Existence And Multiplicity Of Periodic Solutions For Time-dependent Planar Hamiltonian Systems

In this paper, we consider the existence and multiplicity of periodic solutions for time-dependent planar Hamiltonian Systems via Poincaré-Birkhoff twist theorem and topological degree theory. We mainly discuss the following three problems.1. Existence and multiplicity of periodic solutions for time-dependent hamiltonian systems with rapidly oscillating solutions;2. Infinitely many subharmonic solutions for time-dependent hamiltonian systems with slowly oscillating solutions;3. Periodic solutions of second order equations with asymptotically non-resonance on the use of time-maps.When the planar Hamiltonian system is a perturbation for an autonomous system, we can analyse behavior of the perturbed system solutions through the energy function for the autonomous system. In above conditions, we may use the phase-plane analysis and apply the appropriate nonlinear analysis tools. But if the planar Hamilton system is not a perturbation for an corresponding autonomous system (such as second order time-dependent weight equations), the above methods are no longer affective. Even a simple second order superlinear Hill equation, The solutions may blow up, therefore, Poincaré map is not defined. This brings great difficulty to phase-plane analysis. Therefore, such models, except the classical results obtained by Jacobowitz and Hartman, have not been widely treated.The first two problems in this thesis, by analyzing spiral properties of the rapidly and slowly oscillating solutions on time-dependent Hamiltonian systems (typical exam-ples are the second order suplinear time-dependent potential equation and p-superlinear one-dimensional p-Laplacian equation and Superlinear Hill equation, p-sublinear one- dimensional p-Laplacian), we obtain the estimation for the solution radius. We modify the original system into a smooth system, so that we can use Poincaré-Birkhoff twist the-orem. The rotation angle estimation of periodic solutions admit these solutions being the periodic solutions of original system. This new method is based on phase-plane geometric analysis, which develops analytical estimate method presented by Jacobowitz and Hart-man. Our results have generalized Jacobowitz and Hartman’s work to one-dimensional p-Laplacian equation and partial superlinear second-order equation.A third problem in this paper, We deal with the periodic solutions of second order equations with a forced perturbation at resonance points. Using the point of view that the force is a perturbation, we prove that the periodic solution of non-autonomous equation can be estimated by using time map of autonomous equation. The existence of periodic solutions is thus proved via qualitative analysis and topological degree theory. Capietto, Mawhin and Zanolin have once presented a problem on using time-map to deal with resonance phenomenon. In this thesis we report on a result which give partial answers to this question and generalize a existence result obtained by them.