Periodic Solutions Of The Mixed Type Weakly Coupled Hamiltonian Systems

In this paper,we use the high-dimensional Poincar(?)-Birkhoff twist theorem to study the existence and multiplicity of periodic solutions for the mixed type weakly coupled Hamiltonian Systems.The following three problems are involved1.Existence and multiplicity of periodic solutions for coupled systems with super-linear and across the resonant points2.Existence and multiplicity of periodic solutioris for coupled systems with fast spiraling and slow spiraling at infinity3.Existence and Multiplicity of periodic solutions for coupled systems with super-linear and sublinear mixing near the originThere have been many important results in applying Poincar(?)-Birkhoff twist theorem to study the existence and multiplicity of periodic solutions for the planar Hamiltonian system.But on the other hand,people are more concerned about the corresponding results for the high-dimensional Hamiltonian system.In the past,there are some high-dimensional extensions of Poincar(?)-Birkhoff twist theorem,but there is no essential im-provementRecently,A.Fonda and A.J.Urena proved a high-dimensional Poincar(?)-Birkhoff twist theorem with the form of the differential systems.Then,their research group ap-plied this high-dimensional Poincar(?)-Birkhoff twist theorem to prove many results for the existence and multiplicity of periodic solutions for Hamiltonian systems with the same type of the coupled equations.The processing method of the coupling of the same type of the coupled equations is relatively consistent for the phase plane analysis of each coupled equation.However,the mixed type coupled Hamiltonian systems not only re-strains each other for phase plane analysis,but also faces different difficulties,so their processing methods are invalid for mixed types of coupled systems.How to apply the high-dimensional Poincar(?)-Birkhoff twist theorem to prove the existence and multiplicity of periodic solutions for the mixed type weakly coupled systems is the motivation of this paperIn this paper we firstly consider the spiral properties of the solution components of the coupled system in each coupled phase plane.By analyzing the spiral property of the solutions,we can estimate the change of the polar angle by the radial change of the solution.We will show that under certain conditions,the solution component will spirally run to infiiiity in eacli coupled phase plane.The first problem in this paper is to study the coupled systems with superlinear and across resonant points components.The difficulty encountered is that superlinear condition will cause the corresponding solution component without the global existence Therefore,it is necessary to use the spiral property of the solution component in a certain period of time to prove that the solution component has sufficient rotation in a certain limited region,so that the system can be modified in a place with a sufficiently large diameter to satisfy the global existence of the solution.This type of modification does not affect the discussion of the twist of the original syste.m.Combined with the discussion of the oscillation of the cross-resonant partial solution,it can be proved that each solution component of the new system can generate enough twist in the corresponding phase plane within 2m?T time.Therefore,constructing a suitable annulus and applying the high-dimensional Poincar(?)-Birkhoff twist theorem to prove the existence and multiplicity of the periodic solution of the new system,and then using the angular characteristics of the resulting periodic solution,the periodic solution of the original system is obtainedThe second problem in this paper is to study the coupling system with fast spiraling and slow spiraling components.At this time,the difficulties encountered in the fast spiraling part and the slow spiraling part are different.In the fast spiraling part,the solution may not exist globally,so you need to use the fast rotation of the solution to give a priori estimate of the solution with a certain number of zeros.Therefore,the system is modified in a place with a sufficiently large diameter to satisfy the global existence of the solution without affecting the rotatory analysis of the original system.There is no problem with the global existence of the slow spiraling part.However,the solution component is rotated slowly,and the twist generated during the 2? time period is weak So we must consider the twist in 2m? time period.If m is large,the solution will run to the origin of the corresponding phase plane,and the twist estimation is still difficult to obtain.It is necessary to analyze the spiral property of the solution component to modify the system of the slow spiraling part near the origin of the corresponding phase plane Then apply the high-dimensional Poincar(?)-Birkhoff twist theorem to prove the existence and multiplicity of the periodic solution of the system.The third problem in this paper studies the fast-slow spiraling coupled Hamiltonian system near the origin.The phase plane analysis near the origin is more detailed.We first demonstrate the spiral properties of the solution components in the corresponding phase plane under strict symbolic conditions and weak coupling conditions.But only the conditions near the origin are not clear about the global existence of the solution.For this reason,we not only assume a quasi-linear condition for the superlinear component,but also need to modify the system near the origin of the corresponding phase plane according to the spiral property of the sublinear solution component.After modifying the ideas in the study of the second problem,we can prove the existence and multiplicity of the periodic solution of the weakly coupled Hamiltonian system with super-sublinear mixing near the origin.