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.     

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 Graff 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 sub manifolds.     

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.     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for 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, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

陀螺仪运动的混沌与KAM理论

本文建立了外环轴水平放置的重力对称陀螺仪的运动方程．将非自由陀螺仪转子的质心位置作为扰动,利用Ｍｅｌｎｉｋｏｖ方法和ＫＡＭ理论研究了陀螺仪的运动,研究结果表明：在 螺仪转子的质心与支架中心不重合且充分接近时,或 螺仪能量充分大时,陀螺仪运动的Ｐｏｉｎｃａｒｅ映射既出现Ｓｍａｌｅ马蹄意义下的混沌,又存在ＫＡＭ不变环面和不变闭曲线。     

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.     

Invariant hyperbolic tori for Gevrey-smooth Hamiltonian systems under Rüssmann’s non-degeneracy condition

In this paper, we prove the persistence of hyperbolic lower dimensional invariant tori for Gevrey-smooth perturbations of partially integrable Hamiltonian systems under Riissmann＇s nondegeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth, with the same Gevrey index as the Hamiltonian.     

KAM tori for higher dimensional beam equation with a fixed constant potential

In this paper, we consider the higher dimensional nonlinear beam equation:utt + △2u + σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t near u=0 and σ is a positive constant. It is proved that for any fixed σ>0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.     

KAM-Type Theorem on Resonant Surfaces for Nearly Integrable Hamiltonian Systems

Summary. In this paper, we consider analytic perturbations of an integrable Hamiltonian system in a given resonant surface. It is proved that, for most frequencies on the resonant surface, the resonant torus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation if the Hamiltonian satisfies a certain nondegenerate condition. The surviving tori might be elliptic, hyperbolic, or of mixed type. This shows that there are many orbits in the resonant zone which are regular as in the case of integrable systems. This behavior might serve as an obstacle to Arnold diffusion. The persistence of hyperbolic lower dimensional tori has been considered by many authors [5], [6], [15], [16], mainly for multiplicity one resonant case. To deal with the mechanisms of the destruction of the resonant tori of higher multiplicity into nonhyperbolic lower dimensional tori, we have to deal with some small coefficient matrices that are the generalization of small divisors. Received December 18, 1997; revised December 30, 1998; accepted June 21, 1999     

Quasiperiodic solutions of higher dimensional Duffing’s equations via the KAM theorem

In this paper, we consider the higher dimensional second order differential equations of the formẍ + ∇F(x,t) = 0,x ∈R ⁿ with a class of weakly coupled potentials F( x, t ), periodically depending on t. We prove the existence of infinitely many quasi-periodic solutions for such equations via the KAM theorem.     

Persistence of lower dimensional tori of general types in Hamiltonian systems

This work is a generalization to a result of J. You (1999). We study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.

     

二维非线性五次Schr(o)dinger方程可约化的KAM环面

在周期边界条件下,本文考虑二维非线性五次Schr(o)dinger方程iut-△u+|u|4u=0(t∈R,x∈T2),证明一个无限维的KAM (Kolmogorov-Arnold-Moser)定理.应用无限维的KAM定理,本文获得这个方程一族Whitney光滑的部分双曲的小振幅拟周期解.     

Constructing Invariant Tori for the Spatial Hill Lunar Problem

In this paper, the spatial Hill lunar problem is investigated, and the existence of invariant tori of hyperbolic type in a neighborhood of its equilibrium is shown. Moreover, the author checks the non-degenerate condition analytically and obtains two-dimensional elliptic invariant tori on its central manifold as well.     

Completely degenerate lower-dimensional invariant tori for Hamiltonian system

We study the persistence of lower-dimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.     

具拟周期摄动的近可积Hamiltonian系统的KAM型定理

In this paper we consider the persistence of invariant tori of an integrable Hamiltonian system with a quasiperiodic perturbation. It is proved that if the unperturbed system satisfies the Rtissmann non-degenerate condition and the perturbed system satisfies the co-linked non-resonant condition, then the majority of invariant tori is persistent under the perturbation.     

Persistence of Floquet invariant tori for a class of non-conservative dynamical systems

In this paper we consider a class of non-conservative dynamical system with small perturbation. By the KAM method we prove existence of Floquet invariant tori under the weakest non-resonant conditions.

     

Normal Form for Families of Hamiltonian Systems

We consider perturbations of integrable Hamiltonian systems in the neighborhood of normally parabolic invariant tori. Using the techniques of KAM-theory we prove that there exists a canonical transformation that puts the Hamiltonian in normal form up to a remainder of weighted order 2d ＋ 1. And some dynamical consequences are obtained.     

Spectral properties and numerical range of off-diagonal infinite-dimensional Hamiltonian operators

The spectral properties of the off-diagonal infinite-dimensional Hamiltonian operator are studied in this article. Characterizations of the spectrum, the point spectrum and the residual spectrum are given. A spectral inclusion property of numerical range is proved.