Ginzburg–Landau equations with random switching and impulsive perturbations

Abstract

This paper is concerned with a class of generalized Ginzburg–Landau equations with random switching and impulsive perturbations. In the model, the nonlinear term is allowed to have higher polynomial growth rate than the usual cubic polynomials. The random switching is modeled by a continuous-time Markov chain with a finite state space. First, an explicit solution is obtained. Then stochastic boundedness and permanence of the solutions are investigated, and effect of impulses on stochastic boundedness is also discussed. Finally, some asymptotic pathwise estimates are obtained. Meanwhile, how impulses affect these properties is investigated.