Is the dynamic chaos a stochastic process in the autonomous systems of differential equations of the Lorenz system type


DOI: 10.34759/trd-2020-112-1

Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

At present, there are no stability criteria similar to Courant-Friedrichs-Lewy criteria for systems of autonomous differential equations (SADE). The instabilities, manifesting themselves as a computational chaos, occur while the numerical integration of SADE. Moreover, the time step decrease does not lead to this instability elimination. Commonly, the studies on determining the sensitivity of solutions to the initial conditions setting are being conducted to explain the deterministic chaos phenomenon. These studies demonstrate exponential divergence of initially close solution trajectories, and impossibility of selecting such small computational error to «conquer» the uncertainty in the Lorenz type SADE.

The conclusion is drawn from this circumstance that since the principal difficulties do not allow achieving the necessary accuracy, there is no need to muse about determinism. However, such approach does not resolve the problem of solutions determinism, irrespectively to the possibility or impossibility of obtaining of the information regarding the evolution of the considered system. The issues of predestination in the closed systems, in particular with such closed system as Universe, conjugate with these issues.

The studies conducted in the presented work demonstrate that the deterministic chaos occurring in SADE of Lorenz type may be associated with the stochastic process and is not, in essence, the deterministic chaos for any finite time step.

The article discusses the issues associated with the possibility of turbulence modelling based on the Navier-Stokes equations via direct numerical simulation technique.

The problems related with the feasibility for modeling of the turbulence on the basis of Navier-Stokes equations via the direct numerical simulations are also addressed in the paper.

Keywords:

chaos, autonomous differentiation equations, Lorenz system of equations, stochastic systems, turbulence

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