Semenyaka S. Finite-dimensional approximation of central manifold and their application in the qualitative analysis of nonlinear differential systems

The dissertation deals with development of constructive scheme of central manifold method for qualitative analysis of many-dimensional differential systems in critical cases. This scheme is based on the application of finite-dimensional approximation of vector-function instead of using this vector-function that represents the central manifold of the system. This approach enables to simplify qualitative analysis of many-dimentional differential systems that makes easier the obtaining of concrete results for stability theory and bifurcation theory. In the dissertation for the nonautonomous quasilinear system of the differential equations the algorithm of the construction of the finite-dimentional approximations of the center manifold is diveloped. The theorem, which formulates conditions, under which the stability problem for the initial system is redused to the analogous problem for the subsystem, which is the restriction of the initial system on the corresponding approximation of the center manifold, is proved. On the basis asymptotic expansions on coordinates method the algorithm of the construction of the finite-dimensional appproximations of the center manifold of the family of many-dimensional dynamical systems with parameter is diveloped and their application in the bifurcation analysis is given.