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Information × Registration Number 0220U104879, 0119U102376 , R & D reports Title Qualitative, asymptotic and numerical analysis of various classes of differential equations and dynamical systems, their classification, and practical application popup.stage_title Head Filipkovskaya Mariya S., Кандидат фізико-математичних наук Registration Date 30-12-2020 Organization B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine popup.description2 The theorems on the Lyapunov stability, asymptotic stability, including asymptotic stability in the large, and the Lyapunov instability of an equilibrium state of time-varying degenerate differential equations (DEs) were obtained. The theorems on the Lagrange stability and instability and on the dissipativity (ultimate boundedness) of time-varying degenerate DEs were obtained. Degenerate DEs (differential-algebraic equations, descriptor equations) are used in the mathematical modeling of the dynamics of complex mechanical and technical systems, the intersectoral balance of the economy, automatic control systems, neural networks, ecological systems, the kinetics of chemical reactions and transient processes in electrical circuits. Using the obtained theorems, evolutionary properties of the mathematical models of electrical circuits for certain types of radio-electronic devices were investigated. The asymptotic analysis of the initial-boundary value problem for the system of the Maxwell-Bloch equations with a periodic boundary condition and trivial initial data was carried out and explicit formulas for the asymptotics of a solution of the problem were obtained. The numerical methods for solving the boundary integral equations of the homogenized problem and a second-kind hypersingular integro-differential equation are constructed. A computer implementation of these methods has been formed. The constructed methods have been used for the numerical analysis of the diffraction and scattering capacities of mesh strip and wire mesh structures. We build examples of Cantor dynamical systems which possess given dynamical properties such as zero entropy and infinite countable number of ergodic invariant measures. With the help of coding, we build an explicit isomorphism between Bratteli-Vershik systems and symbolic dynamical systems, we study the relation between the properties of the coding and the number of ergodic invariant measures for Bratteli-Vershik systems. Product Description popup.authors popup.nrat_date 2020-12-30 Close