The random dynamical pitchfork bifurcation with additive L\'evy noises

Abstract

This paper concerns the effects of additive non-Gaussian L\'evy noises on the pitchfork bifurcation. We consider two types of noises, $\alpha$-stable process and the truncated process. Under both $\alpha$-stable process and the truncated process, the classical pitchfork bifurcation model exists a unique invariant measure. The Lyapunov exponent associated with the invariant measure is always negative for the system under the truncated case. While the stochastic pitchfork bifurcation still occurs. In both cases, the attractivity uniformity, the finite-time Lyapunov exponent, and the dichotomy spectrum behave varies with the bifurcation parameter changing. Compared with Brownian motion, there is two key difficulties for the L\'evy processes. The stationary density can not be solved explicitly, thus we have to estimate it properly. This is overcome by the strong maximum principle. The bilateral suprema for L\'evy processes need to be analyzed. This is acquired by the strong Markov property. Based on them, we establish the main results indicating stochastic pitchfork bifurcation.