A billet on multi-pantograph ordinary differential equations with constant deviating arguments: Issues on the role of Lévy noise in the almost sure exponential stochastic-self stabilization

Abstract

This letter considers the role played by Lévy noise in the almost sure exponential stochastic self-stabilization of solution to a multi-pantograph delay differential equation (MPDDE). The equation is nonlinear and contains multi-pantograph terms as well as several constant time lags and as such, the solution is generally unstable. The system is perturbed by random noise disturbance of Lévy -type. It is established that under certain conditions, the presence of Lévy noise would stochastically self-stabilize the resulting stochastic multi-pantograph differential equation in an almost sure exponential sense, whereas the comparable deterministic multi- pantograph delay differential equation, in which Lévy noise is absent, still remains unstable.