Lagrange Stability For Quasi-periodic Impact Oscillators

In this thesis, we study the Lagrange stability of quasi-periodic impact oscillators. Impact oscillator is the one of important models of nonlinear oscillation and non-smooth Hamiltonian systems. The dynamics of the impact oscillators relate to the research of the Fermi-Ulam accelerator, dual billiards, the fracture mechanics of metal and celestial mechanics. The study of the Lagrange stability for quasi-periodic impact oscillators is based on the KAM techniques.In this paper, we give the invariant curves theorems of quasi-periodic twist mappings under analytic or smooth conditions. As applications of these theorems, we obtain the Lagrange stability of impact motion for quasi-periodic elastic impact oscillators. The thesis is consisted of three main parts.1. We prove three theorems of invariant curves for planar quasi-periodic twist map.At first, by using Moscr's classical KAM technique, we give invariant curves theorem of analytic quasi-periodic twist mappings. Then, according to the different situations of quasi-periodic frequencies, we generalize above invariant curves theorem to the version with averaged small twist and the version with the frequencies being rational dependent, respectively.In the proof of the invariant curves theorem of mapping with averaged small twist, because the frequencies are only rational independent and don't satisfy Diophantine condition in this case, we can't deal with the problem of small divisors. Thus, we solve the homological equations by approximate method. We define the transformation of variables and take the errors into the small perturbation terms, and then we get the new mapping which satisfies the conditions of quasi-periodic small twist mapping and obtain the existence of invariant curves.In the proof of the invariant curves theorem with the frequencies being rational dependent, at first, we separate the terms of rational dependent and the terms of rational independent in the mapping. Then, we use the same method to deal with the part of rational independent and give the errors estimation. The part of rational dependent is the main part of the transformed mapping; we use the energy-angle variable in Hamiltonian system. In the following, we give a transformation exchanging the angle in the original system and time in the new system. Then the transformed mapping meets all the assumptions of Moser's small twist theorem of the quasi-periodic mappings. So the existences of invariant curves are obtained.2. We study the boundedness of the solutions for the analytic quasi-periodic impact oscillators (Lagrange stability).We will study them in two situations.In the first situation, we consider the boundedness for the bouncing solution of the asymptotically linear quasi-periodic oscillators, non-resonance or near resonance. First of all, we transform the impact system to the Hamiltonian system with impacts. After carrying out some transformations, the Hamiltonian function is reduced to a nearly integrable one. The vector field generated by the near integrable Hamiltonian induces a nearly integrable mapping on the surface of the section corresponding to the boundary (Poincare map). This map is analytic and satisfied the conditions of the generalized Moser's small twist theorems in Chapter 2, which implies the existence of invariant curves. So we obtain the boundedness of bouncing solutions.In the second situation, we consider the boundedness of solutions for the superlincar quasi-periodic impact oscillators. Similarly, we transform the impact system to the Hamiltonian system with impacts. After carrying out action-angle variables and canonical rescaling, we obtain a nearly integrable Hamiltonian system. The rest of the proof is similar as the case of the asymptotically linear one.3. We study the existence of invariant curves for smooth quasi-periodic twist map.Under some smoothness conditions, using the analytic approximation lemma of Jackson, Moser and Zchnder, we construct a sequence of real-analytic functions. Applying the KAM iterate techniques, we estimate the errors in direct way and prove the existence of the invariant curves. Then we can obtain the boundedness of solutions for the asymptotically linear and superlinear impact oscillators if the oscillators arc smooth enough.