Periodic Solutions And Lagrange Stability Of Elastic Impact Oscillators And Related Models

In this thesis, we study the periodic solutions and Lagrange stability of elastic impact oscillators and related models. Impact oscillator is one of the important models of nonlinear oscillation and non-smooth Hamiltonian systems. It is associated with the research of the Fermi-Ulam accelerator, dual billiards, the fracture mechanics of metal and celestial mechanics. The research on its dynamics behavior helps to understand these problems.The thesis is consisted of three main parts.1. We prove the existence of the elastic periodic solutions of the sub-linear Hamilton impact oscillators.According to plane Hamilton equations or Hamilton impact oscillators, Pioncar′eBirkho? twist theorem is a basic tool to prove the existence of periodic solutions or infinitely many subharmonic solutions. The key step in proof is to construct an annular domain on the phase plane which satisfies the twist conditions. The twist on the phase plane corresponding to the sub-linear equation is so weak that we have to consider the iterations of its Pioncar′e map, of which the twist can be enough. But the side e?ect of iterations is that some solutions will ran to the origin, then we can’t estimate the angle of rotation. Di?erent from the researches on oscillators without impact, we analyze the spiral properties of the solutions and adopt the method of ”successor map”. Then prove the twist property of impact oscillators directly. We give a general existence theorem under the sign condition. By analysis of the phase plane, we generalize this theorem to subharmonic bouncing solutions of impact oscillator with sub-quadratic potential or weak sub-quadratic potential. Moreover, we consider equations such as pendulum type equation, for this kind of equation doesn’t meet sign conditions and the success map of it is also undefined. We introduce new transformation of coordinates, then the oscillator on the right phase plane is now on the whole plane except origin.2. We study the boundedness of the solutions for the quasi-periodic impact oscillators(Lagrange stability).As an application of the invariant curves theorems of Moser type, smooth, quasiperiodic small twist mappings, we study the boundedness(Lagrange stability) for the bouncing solution of the sub-linear, bounded and semi-linear quasi-periodic oscillators.First of all, we transform the impact system to the Hamiltonian system with the central symmetry vector field. In order to overcome the non-smoothness of the angle variable,we exchange time variable with angle variable, smooth it by integration and polishing,carry out some transformations, and then the Hamiltonian function is reduced to a nearly integrable one with small perturbations. The vector field generated by the near integrable Hamiltonian must be kept central symmetry, so we adopt some certain skills to keep it symmetric in every coordinate transform. The Poincar′e map meets the conditions of the generalized Moser.s small twist theorems, which implies the existence of invariant curves.So we obtain the boundedness of bouncing solutions.3. We study the existence of periodic solutions for singular resonant equations without impact, and apply it to the research for periodic solutions and quasi-periodic solutions of radially symmetric systems.Some singular resonant equations do not have bouncing solutions. But researches on singular equations without impact are relevant to high-dimensional radially symmetric systems. The method of estimation for the Poincar′e map is invalid for the singular resonant equations with unbounded perturbations. So we estimate the successor map on phase plane by detailed qualitative analysis, and then we prove the existence of continuum for periodic solutions by topological degree theory. Consequently, we prove the existence of periodic and quasi-periodic solutions for the radially symmetric systems. Meanwhile,we give an example of radially symmetric systems, which has no periodic solution at resonance point.