Bifurcations in stochastic dynamical systems with simple singularities
The paper deals only with 1-dimensional stochastic dynamical system. In Section 2, some necessary preliminaries concerning stochastic dynamical systems driven by the dichotomous Markov noise are given and a geometric method to investigate their probability densities is proposed. Section 3 is devoted to a presentation of various physical systems with stochastic noise, for example, the generalized Langevin system, the dye laser system with pre-Gaussian pump fluctuations. Equations for the stationary probability are shown. In Section 4, the theory for stable dynamical systems with a stochastic control space is developed and some properties are proved. The equations for stationary probability densities are reduced to equations with so-called singular points. In Section 5, the analysis of bifurcation diagrams for stable dynamical systems is provided. The fold model and the cusp model with dichotomous noise are investigated and the corresponding bifurcation diagrams are obtained.
scientific article; zbMATH DE number 4176159