Discontinuity, Nonlinearity, and Complexity
Dynamical System Model with the use of Liouville Equation for Empirical Distribution Function Densities
Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 529540  DOI:10.5890/DNC.2020.12.006
Alexey A. Kislitsyn$^1$ , Yurii N. Orlov$^{1,2}$
$^1$ Keldysh Institute of Applied Mathematics of RAS, Miusskaya Sq., 4, Moscow 125047, Russia
normalsize
$^2$ Institute of Machines Sciences named after A.A. Blagonravov of RAS, Maly Kharitonyevsky Pereulok, Moscow,
101990, Russia
Download Full Text PDF
Abstract
The difference approximation of the onedimensional Liouville equation for the sample distribution function density of the nonstationary time series estimated by the histogram is considered. The scheme with semigroup property conservation is constructed for evolution model of this sample distribution function density. We investigate the problem of appropriate Liouville equation construction for given initial and final distributions. We prove the necessary and sufficient condition of such a representation, which is a strong positivity of the initial density distribution in the inner class intervals. The determination of the corresponding Liouville velocity algorithm is constructed and its mechanicalstatistical meaning is shown. The dynamical system, associated with this Liouville equation, is considered. We interpret the Liouville statistical velocity as a corresponding velocity of dynamical system, according to representation of statistical mechanics. We show, that this interpretation leads to monotonic discrete dynamical system with stationary point, corresponding to equality of initial and final distribution functions.
References

[1] 
Koroliuk, V.S., Portenko, N.I., Skorokhod, A.V., and
Turbin, A.F. (1985), Handbook of probability theory and mathematical statistics, 640, Nauka: Moscow.


[2] 
Kremer, N.Sh. and Putko B.A. (2002), Econometrics, 311, UnityDANA: Moscow.


[3] 
Orlov, Y.N. and Osminin, K.P. (2008), Construction of a sample distribution function for forecasting a nonstationary time series, Mathematical Modeling,
9, 2333.


[4] 
Bosov, A.D., Kalmetiev, R.Sh., and Orlov, Y.N. (2014), Simulation of a nonstationary time series with the specified properties of the sample distribution, Mathematical Modeling, 3, 97107.


[5] 
Orlov, Y.N. (2014), Kinetic methods of investigation of nonstationary time series, 276, MIPT: Moscow.


[6] 
Kalitkin, N.N. (1978), Numerical methods, 512, Nauka: Moscow.


[7] 
Samarsky, A.A. and Vabischevich, P.N. (1998), Nonlinear monotone schemes for the transfer equation, Doklady RAS: Moscow, 361(1), 2123.


[8] 
Galanin, M.P. (2005), Numerical solution of the transfer equation. In collector: The future of applied mathematics. Lectures for young researchers, Editorial URSS: Moscow, 78116.


[9] 
Vlasov, A.A. (1966), Statistical distribution functions., 356, Nauka: Moscow.


[10] 
Bogolubov, N.N. (1946), Problems of dynamic theory in statistical physics, 119, GITTL: Moscow.


[11] 
Lukashin, Y.P. (2003), Adaptive methods for shortterm time series forecasting, 416, Finance and statistics: Moscow.
