On the stochastic stability of some two-dimensional dynamical systems
The stochastic stability for a two-dimensional diffusion process is analyzed. The process is described by two nonlinear Itô stochastic differential equations with some minimal constraints. stochastic differential equation (SDE) is a differential equation in which one or more terms are stochastic processes, thus resulting in a solution which is itself a stochastic process. SDEs are used to model diverse phenomena such as fluctuating stock prices or physical systems subject to thermal fluctuations. Basically, the author analyzes a particular type of a stochastic system whose deterministic part is equivalent to the classical Lienard equation (which is a second order differential equation). The paper has 4 pages (the references are not included) and starts by defining a system of two-nonlinear stochastic differential equations. The system's stability is analyzed by obtaining sufficient conditions for the asymptotic stability in the whole and the mean square exponential stability of the zero solution. Starting with the third section until the last one (the 8th), nine theorems are presented (without proof) that detail different situations of the system stability. The overall conclusion is that there are sufficient conditions for the stochastic stability for such a system.
scientific article; zbMATH DE number 5824141
scientific article