On the numerical optimization of KAM estimates by classical perturbation theory

A recently proposed algorithm for the estimate of the threshold above which a certain torus disappears, which combines classical Birkhoff normalization procedure with KAM theory, is reconsidered and improved. This is done by studying the particular case of the forced pendulum Hamiltonian , and considering the golden torus. We find that there exists an optimal orderN of normalization, and the ratio between the estimate thus obtained and the experimental value turns out to be 19. A possible explanation of such result is also suggested.     

Arnol′d Diffusion in a Pendulum Lattice

The main model studied in this paper is a lattice of pendula with a nearest‐neighbor coupling. If the coupling is weak, then the system is near‐integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way. © 2014 Wiley Periodicals, Inc.     

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.     

New Identities for Weak KAM Theory

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.     

ON NON-ISOTROPIC JACOBI PSEUDOSPECTRAL METHOD

In this paper, a non-isotropic Jacobi pseudospectral method is proposed and its appli- cations are considered. Some results on the multi-dimensional Jacobi-Gauss type interpolation and the related Bernstein-Jackson type inequalities are established, which play an important role in pseudospectral method. The pseudospectral method is applied to a twodimensional singular problem and a problem on axisymmetric domain. The convergence of proposed schemes is established. Numerical results demonstrate the efficiency of the proposed method.     

Liouville type theorems for a system of integral equations on upper half space

In this paper,we consider the following system of integral equations on upper half space {u(x) = ∫Rn + (1/|x-y|n-α-1/|-y|n-α) λ1up1(y) + μ1vp2(y) + β1up3(y)vp4(y) dy;v(x) = ∫Rn + (1/|x-y|n-α-1/|-y|n-α)(λ2uq1(y) + μ2vq2(y) + β2uq3(y)vq4(y) dy,where Rn + = {x =(x1,x2,...,xn) ∈ Rn|xn 0}, =(x1,x2,...,xn-1,-xn) is the reflection of the point x about the hyperplane xn= 0,0 α n,λi,μi,βi≥ 0(i = 1,2) are constants,pi≥ 0 and qi≥ 0(i = 1,2,3,4).We prove the nonexistence of positive solutions to the above system with critical and subcritical exponents via moving sphere method.     

NORM CALCULATIONS OF A CLASS OF INTEGRAL OPERATORS

In this paper we calculate the norm of a special class of integral operators acting on LP （C^n, dvs）, where dvs is the Gaussian measure on C^n.     

Approximation of Generalized Bernstein Operators

This paper is devoted to study direct and converse approximation theorems of the generalized Bernstein operators Cn( f,sn,x) via so-called unified modulus ω2φλ( f,t), 0 ≤λ≤1. We obtain main results as follows ω2φλ( f,t) =O(tα)|Cn( f,sn,x)- f(x)| =O(n-12 δ1-λn(x))α,where δ2n(x) =max{φ2(x),1/n} and 0 α 2.     

KAM dynamics and stabilization of a particle sliding over a periodically driven curve

The dynamics of a bead sliding without friction along a periodically pulsating wire is under consideration. If the arc length of the wire is taken as the relevant coordinate, the motion of the bead is described by a periodic newtonian equation. Sufficient conditions are derived assuring that a given equilibrium is of twist type, a property that implies its nonlinear stability as well as a KAM scenario around it. Special attention is paid to the stabilization of unstable equilibria, in parallel with the stabilization of the inverted pendulum.     

Fixed Point Theorem of {a,b,c} Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following： 1. If T is {a,b,c}-ntype mapping, then inf{ ｜｜ T（x）-x｜｜ ：x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞｜｜T（xn） - Xn｜｜ = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn （x）}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].     

Weighted Estimates for the Maximal Commutator of Singular Integral Operator on Spaces of Homogeneous Type

Weighted estimates with general weights are established for the maximal operator associated with the commutator generated by singular integral operator and BMO function on spaces of homogeneous type, where the associated kernel satisfies the HSlder condition on the first variable and some condition which is fairly weaker than the Holder condition on the second variable.     

Asymptotic Property of Approximation to xαgnx by Newman Type Operators

The approximation of ｜x｜ by rational functions is a classical rationalproblem. This paper deals with the rational approximation of the function xasgnx, which equals ｜x｜ if α=1. We construct a Newman type operator rn（x） and show max｜x｜≤1{｜x^αsgnx-rn（x）｜}-Cn-α/2e-√2nα where C is a constant depending on α.     

Uncertainty Principles for the Generalized Fourier Transform Associated with Spherical Mean Operator

The aim of this paper is to establish an extension of qualitative and quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator.     

Almost periodic solutions for a class of higher dimensional Schrödinger equations

In this paper, we show that there are almost periodic solutions corresponding to full dimensional invariant tori for higher dimensional Schr?odinger equations with Fourier multiplier iu_t-Δu+M_ξu+f(|u|²)u = 0, subject to periodic boundary conditions, where the nonlinearity f is a realanalytic function near u = 0 with f(0) = 0.The proof is based on an improved infinite dimensional KAM theorem.     

Boundedness for Hardy Type Operators on Herz Spaces with Variable Exponents

In this paper, we will prove the boundedness of Hardy type operators Hβ(x) and H*β(x) of variable order β(x) on Herz spaces Kα(·)p(·),q and Kα(·)p(·),q' where α(·) and p(·)are both variable.     

A Remark on Pal Type Interpolation on Non-Uniformly Distributed Nodes on the Unit Circle

In this paper we study the problem of explicit representation and convergence of Pal type （0;1） interpolation and its converse, with some additional conditions, on the non-uniformly distributed nodes on the unit circle obtaIned by projecting the interlaced zeros of Pn （x） and Pn′ （x） on the unit circle. The motivation to this problem can be traced to the recent studies on the regularity of Birkhoff interpolation and Pal type interpolations on non-uniformly distributed zeros on the unit circle.     

Numerical simulation and geometrical analysis on the onset of chaos in a system of two coupled pendulums

By means of two geometrical approaches: (i) the potential landscape, and (ii) the Hamiltonian manifold method, we have found the relations that predict the transition to chaos in a two-coupled pendulum system due to a shift from positive to negative curvature. Our scrutiny from numerical simulations show that the splitting of the KAM island at the symmetric mode is another scenario that hints at the incidence of chaotic behaviour. We have uncovered two regimes on the onset of chaos, depending on whether the coupling parameter is greater or less than a critical point. In the first regime, chaos is observed to first occur after the splitting of the KAM island at the symmetric mode and before the switching of the dynamical curvature. On the other hand, an interchange of scenarios arises in the second regime, with chaos now appears after curvature switching and before KAM island splitting at the symmetric mode.     

The Willmore functional of surfaces

The Whitney immersion theorem asserts that every smooth n-dimensional manifold can be immersed in R2n-1, in particular, smooth surfaces can be immersed in R3, and for embeddings the ambient dimension needs to go up by 1. Immersions or embeddings carry both intrinsic and extrinsic information of manifolds, and the latter determines how the manifold fits into the Euclidean space. The key geometric quantity to capture the extrinsic geometry is the second fundamental form.     

KAM TORI FOR DEFOCUSING KDV-MKDV EQUATION

In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u~2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.     

Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

We consider small perturbations of analytic non-twist area preserving mappings, and prove the existence of invariant curves with prescribed frequency by KAM iteration. Generally speaking, the frequency of invariant curve may undergo some drift, if the twist condition is not satisfied. But in this paper, we deal with a degenerate situation where the unperturbed rotation angle function r → ω + r²ⁿ⁺¹ is odd order degenerate at r = 0, and prove the existence of invariant curve without any drift in its frequency. Furthermore, we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders.