Stochastic dynamics of time correlation in complex systems with discrete time

Abstract

In this paper we present the concept of description of random processes in complex systems with 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 (TCFs). We have introduced the dynamic (time dependent) information Shannon entropy S(i)(t) where i=0,1,2,3,ellipsis, as an information measure of stochastic dynamics of time correlation (i=0) and time memory (i=1,2,3,ellipsis). 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,ellipsis) in complex system. The theory developed started from the careful analysis of time correlation involving dynamics of vectors set of various chaotic states. We examine two stochastic processes involving the creation and annihilation of time correlation (or time memory) in details. We carry out the analysis of vectors' dynamics employing finite-difference equations for random variables and the evolution operator describing their natural motion. The existence of TCF results in the construction of the set of projection operators by the usage of scalar product operation. 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 TCFs and memory functions (MFs). The solution of the equations above thereof brings to the recurrence relations between the TCF and MF of senior and junior orders. This offers new opportunities for detecting the frequency spectra of power of entropy function S(i)(t) for time correlation (i=0) and time memory (i=1,2,3,ellipsis). 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.

Universal approach to overcoming nonstationarity, unsteadiness and non-Markovity of stochastic processes in complex systems

“…It was shown in Refs. [2,10,11] that the finite-difference kinetic equation of a nonMarkov type for TCF M 0 ðtÞ can be written by means of the technique of projection operators of Zwanzig'-Mori's type [14,15] …”

Regular and stochastic behavior of Parkinsonian pathological tremor signals

“…The first method is based on the notions and concepts of the statistical theory of discrete non-Markov stochastic processes [1,2]. The method is connected with the studies of statistical non-Markov effects, long-and short-range statistical memory effects, regularity and stochastic behavior effects, and dynamic alternation of relaxation modes in the patient in various dynamic states.…”

“…We use the dependences a(t) and M 1 (t) to analyze the amplitude of Parkinsonian tremor velocity. We also use these dependences to calculate the non-Markovity parameter [1,2] which characterizes the degree of correlativity of the signal. The studies, which have been carried out earlier [2,12,13], show that this parameter contains detailed information about the physiological state of a system.…”

“…By using the Zwanzig-Mori's technique of projection operators [49,50,51,52] it is possible to receive an interconnected chain of finite-difference equations of a non-Markovian type [1,2] for the initial TCF a(t) and the normalized memory functions in the following way:…”

“…In particular, the signals of PD pathological tremor are physically interpreted to answer the following questions: The theory of discrete non-Markov stochastic processes [1,2] is based on the finitedifference representation of the kinetic Zwanzig-Mori's equations [49,50,51,52] for condensed matters, which are well known in the statistical physics of nonequilibrium processes.…”