A Brief on Multi-Pantograph Ordinary Differential Equations with Constant Deviating Arguments: The Role of Lévy Noise in Almost Sure Exponential Stochastic Self-Stabilization

This study investigates the role of Lévy noise in the almost sure exponential stochastic self-stabilization of multi- pantograph ordinary delay differential equations (MPODDEs). The equation is nonlinear and contains multi-pantograph terms as well as several constant time lags and as such, are typically unstable. The method applied involves the use of Lyapunov sample exponent function and a specialized convergence rate function technique suggested by Mao, (1997). It is demonstrated that if the noise driving force parameter of the stochastically perturbed equation is finite, then the new stochastic multi-pantograph ordinary delay differential equation (SMPODDE) is self - stabilized in an almost sure exponential sense. This phenomenon does not occur in the deterministic multi-pantograph ordinary delay differential equation where noise is absent.