Grechko A. The criterions of existence of the bounded solutions of inhomogeneous linear extention of the dynamical sys-tems.

The thesis deals with comprehensive research of bounded solutions on all axis of the inhomogeneous linear extension of the dynamical system. Sufficient conditions for existence of bounded solutions of the inhomogeneous linear extension of the dynamical system are obtained. For monotone linear extensions of dynamical systems on nontrivial bundles the existence of two complementary, dimension-one and codimension-one invariant subbundles and exponential separation of the linear extension is established. Applications of Lyapunov functions with alternating sign in the form of quadratic form to the solution of problems of a regularity block-triangular linear extensions of a dynamical systems and diagonal perturbations of some regular classes of linear extensions of dynamic systems are described. The necessary condition for weak regularity and block diagonalization of some linear extension of the dynamical system is established.