Periodic Solutions Of Elastic Impact Oscillators Of Superlinear Time-Weight Equations

Impact oscillator is one of the important models of nonsmooth dynamical system. In this article, we study the dynamics of elastic impact oscillators. We will consider the super-linear oscillator having a potential with weight and study it in two parts: the existence of periodic bouncing solutions of oscillator having a potential with definite sign; the existence of periodic bouncing solutions of oscillator having a potential with indefinite sign.In order to prove the existence of the periodic boucing solutions, firstly we will introduce a new coordinate transformation, transform the system from right half plane to the whole plane. And we translate the impact system into a new equal system. So we can obtain the existence of periodic solutions of impact system if we have them in the new system.In the study of the impact solutions of oscillator having definite sign potential, we make an analysis of the solutions on the new phase plane. We can prove that the Poincare-map has the property of boundry twist. And we obtain the existence of infinite w-periodic solutions by using the Poincare-Birkhoff theorm.In the study of the impact solutions of oscillator having indefinite sign potential, we have a topology analysis on certain solutions set in the space of solutions, then we can tranceform the proplem of the existence of periodic solutions to the problem of existence of certain fixed points. Finally, the existence of infinite w-periodic solutions is proved by an application of a topological lemma.