Time-Varying Statistics of Oscillators Endowed with Fractional Derivative Elements

Abstract

An approximate technique is presented for determining the statistics of the response envelope of lightly-damped nonlinear oscillators endowed with fractional derivative elements. Specifically, via the stochastic averaging concept, a one-dimensional first order stochastic differential equation governing the response envelope of the oscillator, and the corresponding Fokker-Planck-Kolmogorov equation are obtained. Next, an analytical solution of the linear systems is considered, and the nonlinear oscillator solution is sought by a Galerkin scheme with base functions selected as the eigenfunctions associated with the solution of the linear oscillator. As an example, a Van der Pol oscillator endowed with fractional derivate elements subject to Gaussian white noise is examined. Comparisons with pertinent Monte Carlo simulation results are made to demonstrate the reliability of the proposed technique.