact or estimated probabilities for stability indices.
- Applied methods to linear differential and difference equations.
- Analyzed stability properties in high-dimensional systems.

## 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.