Infinitely Many Periodic Trajectories of the Generalized Lienard Differential System

摘要

Several recent theorems provide sufficient conditions for the existence or for the existence and stability of infinitely many periodic trajectories of the the generalized Lienard system dx/dt = muF(x) - y, dy/dt = g(x), mu,x,y real. First by applying a diffeomorphism of the plane two of these theorems which require that g(x) identically equal to x are extended to the case where g is spring-like, i.e. g satisfying the hypothesis xg(x) > 0 for x not = 0. For a third theorem which requires g to be spring-like, the conditions on F are relaxed by using a diffeomorphism of the plane. Next the generalized Lienard system is considered under the assumption that there exists an interval (x(1), x(2)) such that g(x) is negative to the left of x(1) and g(x) is positive to the right of x(2). Sufficient conditions are established which, when is sufficiently small, guarantee that infinitely many periodic trajectories exist and that those trajectories are alternately stable and unstable. A useful asymptotic expansion for a function involved in these sufficient conditions is obtained. The results are applied to the system with F(x) periodic of mean zero and g(x) asymptotic to the identity function. (Author)