مشخصات پژوهش

‎In this work we use qualitative theory of differential equations to study the qualitative behavior of the solutions of a generalized Liénard system‎. ‎First‎, ‎we study the problem whether all trajectories of this system intersect the vertical isocline‎, ‎which is very important in the global asymptotic stability of the origin‎, ‎oscillation theory‎, ‎and existence of periodic solutions‎. ‎Also‎, ‎some implicit necessary and sufficient conditions and some explicit conditions will be presented for solutions of the system to be oscillatory or nonoscillatory‎. ‎The other object of this work is to study the orbit structure of the system in a neighborhood of a trajectory which is doubly asymptotic to an equilibrium solution‎, ‎i.e.‎, ‎an orbit which lies in the intersection of the stable and unstable manifolds of a critical point‎. ‎Such an orbit is called a homoclinic orbit‎. ‎This paper is also a study of one of the most beautiful phenomena in dynamical system‎: ‎Limit Cycle‎. ‎In mathematics‎, ‎in the study of dynamical systems with two-dimensional phase space‎, ‎a limit cycle is a closed trajectory having the property that at least one other trajectory spirals into it as time approaches either positive or negative infinity‎. ‎In this paper‎, ‎existence and uniqueness of limit cycles for a generalized Liénard system will be studied‎.