A Kolmogorov-Fokker-Planck approach for a stochastic Duffing-van der Pol system

Abstract

The stochastic Duffing-van der Pol (SDvdP) system is an appealing case in stochastic system theory, since it involves linear and non-linear vector fields, i.e. system non-linearities, state-independent and state-dependent stochastic accelerations. The equation describing the Duffing-van der Pol system is a second-order non-linear autonomous differential equation. Stochastic estimation procedures, without accounting for possible small stochastic accelerations, may cause an inaccurate estimation of positioning of dynamical systems. In this paper, the estimation-theoretic scenarios of the stochastic version of the Duffing-van der Pol system, which accounts for a state-independent perturbation as well as a state-dependent stochastic perturbation of the order n, where n ≥ 1, is the subject of investigation. This paper discusses the notion of a qualitative analysis for the non-linear stochastic differential system using the Ito differential rule, and subsequently the method is applied to the stochastic system of this paper. This paper develops and explores the efficacy of three different approximate estimation procedures for analyzing the non-linear problem of concern here as well. The theory of the estimation procedures of this paper is developed using the Kolmogorov-Fokker-Planck equation. Numerical simulations involving two different sets of initial conditions are given, since approximate estimation procedures are hard to evaluate theoretically. This paper suggests that the higher-order approximate estimation procedure will lead to the more accurate state estimates.