Weak Torsion Properties And The Existence Of Periodic Solutions For Collision Hamiltonian Systems

Impact oscillator is one of the important models of non-smooth dynamical sys-tem. In this article, we study the dynamics of elastic impact oscillators. At first, we consider the existence of periodic bouncing solutions of second-order Hamiltonian os-cillator. Then, we discuss the existence of infinite periodic solutions asymptotically linear impact oscillators.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 the impact system if we have them in the new system.In the study of the impact solutions of second-order Hamiltonian oscillator, we will present a new property:weak twist properties. We transform the system when the impact Hamiltonian system satisfy the weak twist properties, we prove that the new system have periodic solutions with Poincare-Birkhoff theorem. We prove that the periodic solutions is a solution of the original system with weak twist properties and the other properties of the solution. Then, we obtain the existence of periodic bouncing solutions.It’s a new method to study the existence of perodic bouncing solutions of impact oscillators.In the discussion of the periodic solutions of asymptotically linear impact oscilla-tors, we analyze the character of the solutions to verify the corresponding weak twist properties.So we can use the conclusion of second-order Hamiltonian system to solve the problem.