一类超线性不对称Duffing方程的Aubry-Mather集

通过引进合适的作用-角变量变换,并运用新的估计方法,对超线性不对称Duffing方程的Poincaré映射,应用推广的Aubry-Mather定理,证明了一类超线性不对称Duffing方程的Aubry-Mather集的存在性.     

一类超线性Duffing方程的Aubry-Mather集

本文通过引进适当的作用-角变量变换并结合新的估计方法,对超线性Duffing方程的Poincaré映射应用推广的Aubry-Mather定理,获得了一类超线性Duffing方程的Aubry-Mather集存在的充分性条件.     

超线性Duffing方程的Mather集

本文把近年来出现的关于单调扭转映射的Aubry-Mather 理论应用到超线性Duffing方程 x+g(x)=p(t)的研究,这里p(t)∈C⁰(R) 以1为周期,g(x)∈C⁰(R)具有超线性增长性:lim g(x)/x=+∞.其结果可以对缺乏高阶光滑性的大量方程仍给出其整体行为,特别是周期解和拟周期解的刻划.     

次线性可逆系统的Aubry-Mather集

通过引进合适的作用-角变量变换并结合新的估计方法,对次线性可逆系统的Poincare映射,应用推广的平面可逆系统的Aubry-Mather定理,在系统光滑性为C~1的情形下,获得了一类次线性可逆系统的Aubry-Mather集存在的充分条件.     

具有不对称超线性项的Duffing方程解的有界性

本文研究如下具有不对称超线性项的Duffing方程解的有界性x"+ax+3-bx-3=p(t),这里a,b是正常数,x+=max(x,0),x-=max(-x,0).当p(t)∈C(5)(S1)时,证明了该方程的所有解都是有界的.     

时间映射和跨共振点的Duffing方程

本文在较弱的条件下证明了跨共振点的Duffing方程至少存在一个调和解和无穷多个次调和解,并在对称情况下给出了对称次调和解的一个稠密性分布定理。主要的讨论建立在对时间映射的分析,并利用由丁伟岳推广的Poincare-Birkhoff 扭转定理。     

p-Laplacian方程的Aubry-Mather集

本文通过引进合适的作用-角变量变换并结合新的估计方法,对p-Laplacian方程的Poincare映射应用推广的Aubry-Mather定理,证明了一类p-Laplacian方程的Aubry-Mather集的存在性.     

一类超线性Hill型对称碰撞方程的周期运动

本文考虑一类超线性Hill型对称碰撞方程的对称碰撞周期解的存在性、重性和分布问题.通过坐标变换的方法把碰撞相平面转化为全平面进行研究,在一类关于时间映射的超线性条件下证明有外力方程无穷多个对称碰撞调和解和对称碰撞次调和解的存在性;同时研究在没有外力时方程的对称碰撞周期解的稠密性分布.本文还给出对称碰撞方程对称碰撞周期解存在的充分条件.     

具有非线性阻尼项和周期强迫项的次线性不对称可逆系统的拟周期解

本文考虑一类具有非线性阻尼项和周期强迫项的次线性不对称可逆系统x′′+α(x~+)~(1/3)-β(x~-)~(1/3)+q(x)g(x′)+f(x)=e(t)的Aubry-Mather集和拟周期解的存在性,其中x~±=max{±x,0},q(x)、g(x)和f(x)均是R上连续可微函数,e(t)是R上连续2π-周期函数.利用周修义(Shuinee Chow)和裴明亮建立的可逆系统的Aubry-Mather定理,在函数q(x)、f(x)和e(t)具有某种奇偶性假设条件下,本文证明了该可逆系统存在无穷多广义拟周期解.     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

MATHER SETS FOR SUBLINEAR DUFFING EQUATIONS

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一类可逆系统的Aubry-Mather集

本文利用推广的Aubry-Mather定理,获得了一类二阶可逆微分方程Aubry-Mather集存在的充分性条件.     

The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations

We investigate the large-time behavior of viscosity solutions of the Cauchy-Dirichlet problem (CD) for Hamilton-Jacobi equations on bounded domains. We establish general convergence results for viscosity solutions of (CD) by using the Aubry-Mather theory.      

Lagrange stability for impulsive Duffing equations

This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity. This result extends some well-known results on Duffing equations to impulsive Duffing equations.     

Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices

We employ KAM theory to rigorously investigate quasiperiodic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices. Toward this end, we apply a coherent structure ansatz to the Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation describing the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather sets to prove that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. We show that this threshold scales with the square root of the inverse of the two-body scattering length, whereas the rotation number of tori above this threshold is proportional to the amplitude. As a consequence, one obtains the same dynamical picture for lattices of all depths, as an increase in depth essentially affects only scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.     

高阶亚线性Duffing方程的周期解

在本文中，二阶亚线性Ｄｕｆｆｉｎｇ方程周期解存在的结果被推广到高阶Ｄｕｆｆｉｎｇ方：ｘ＾（２ｎ）＋ｇ（ｘ）＝ｐ（ｔ）＝ｐ（ｔ＋２π）（ｎ≥１）和ｘ＾（２ｎ＋１）＋ｇ（ｘ）－ｐ（ｔ）＝ｐ（ｔ＋２π）。     

Hamilton-Jacobi methods for Vakonomic Mechanics

We extend the theory of Aubry-Mather measures to Hamiltonian systems that arise in vakonomic mechanics and sub-Riemannian geometry. We use these measures to study the asymptotic behavior of (vakonomic) action-minimizing curves, and prove a bootstrapping result to study the partial regularity of solutions of convex, but not strictly convex, Hamilton-Jacobi equations.      

Lazer–Leach type conditions on periodic solutions of semilinear resonant Duffing equations with singularities

In this paper, we study the existence of positive periodic solutions of resonant Duffing equations with singularities. Some Lazer–Leach type conditions are given to ensure the existence of positive periodic solutions of singular resonant Duffing equations.     

On a theorem by Mather and Aubry-Mather sets for planar Hamiltonian systems

A result due to Mather on the existence of Aubry-Mather sets for superlinear positive definite Lagrangian systems is generalized in one-dimensional case. Applications to existence of Aubry-Mather sets of planar Hamiltonian systems are given. Project supported by the National Natural Science Foundation of China (Grant No. 19631020).