Periodic And Chaotic Dynamics Of Hamiltonian Systems:a Twist Approach

In this thesis,by means of variational method and topological method,the twist property of the solutions of the Hamiltonian system in phase space are characterized,and the existence of periodic solutions and subharmonic solutions,multiplicity of solutions and chaotic dynamics of some important models are studied.This paper mainly discusses the following three problems:1.The Poincare-Birkhoff theorem for higher dimensional Hamiltonian systems coupling resonant linear components with twisting components and its application to a mixed type Hamiltonian systems coupling superliner-sublinear components with Ahmad-LazerPaul type resonance components;2.Subharmonic solutions of sublinear Hamiltonian systems with degenerate properties;3.The periodic solution and chaotic dynamics behavior of elastic impact oscillator with superlinear indefinite weight.The first problem A higher dimensional version of Poincaré-Birkhoff Theorem for systems coupling resonance is studied by means of variational method.The Hamiltonian system coupling resonance and twist is transforms,via a series of symplectic changes,into an unbounded perturbation of a degenerate linear functional.There is no critical point theorem of multiple solution for this kind of problem.Using a generalized saddle point theorem by J.Liu[17],we obtain a multi-solution type saddle point theorem of degenerate linear functionals with unbounded perturbation corresponding to Hamiltonian systems.As an application of this saddle point theorem,we provide an extension of the Poincaré-Birkhoff Theorem for systems coupling resonance and twist,which enables us to deal with many mixed coupled Hamiltonian systems,such as multiplicity of periodic solutions and subharmonic solutions for Hamiltonian systems coupling superliner-sublinear components with Ahmad-Lazer-Paul type resonance components.This isn’t previously studied in the literature.Moreover,our multi-solution type saddle point theorem of degenerate linear functionals with unbounded perturbation provides an effective variable tool for dealing with multi-solution problems in case of "degenerate+unbounded".The second problem We discuss the existence of subharmonic solutions for two types of sublinear systems.The common feature of these systems is that the lack of signpreserving property.It brings many difficulties to characterize twist properties.One is the sublinear bounded coupled Hamiltonian systems with Landesman-Lazer conditions.The difficulty of this system is the characterization of the spiral properties of the solution in the component phase plane.We cannot obtain the upper and lower bounds of spiraling solutions by using Hamiltonians.Therefore,we need more detailed phase plane analysis to discuss the spiral properties of the solutions.In addition,the components of the coupled system restrict and influence each other.For this reason,we construct a truncation function,which makes the modified component system independent of each other near the origin and then the angle estimation of the solution near the origin is meaningful.The other is the degenerate Lotka-Volterra type Hamiltonian system.The coefficient function of the corresponding Hamilton system may vanish on a subinterval mimics the possibility that,seasonally,the predation/hunting is absent.However,the twist property of the solution on the phase plane requires the definite sign property of the weight function under some conditions.So this degradation condition brings great difficulties to the phase plane analysis.By qualitative analysis,we still obtain the twist properties in a given period time and then we can use the Poincaré-Birkhoff theorem to prove multiplicity of subharmonic solutions.The third problem The periodic and chaotic dynamics behavior of elastic impact oscillator with superlinear indefinite weight are studied.The superlinearity implies a blow-up in the intervals of negativity and positivity of weight,so the Poincaré-Birkhoff theorem is not applicable.However,in the intervals where the weight is positive,the solutions of the system have the elastic property and the twist property by superlinear condition and in the intervals where the weight is negative,the solutions present blow-up,which shows the bend-twist dynamic behavior on the topology of the phase plane.Previous studies focused on the Hill type equation,which differs from our case in that the oscillator has a forcing term and generally does not have a zero solution.So,it is not known whether the corresponding oscillator has a globally defined solution,resulting in the complicated phase plane analysis.Our method is to analyze the bend-twist behavior of the solution on the phase plane and find a stable manifold.Then,a closed topological quadrangle is constructed by phase plane analysis and inverse mapping of Poincare mapping.Finally,by using the bend-twist fixed point theorem,the existence of periodic solution and subharmonic solution can be obtained.Further,by studying the bend-twist property of the solutions,we prove the existence of bounded non-periodic solutions with a given sequence of zeros and discuss the chaotic behavior of elastic impact oscillator with superlinear indefinite weight via symbolic dynamics.