A KAM-type Theorem for Generalized Hamiltonian Systems

In this paper we mainly concern the persistence of invariant tori in generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, system under consideration can be odd dimensional. Under the Riissmann type non-degenerate condition, we proved that the majority of the lower-dimension invariant tori of the integrable systems in generalized Hamiltonian system are persistent under small perturbation. The surviving lower-dimensional tori might be elliptic, hyperbolic, or of mixed type.

Aubry-Mather theory for contact Hamiltonian systems Ⅲ

By exploiting the contact Hamiltonian dynamics(T*M×R,Φ_(t))around the Aubry set of contact Hamiltonian systems,we provide a relation among the Mather set,theΦ_(t)-recurrent set,the strongly static set,the Aubry set,the Ma?éset,and theΦ_(t)-non-wandering set.Moreover,we consider the strongly static set,as a new flow-invariant set between the Mather set and the Aubry set in the strictly increasing case.We show that this set plays an essential role in the representation of certain minimal forward weak Kolmogorov-Arnold-Moser(KAM)solutions and the existence of transitive orbits around the Aubry set.

KAM Type-Theorem for Lower Dimensional Tori in Random Hamiltonian Systems

In this paper, we study the persistence of lower dimensional tori for random Hamiltonian systems, which shows that majority of the unperturbed tori persist as Cantor fragments of lower dimensional ones under small perturbation. Using this result, we can describe the stability of the non-autonomous dynamic systems.

Multiple Periodic Solutions for Some Classes of First-Order Hamiltonian Systems

Considering a decomposition R2N=A⊕B of R2N , we prove in this work, the existence of at least (1+dimA) geometrically distinct periodic solutions for the first-order Hamiltonian system Jx'(t)+H'(t,x(t))+e(t)=0 when the Hamiltonian H(t,u+v) is periodic in (t,u) and its growth at infinity in v is at most like or faster than |v|a, 0≤ae is a forcing term. For the proof, we use the Least Action Principle and a Generalized Saddle Point Theorem.

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first- passage time （MFPT） of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kraraers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.

Persistence of Hyperbolic Tori in Generalized Hamiltonian Systems

In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graft and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on submanifolds.

Eigenvalue Problem of Doubly Stochastic Hamiltonian Systems with Boundary Conditions

In this paper, we investigate the eigenvalue problem of forward-backward doubly stochastic dii^erential equations with boundary value conditions. We show that this problem can be represented as an eigenvalue problem of a bounded continuous compact operator. Hence using the famous Hilbert-Schmidt spectrum theory, we can characterize the eigenvalues exactly.

P-symmetric Subharmonic Solutions for Nonlinear Hamiltonian Systems

In this paper,we prove that for each positive k≡1 mod m there exists a P-symmetric kmτ-periodic solution xk for asymptotically linear mτ-periodic Hamiltonian systems,which are nonautonomous and endowed with a P-symmetry.If the P-symmetric Hamiltonian function is semi-positive,one can prove,under a new iteration inequality of the Maslov-type P-index,that xk_(1) and xk_(2) are geometrically distinct for k_(1)/k_(2)≥(2n+1)m+1;and xk_(1),xk_(2) are geometrically distinct for k_(1)/k_(2)≥m+1 provided xk_(1) is non-degenerate.

A KAM theorem of degenerate infinite dimensional Hamiltonian systems (II)

A class of weaker nondegeneracy conditions is given and an existence theorem of invariant tori is prove n for small perturbations of degenerate integrable infinite dimensional Hamiltonian systems under the weaker nondegeneracy conditions. The measure estimates of the parameter set are also given for which invariant tori exist. It is valuable to point out that by the motivation of finite dimensional situation the nondegeneracy conditions may be the weakest. Mainly KAM machine is used to prove the existence of invariant tori. The measure estimates for small divisor conditions, on which the measure estimates of the parameter set are based, will be given in the second paper.

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.

Periodic Solutions of Nonautonomous Second Order Hamiltonian Systems

In this paper, we develop the local linking theorem given by Li and Willein by replacing the Palais-Smale condition with a Cerami one, and apply it to the study of the existence of periodic solutions of the nonautonomous second order Hamiltonian systems （H） ü＋A（t）u＋∨V（t, u）=0, u∈R^N, t∈R. We handle the case of superquadratic nonlinearities which differ from those used previously. Our results extend the theorems given by Li and Willem.

KAM Theory for Partial Differential Equations

In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations.We provide an overview of the state of the art in this field.

HOMOCLINIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.

Some applications of stochastic averaging method for quasi Hamiltonian systems in physics

Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response sta-tistics.In the present paper,the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced.The applications of the stochastic averaging method in studying the dynamics of active Brownian particles,the reaction rate theory,the dynamics of breathing and denaturation of DNA,and the Fermi resonance and its effect on the mean transition time are reviewed.

New Identities for Weak KAM Theory(For Haim Brezis,in continuing admiration)

This paper records for the Hamiltonian H =1/2|p|~2+ W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.

Lyapunov Center Theorem of Infinite Dimensional Hamiltonian Systems

In this paper we reformulate a Lyapunov center theorem of infinite dimensional Hamiltonian systems arising from PDEs.The proof is based on a modified KAM iteration for periodic case.

Homoclinic Orbits for First Order Hamiltonian Systems with Some Twist Conditions

In this paper, we study the nonperiodic first-order Hamiltonian system u = JL（t）u ＋ JH＇（t,u）, where HεCl（RxR2n）. With some assumptions on L, the corresponding Hamiltonianoperator has only discrete spectrum. By using the index theory for self-adjoint operator equation, we establish the existence of multiple homoclinic orbits for the asymptotically quadratic nonlinearty satisfying some twist conditions between infinity and origin.

含裂纹纳米板振动问题的哈密顿体系方法

基于裂纹处范德华力效应,采用非局部弹性理论构造纳米板模型,并通过导入哈密顿体系建立含裂纹纳米板振动问题的对偶正则控制方程组。在全状态向量表示的哈密顿体系下,将含裂纹纳米板的固有频率和振型问题归结为广义辛本征值和本征解问题。利用哈密顿体系具有的辛共轭正交关系,得到问题解的级数解析表达式。结合边界条件,得到固有频率与辛本征值的代数方程关系式,进而直接给出固有频率的表达式。数值结果表明,非局部尺寸参数和裂纹长度对纳米板振动的各阶固有频率有直接的影响。对比表明,辛方法是准确且可靠的,可为工程应用提供依据。

复杂双摆的KAM定理

本文建立了复杂双摆的Ｈａｍｉｌｔｏｎｉａｎ运动方程，并将重力作为扰动，利用ＫＡＭ理论研究了复杂双摆的运动规律．研究结果表明：复杂双摆运动规律与传统双摆运动规律类似，当重力能量与总能量相比很小时，ＫＡＭ不变闭曲线存在表明无重力系统的“总动量”守恒的特点．

基于互联与阻尼分配无源控制(IDA-PBC)的汽轮机调速系统控制器设计方法

为了保证新型电力系统下机组及系统的稳定性,提出一种针对汽轮机调速系统的互联与阻尼分配无源控制器设计方法。首先,以哈密顿系统理论为基础,对系统能量的动态过程进行描述,分析了哈密顿系统结构对系统能量的影响。然后,采用变量梯度法构建包含调速系统的系统能量函数,以注入能量的导数最小为约束条件确定了系统的阻尼矩阵,克服了互联与阻尼分配无源控制器设计过程中确定系统阻尼矩阵困难的问题,完成了汽轮发电机调速系统互联与阻尼分配无源控制器的设计。最后,在MATLAB/Simulink中搭建单机无穷大系统的模型,验证了本文设计控制器的有效性。