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Information × Registration Number 0119U100334, ( 0220U100628  0221U105325  0222U003410  ) R & D request Title Development of new analytic-geometric, asymptotic and qualitative methods for investigating invariant sets of divergent-objective equations. Head Perestiuk Mykola O., Доктор фізико-математичних наук Registration Date 31-01-2019 Organization Taras Shevchenko National University of Kyiv popup.description1 The use of the modern mathematical apparatus of the Riemann and Symplectic geometries as a whole and the modification of nonlocal topological and geometric methods in combination with analytical methods (iterative, numerically analytic, CAM-theory methods, etc.) will allow obtaining new nonlocal results in relation to the existence of invariant varieties of nonlinear dynamical systems on phase spaces with a nontrivial topological structure, the theory of bifurcations, and a qualitative analysis of the trajectories of such systems. In the study of invariant sets and attractors of pulsed dynamical systems, the theory of stability of differential equations with impulse disturbance, the theory of global attractors of m-semiconducts in general metric spaces, and the theory of nonlinear evolution equations will be used. Modern methods of nonlinear analysis, such as a multi-valued analogue of the implicit function theorem, will enable us to investigate the behavior of the pulsed stream in the vicinity of the attractor, which will make it possible to establish the invariance and stability of the nonimpulse part of the global attractor. In the study of the behavior of solutions of stochastic differential equations in partial derivatives, the abstract scheme of Krylov-Bogolyubov will be used. In the study of a two-domain equation, a functional approach with the use of the stationary point theorem in special Banach spaces for fractional type operators and the Galerkin method adapted to stochastic systems will be used. In the study of equations of hydrodynamic type with a small dispersion and asymptotic behavior of solutions of nonlinear boundary and spectral problems in thick fractal compounds and in thin star-domains, methods of modern asymptotic analysis will be used and the method of matching of asymptotic series is modified. The application of this method will investigate the properties of solutions in unbounded domains. For spectral problems in a dense fractal compound a popup.nrat_date 2024-12-10 Close