Stochastic stability criteria for two-dimensional linear autonomous systems perturbed by white noise

TL;DR

This paper establishes criteria for the stochastic stability of second-order linear autonomous systems affected by white noise, using Lyapunov functions and analyzing bifurcation points for instability.

Contribution

It provides necessary and sufficient conditions for stochastic stability of such systems, including explicit bifurcation values of noise intensity, in both Ito and Stratonovich forms.

Findings

Derived analytical expressions for bifurcation noise levels.

Established Lyapunov-based stability criteria.

Analyzed a damped harmonic oscillator example.

Abstract

We give necessary and/or sufficient conditions for stochastic stability of second-order linear autonomous systems with parameters, which are perturbed by a random process of the "white noise" type. The Ito's and Stratonovich's forms of stochastic differential equations representing the system with white noise are considered. The investigation of stochastic stability of the systems considered is based on the construction of special Lyapunov functions in the quadratic forms. The bifurcation value of the white noise intensity at which a system first becomes unstable is presented in the analytical expression. As an example a damped harmonic oscillator with randomly perturbed parameters is considered.