Stochastic Dynamics of Time Correlations in Complex Systems with Discrete Time

Abstract

In this paper we present the concept of description of random processes in complex systems with the discrete time. It involves the description of kinetics of discrete processes by means of the chain of finite-difference non-Markov equations for time correlation functions (TCF). We have introduced the dynamic (time dependent) information Shannon entropy $S_i(t)$ where i=0,1,2,3,... as an information measure of stochastic dynamics of time correlation $(i=0)$ and time memory (i=1,2,3,...). The set of functions $S_i(t)$ constitute the quantitative measure of time correlation disorder $(i=0)$ and time memory disorder (i=1,2,3,...) in complex system. Harnessing the infinite set of orthogonal dynamic random variables on a basis of Gram-Shmidt orthogonalization procedure tends to creation of infinite chain of finite-difference non-Markov kinetic equations for discrete TCF and memory function.The solution of the equations above thereof brings to the recurrence relations between the TCF and MF of senior and junior orders. The results obtained offer considerable scope for attack on stochastic dynamics of discrete random processes in a complex systems. Application of this technique on the analysis of stochastic dynamics of RR-intervals from human ECG's shows convincing evidence for a non-Markovian phenomemena associated with a peculiarities in short and long-range scaling. This method may be of use in distinguishing healthy from pathologic data sets based in differences in these non-Markovian properties.