Ordinary differential equations are crucial in science and engineering as they offer mathematical models for various real-world phenomena. This significance underscores the need to enhance an advanced study of differential equations. The course focuses on three essential topics within the modern theory of motion: dynamical systems, emphasizing periodic motions; integral manifolds, which include stable, unstable, and center manifolds; and recurrent motions, such as almost periodic and Poisson stable motions, as well as bifurcations.