Main scientific results

Russian

The statistical theory of gravitating cosmogonic bodies formation has been developed, and namely, within framework of this theory the models as well as evolution equations of the statistical mechanics of formation of gravitating cosmogonical bodies have been proposed:

– the known problem of gravitational condensation of infinite distributed cosmic substance is solved on the basis of the proposed statistical model of the antidiffusion processes which allows to solve the gravitational paradox for infinite homogeneous spatial substance. It is known that the Jeansʼ gravitational instability theory does not explain the reason for obtaining the critical value of the disturbation wavelength, i.e. the mechanism for increasing the wavelengths of disturbations in a stationary one-component gas medium. Within the framework of the proposed statistical model it is shown that the gravitational instability of a gas-dust cloud arises with increasing the mass density of condensation disturbations of a one-component cosmic gas medium;

– within the framework of the developed statistical model of the antidiffusion process a new law for the planetary distances distribution in our Solar system is proposed depending on the average value of the specific angular momentum of a continuous medium in the protoplanetary cloud. The proposed law of planetary distances in the Solar system generalizes the famous law of Academician O.Y. Schmidt. In the comparison with other known laws of planetary distances the new law gives a fairly good estimation of the observed planetary distances in the Solar system;

– a nonlinear time-dependent Schrödinger-like undulatory equation that describes the arising nonlinear phenomena due to the processes of self-organization in the forming cosmogonical body has been derived;

– the universal stellar law (USL) for extrasolar planetary systems connecting the temperature, size and mass of each star has been derived and justified. The analysis of the modified USL has shown that most part of the stars corresponds to the category of ideal (or classical) stars (respectively, extrasolar planetary systems) outside the concrete dependence on their belonging to the spectral classes O, B, A, F, G, K and M, although there is some error in estimating the parameters of the stars depending on their spectral belonging: a lower accuracy of the modified USL is observed for very bright stars belonging to the higher order spectral classes O, B and A, or for the more dim stars belonging to the lower spectral class M or group of red giants;

The theory of matrix decomposition of chaotic attractors into state-space of the complex dynamical systems has been proposed. Within framework of this theory new methods of nonlinear analysis of dynamical systems have been developed:

– analytical expressions for the decomposition of operators of nonlinear dynamical systems on the basis of vector-matrix series in state-space have been derived;

– a representation of the shift operator on the trajectories of a nonlinear dynamical system through the corresponding shift operators on the trajectories of multidimensional linear dynamical systems has been proposed;

– quantitative analysis of known attractors (Lorenz, Hēnon, Rössler, Fitz-Hugh, etc.) of complex dynamic systems based on the proposed matrix series has been developed;

– a method for estimating Lyapunov characteristic exponents on the basis of matrix decomposition theory has been proposed;

– an analysis of dynamical states of the Hopfield neural network using the matrix decomposition has been carried out.

An algorithm for defining the minimal embedding dimension of a nonlinear dynamical system attractor by means of the local topological analysis of phase trajectories has been developed. The necessary and sufficient conditions for the existence of the topological stabilization for the restored attractor have been formulated. The efficiency of the proposed algorithm in the computational aspect (on average by an order) has been shown in comparison with the known correlative-topological algorithms (Grassberger-Procaccia, Schuster, etc.).

An original approach for the recognition of chaotic signals using nonlinear signal decomposition by measuring Wiener kernels has been developed. In particular, a new solution to the problem of recognizing speech phonemes based on the evaluatin of Wiener kernels has been proposed. The nonlinear filter bank structure for the phoneme recognition has been synthesized.

A new analytical model based on the nonlinear ordinary differential equations system describing the transition from the laminar to the turbulent regime through the periodic one has been constructed. This model also describes the dynamical fluid states arising in such transition in the Couette-Taylor hydrodynamic system (rotating coaxial cylinders). Its application for the study of aerodynamic regimes of rotor mechanisms functioning in the setting parts of machine-building constructions has been shown.

The eigenvalues problem of the linearized operator of a differential equations system of the reaction-diffusion type has been analytically solved. It need for modelling the self-organization processes in biological and chemical active media. The obtained results are useful to develop methods for control the arising autowave processes.

During 2006-2017 years the co-workers of the LSSM carried out a number of new tools including software and methodical recomendations for computer modelling and non-linear analysis of aerohydrodynamic flows in engineering constructions designed at industrial enterprises of the Republic of Belarus:

– for computer simulation of the mass (solid particles) movement process in air flows within the setting parts of mechanisms of the forage harvester KVK-800 at the “Gomselmash”, in particular “GSKB for the grain harvesting and forage harvesting equipment” (Gomel);

– for computer simulation of hydrodynamic flows with usage of rotating computational meshes arising in the working region of the hydrotransformer TGD-340A at “Amkodor”, Ltd. (Minsk);

– for computer modelling and analysis of continuous medium flows in order to improve the grain cleaning system in the grain harvester at the “GSKB for grain harvesting and forage harvesting equipment” (Gomel);

– for computer modelling of steam-water flows inside the working chamber of a microturbine and full-scale tests with the purpose of development and implementation of integrated information technology supporting the design of new types of power machines (microturbines) at “Promprivod”, Ltd. (Minsk);

– for computer modelling and analysis of particle flows in a continuous medium for the improvement of the threshing-separating device and the transportation of the grain mass in the combine harvester at the “GSKB for grain harvesting and forage harvesting equipment” (Gomel).

The results of the LSSM innovative activity on computer modelilng and nonlinear analysis of aerohydrodynamic processes in the setting parts of engineering constructions were presented at the XIV International Specialized Exhibition TIBO-2007, April 24-27, 2007, Minsk. Exhibit – "Computational aerohydrodynamics of the setting parts of machine-building constructions".

Based on the theoretical results obtained in the LSSM, the following software and hardware complexes were developed:

– a mobile robot control system as a part of working model for monitoring to reveal the facts of unauthorized intrusions (the mock-up was demonstrated at the Innovative Forum of the Republic of Belarus "INTRI" – 2010).