Manifestation of Chaos in Real Complex Systems: Case of Parkinson's Disease

In this chapter we present a new approach to the study of manifestations of chaos in real complex system. Recently we have achieved the following result. In real complex systems the informational measure of chaotic chatacter (IMC) can serve as a reliable quantitative estimation of the state of a complex system and help to estimate the deviation of this state from its normal condition. As the IMC we suggest the statistical spectrum of the non-Markovity parameter (NMP) and its frequency behavior. Our preliminary studies of real complex systems in cardiology, neurophysiology and seismology have shown that the NMP has diverse frequency dependence. It testifies to the competition between Markovian and non-Markovian, random and regular processes and makes a crossover from one relaxation scenario to the other possible. On this basis we can formulate the new concept in the study of the manifestation of chaoticity. We suggest the statistical theory of discrete non-Markov stochastic processes to calculate the NMP and the quantitative evaluation of the IMC in real complex systems. With the help of the IMC we have found out the evident manifestation of chaosity in a normal (healthy) state of the studied system, its sharp reduction in the period of crises, catastrophes and various human diseases. It means that one can appreciably improve the state of a patient (of any system) by increasing the IMC of the studied live system. The given observation creates a reliable basis for predicting crises and catastrophes, as well as for diagnosing and treating various human diseases, Parkinson's disease in particular.Comment: 20 pages, 8 figures, 3 tables. To be published in "The Logistic Map and the Route to Chaos: From the Beginnings to the Modern Applications", eds. by M. Ausloos, M. Dirickx, pp. 175-196, Springer-Verlag, Berlin (2006

Diffusion Time-Scale Invariance, Markovization Processes and Memory Effects in Lennard-Jones Liquids

We report the results of calculation of diffusion coefficients for Lennard-Jones liquids, based on the idea of time-scale invariance of relaxation processes in liquids. The results were compared with the molecular dynamics data for Lennard-Jones system and a good agreement of our theory with these data over a wide range of densities and temperatures was obtained. By calculations of the non-Markovity parameter we have estimated numerically statistical memory effects of diffusion in detail.Comment: 10 pages, 3 figure

Microscopic vortices in classical liquids

In the present article we introduce the notions about the microscopic vortices (MV) in classical liquids. The infinite exact chain of engaging kinetic equations of non-Markov type were obtained for the time correlation function (TCF) MV. For its closing and solving the so-called orthogonal dynamic variables of the first, second, third and higher levels are introduced. The consequent usage of this variables let the "quasi-hydrodynamic" approximation for the memory function of the third level M3(t) be used. In the case M3(t) is presented as the linear combination of memory functions of the lowest levels. The coefficients in this expansion may be described by means of the relaxation frequency and even moments TCF MV. The present theory can be compared with the molecular-dynamic (MD)-data of different authors for the transverse currents in liquid argon. It is in accordance with the experimental MD-data frequency spectrum MV the liquid argon; it gives an opportunity to determine a spectrum of vortex excitation and relaxation parameters (the lifetime and the excitation relaxation time), a spectrum of non-Markov's MV parameter and its spatial dispersion. We obtained data to prove the existence of considerably fluctuating MV in liquids. Their relaxation is characterized by considerably expressed non-Markov's kinetic properties. © 1994

Time-scale invariance of relaxation processes of density fluctuation in slow neutron scattering in liquid cesium

The realization of idea of time-scale invariance for relaxation processes in liquids has been performed by the memory functions formalism. The best agreement with experimental data for the dynamic structure factor $S(k,\omega)$ of liquid cesium near melting point in the range of wave vectors (0.4 \ang^{-1} \leq k \leq 2.55 \ang^{-1}) is found with the assumption of concurrence of relaxation scales for memory functions of third and fourth orders. Spatial dispersion of the four first points in spectrum of statistical parameter of non-Markovity $\epsilon_{i}(k,\omega)$ at $i=1,2,3,4$ has allowed to reveal the non-Markov nature of collective excitations in liquid cesium, connected with long-range memory effect.Comment: REVTEX +3 ps figure

Quasihydrodynamic approximation for memory functions in non-Markovian relaxation processes in condensed matter

A quasihydrodynamic approximation is introduced for senior memory functions in the chain of non-Markovian kinetic equations for time correlation functions in many-body systems. This approximation allows to describe complicated high-frequency and short wavelength spectra for a wide range of phenomena: from the molecular scattering and the scattering of slow neutrons in liquids to the description of the decay of magnetic induction signal in crystals

The structure of the kinetic equation for time correlation functions

The general kinetic equation for time correlation functions of statistical systems is constructed with the aid of four projection operators. © 1973

Simple model for the calculation of the coefficient of self-diffusion in a liquid

The coefficient of self-diffusion in three-dimensional classical liquid is computed approximately from the hierarchy of kinetic equations for the time-correlation functions (TCF). © 1976

Application of methods of the scattering theory in the statistical theory of liquids

This paper is devoted to the application of the projection methods of the nonrelativistic quantum theory of scattering (the method of Petrov-Bubnov-Galerkin (PBG) and the Bubnov-Galerkin (BG) method) in the statistical theory of liquids. By means of the projection PBG method we have found a new family of equations both for the correlation functions and for the radial distribution function (RDF). In the generalized equation for the RDF we have obtained new terms which are linear and quadratic in the density and the latter are absent in all the previous theories. By means of the projection BG principle the approximate eigenfunctions of the Liouville operator in a liquid were obtained as a linear combination of the Kihara functions. It was shown that the spectrum of the collective excitations is determined by the complex Fourier transformation of the force acting on an arbitrary particle in a liquid. © 1976

Possibility between earthquake and explosion seismogram differentiation by discrete stochastic non-Markov processes and local Hurst exponent analysis

The basic purpose of the paper is to draw the attention of researchers to new possibilities of differentiation of similar signals having different nature. One of examples of such kind of signals is presented by seismograms containing recordings of earthquakes (EQ's) and technogenic explosions (TE's). We propose here a discrete stochastic model for possible solution of a problem of strong EQ's forecasting and differentiation of TE's from the weak EQ's. Theoretical analysis is performed by two independent methods: with the use of statistical theory of discrete non-Markov stochastic processes (Phys. Rev. E62,6178 (2000)) and the local Hurst exponent. Time recordings of seismic signals of the first four dynamic orthogonal collective variables, six various plane of phase portrait of four dimensional phase space of orthogonal variables and the local Hurst exponent have been calculated for the dynamic analysis of the earth states. The approaches, permitting to obtain an algorithm of strong EQ's forecasting and to differentiate TE's from weak EQ's, have been developed.Comment: REVTEX +12 ps and jpg figures. Accepted for publication in Phys. Rev. E, December 200