An extension of Dragilev’s theorem for the existence of periodic solutions of the Liénard equation

Asymptotic behavior of periodic solutions in one-parameter families of Liénard equations

“…> 0. (4) Let AA and BB be the line segments joining A to A and B to B , respectively. Let A B and B A be the pieces of Gr(F ) joining A to B and B to A, respectively.…”

“…For a proof see Theorem 5.1 of [19]. It is worthwhile to mention also that, in [4], the authors provide an extension of Dragilev's theorem.…”

On the uniqueness of the limit cycle for the Liénard equation with f(x) not sign-definite

“…Following the elegant and concise description of Lefschetz [12, p.266], if we consider the level curve 1 2 8) in the dynamical interpretation as motion of a particle, the first term represents its kinetic energy and (1.7) expresses the law of conservation of energy as applied to the particle. For this reason, we may consider the level curves of the function E(x, y) as energy levels (see [3] for a similar discussion). Coming back to the Liénard system (1.3), if we evaluate the derivative of the energy function along the trajectories of system (1.3) we obtaiṅ…”

“…Our approach to gain uniqueness via condition (D), is based on a comparison method with the energy level lines of the associated Duffing equation. Such an approach which has been already mentioned above, will be called method of energy, following [3].…”

On the Qualitative Behavior of a Class of Generalized Liénard Planar Systems

“…In this paper we study the qualitative behavior of the trajectories of (see the monographs [14,24,34,44,47], as well as the articles [8,15,16,25,27,29,[36][37][38][39][40] and the quotations therein) or ẋ = h(y) − F(x)…”