^{1}and Radosław Iwankiewicz

^{1,2}

^{1}West Pomeranian University of Technology, Al. Piastow 50, 70-311 Szczecin, Poland

^{2}Hamburg University of Technology, Eissendorfer Str. 42, 21-073 Hamburg, Germany

The moment equations technique together with closure approximations is developed for a nonlinear dynamic system subjected to a random train of impulses driven by an Erlang renewal counting process. The original non-Markov problem is converted into a Markov one by recasting the excitation process with the aid of an auxiliary, pure-jump stochastic process characterized by a Markov chain. Hence the conversion is carried out at the expense of augmentation of the state space of the dynamic system by auxiliary Markov states. The problem is characterized by the set of joint probability density - discrete distribution function, which are the joint probabilities of the original state vector and of the Markov chain being in the particular j-th state. Accordingly the statistical moments of the state variables are defined as integrals with respect to the mixed-type, probability density - discrete distribution function. The differential equation for moments are obtained with the aid of the forward integro-differential Chapman-Kolmogorov operator [1]. Three closure approximation techniques are developed. The first one is the result of application of cumulant-neglect closure directly to unconditional moments. The second one is the quasi-moment closure applied to unconditional quasi-moments. The third one is the modified cumulant-neglect closure approximation technique based on the representation of the joint probability density - discrete distribution function in terms of conditioning on two mutually exclusive and exhaustive events: that the Markov chain is in the j-th state while the system is at rest (it is in the j-th state for the first time) and that the Markov chain is in the j-th state while the system is not at rest (it is in the j-th state for any subsequent time). Thus the joint probability density function consists of a spike and of the continuous part (cf [2]). The closure approximations are first formulated for the conditional moments resulting from the continuous part of the probability density function and next for the unconditional moments. The considered nonlinear system is the oscillator with cubic restoring force term. The equations for moments up to the fourth-order are considered. Hence the closure approximations are derived for redundant fifth- and sixth order moments, both ordinary and centralized. The developed moment equations with closure approximation technique is illustrated by an example.