The fractional Lévy-driven a simple dynamics system: Transcritical bifurcation

Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies. In this article, we consider random dynamical systems induced parametrized one dimensional stochastic differential equation driven by the fractional Lévy Process(FLP). We address some conditions on invariant measure of Markov semigroups which ensures stochastic bifurcations of a wide class of stochastic differential equations with FLP around singular points where both the drift and diffusion functions vanish. This leads to sufficient conditions on drift and diffusion coefficients for a stochastic transcritical bifurcations of the families of random dynamical systems.