Dissipativity in the plane

A classical method is considered for studying dissipativity of systems of ordinary differential equations on the plane, consisting in the construction of a system of compact sets covering the plane, whose boundaries are given by trajectories of a certain auxiliary system, and the trajectories of the given system intersect them from outside in (Theorem 1). In this connection the problem of coincidence (Theorem 3) and intersection (Theorems 4–6) of trajectories of two differential inclusions is solved. In conclusion, this method is used to prove Theorem 8, which generalizes certain known results, in particular, theorems by Dragilev, Opial, Reissig, Filippov, and others, on the existence of a periodic solution and the dissipativity of the Liénard and Rayleigh equations, as well as a result of Cartwright and Swinnerton-Dyer, close to Theorem 8.