Stepanenko N. Variable-polarity Ljapunov's functions in the theory of differential equations.

The thesis devoted to study the questions of existence and integral conception of Ljapunov's functions for linear systems of differential equations and for linear expansions of dynamical systems. It was established that there exist linear expansions of dynamical systems on torus regular at any constant vector function and at the same time there is no quadratic form with constant coefficient which would have derivative of fixed sign. There were explored the sets of quadratic forms represented in integral form depend on two different symmetric matrixes. The property of regularity of the linear expansions of dynamical systems on torus, normal arguments of which are represented in canonical form were explored.