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.

THE KAM THEORY OF DOUBLE PENDULUM

The KAM theory can be used properly to describe the motion of the double pendulum if the gravity is treated as a perturbation. The existence of the KAM invariant closed curves represents that some characters of the “total generalized momentum” conservation of the gravity free system can be kept when the gravity is small in comparison with the total enery.

On Reducibility of Beam Equation with Quasi-periodic Forcing Potential

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation utt ＋ △^2u ＋ αu ＋ εφ（t）（u ＋ u^3） = 0 , α 〉 0 in the dimension one is considered, where u（t,x） and φ（t） are analytic quasi-periodic functions in t, and e is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.

Reducibility of Three Dimensional Skew Symmetric System with High Dimensional Weak Liouvillean Frequencies

In this paper,we consider the reducibility of three-dimensional skew symmetric systems.We obtain a reducibility result if the base frequency is high-dimensional weak Liouvillean and the parameter is sufficiently small.The proof is based on a modified KAM theory for 3-dimensional skew symmetric systems.

Non-existence of KAM Torus

Given an integrable Hamiltonian ho with n-degrees of freedom and a Diophantme trequency ω, then, arbitrarily close to h0 in the C^r topology with r 〈 2n, there exists an analytical Hamiltonian he with no KAM torus of rotation vector w. In contrast with it, KAM tori exist if perturbations are small in C^T topology with r 〉 2n.

Response Solutions for Degenerate Reversible Harmonic Oscillators with Zero-average Perturbation

In this paper,we consider a class of normally degenerate quasi-periodically forced reversible systems,obtained as perturbations of a set of harmonic oscillators,{x˙=y+∈f1(ωt,x,y),y˙=λx^(l)+∈f2(ωt,x,y),where 0≠λ∈R,l>1 is an integer and the corresponding involution G is(−θ,x,−y)→(θ,x,y).The existence of response solutions of the above reversible systems has already been proved in[22]if[f2(ωt,0,0)]satisfies some non-zero average conditions(See the condition(H)in[22]),here[·]denotes the average of a continuous function on T^(d).However,discussing the existence of response solutions for the above systems encounters difficulties when[f_(2)(ωt,0,0)]=0,due to a degenerate implicit function must be solved.This article will be doing work in this direction.The purpose of this paper is to consider the case where[f2(ωt,0,0)]=0.More precisely,with 2p<l,if f_(2)satisfies[f_(2)(ωt,0,0)]=[∂f_(2)(ωt,0,0)/∂x]=[∂^(2)f_(2)(ωt,0,0)/∂x2]=···=[∂p−1f2(ωt,0,0)∂xp−1]=0,eitherλ−1[∂pf2(ωt,0,0)∂xp]<0 as l−p is even orλ−1[∂pf2(ωt,0,0)∂xp]=0 as l−p is odd,we obtain the following results:(1)Forλ>˜0(seeλ˜in(2.2))and sufficiently small,response solutions exist for eachωsatisfying a weak non-resonant condition;(2)Forλ<˜0 and∗sufficiently small,there exists a Cantor set E∈(0,∗)with almost full Lebesgue measure such that response solutions exist for each∈E ifωsatisfies a Diophantine condition.In the remaining case whereλ−1[∂pf2(ωt,0,0)∂xp]>0 and l−p is even,we prove the system admits no response solutions in most regions.

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.

Reducibility for Schrodinger Operator with Finite Smooth and Time-Quasi-periodic Potential

In this paper, the author establishes a reduction theorem for linear Schr?dinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM(Kolmogorov-Arnold-Moser) technique. Moreover, it is proved that the corresponding Schr?dinger operator possesses the property of pure point spectra and zero Lyapunov exponent.

BOUNDEDNESS OF SOLUTIONS FOR SUPERLINEAR REVERSIBLE SYSTEMS

This paper is concerned with the boundedness of solutions for second order differential equations x + f(x, t)x + g(x, t) = 0, which are neither dissipative nor conservative, and where the functions f and g are odd in x and even in t, which are 1-periodic in t, and the function g satisfies g(x,t/x+, as|x| - +. Using the KAM theory for reversible systems, the author proves the existence of invariant tori and thus the boundedness of all the solutions and the existence of quasiperiodic solutions and subharmonic solutions.

Exponential Convergence to Time-Periodic Viscosity Solutions in Time-Periodic Hamilton-Jacobi Equations

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold,where the Hamiltonian satisfies the condition:The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit.It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.

REDUCIBILITY OF REVERSIBLE SYSTEM WITH SMALL PERTURBATION

Reversible system following Hamiltonian system turns out to be another type of equation which has aroused wide interest in recent years. One open problem is whether the celebrated KAM theory and Nekhoroshev method can be extended to reversible system. As showed in [5], there exist KAM tori in reversible system. In this paper, we give a preliminary discussion on whether there is Nekhoroshev type result in reversible system. In particular, we try to apply this method to the reducibility of a type of common reversible system.

Lasry-Lions, Lax-Oleinik and generalized characteristics

In the recent works, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental solutions of the associated Hamilton-Jacobi equations. In the present paper, we exploit the relations among Lasry-Lions regularization, Lax-Oleinik operators(or inf/sup-convolution) and generalized characteristics, which are discussed in the context of the variational setting of Tonelli Hamiltonian dynamics, such as Mather theory and weak KAM(Kolmogorov-Arnold-Moser) theory.

Large time behavior of solutions for a class of time-dependent Hamilton-Jacobi equations

We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.

Quasi-periodic Solutions for the Derivative Nonlinear Schrdinger Equation with Finitely Differentiable Nonlinearities

The authors are concerned with a class of derivative nonlinear Schr¨odinger equation iu_t + u_(xx) + i?f(u, ū, ωt)u_x=0,(t, x) ∈ R × [0, π],subject to Dirichlet boundary condition, where the nonlinearity f(z1, z2, ?) is merely finitely differentiable with respect to all variables rather than analytic and quasi-periodically forced in time. By developing a smoothing and approximation theory, the existence of many quasi-periodic solutions of the above equation is proved.