The book is devoted to the problem of constructing certain classes of solutions of systems of ordinary differential equations. For this purpose, a procedure has been developed for constructing solutions in the form of series, which are similar to the series used in the first Lyapunov method. A special place in the book is given to asymptotic solutions that strive for equilibrium positions with an unlimited increase or decrease in the independent variable .. In this case, we consider the so-called highly nonlinear case, when the existence of such solutions cannot be deduced based only on the analysis of the first approximation system. The book is illustrated by a large number of specific examples in which the presence of private solutions of a particular class indicates some features of the dynamic behavior of the system. For specialists in the field of mechanics, mathematics, theoretical physics, engaged in the theory of dynamic systems, for students and postgraduates of universities and technical universities studying in the specialty "Applied Mathematics"