二维各向同性变频率谐振子的超对称性及其不变量的超对称量子力学精确解

采用超对称量子力学与不变量相结合的方法讨论了二维各向同性变频率谐振子,给出了二维各向同性变频率谐振子的不变量,采用超对称量子力学方法精确求解了不变量的本征值和本征函数,并且给出了当频率恒定时,二维常频率谐振子的本征值和本征函数的精确解.最后对不变量的超对称性进行了讨论.     

声强谱频移补偿的波导不变量和距离估计方法

基于声强的距离一频率干涉结构,利用对水平阵不同阵元信号的声强谱进行频移补偿的方法估计波导不变量和距离。在波导不变量和距离其中一项已知的情况下,可以通过该方法估计另外一项。利用2011年12月北黄海水声实验中拖曳声源发射和海底水平阵接收的线性调频信号,通过上述方法估计了不同距离处的波导不变量。结果显示波导不变量的估计值在0.8到1.2之间变化,其均值与利用实测海洋环境计算的理论值符合良好。同时,利用相同的信号和波导不变量的理论计算值,通过上述方法在不同时刻进行距离估计,结果与GPS测量距离基本一致。      

广义Birkhoff系统的积分不变量

研究广义Birkhoff系统的积分不变量.给出系统存在积分不变量的条件,在此条件下导出系统的线性积分不变量、通用积分不变量和二阶绝对积分不变量.举例说明结果的应用. 关键词： 广义Birkhoff方程 线性积分不变量 通用积分不变量 二阶绝对积分不变量     

CDG方程和耦合KdV-MKdV方程的微分不变量

本文利用了Olver提出的等价活动标架方法,通过构造合适的活动标架,得到了CDG方程和耦合KdV-MKdV方程的微分不变量,并推得了微分不变量代数.     

用不变量方法解具有非线性场和原子与场非线性耦合的Jaynes-Cummings模型

采用Lewis-Riesenfeld不变量方法研究了具有非线性场和任意形式原子与场相互作用的Jaynes-Cummings模型.该模型由于具有超对称代数结构,因此其Hamiltonian量可用超对称算符的线性组合表示.在算符N′的本征值子空间,用生成元算符构造出系统的不变量后,利用不变量方法求得了系统的一般波函数和时间演化算符,同时也计算了原子布居数和光子数的时间演化表达式.     

Lagrange系统一类新型的非Noether绝热不变量——Lutzky型绝热不变量

研究了Lagrange系统的Lie对称性摄动与新型的非Noether绝热不变量. 列出了未受扰Lagrange系统的Lie对称性导致的Lutzky型精确不变量；基于力学系统的高阶绝热不变量的定义，研究在小扰动作用下Lagrange系统Lie对称性的摄动，得到了系统的一类Lutzky形式的绝热不变量.举例说明方法和结果的应用.     

相空间中离散力学系统对称性的摄动与Hojman型绝热不变量

研究相空间中离散力学系统对称性的摄动与绝热不变量.列出相空间中未受扰离散力学系统的特殊Lie对称性导致的Hojman型精确不变量.基于相空间中力学系统的高阶绝热不变量的定义，研究在小扰动作用下系统Lie对称性的摄动，得到了相空间中离散力学系统的一类新的绝热不变量——Hojman型绝热不变量.举例说明结果的应用. 关键词： 相空间 Lie对称性 摄动 绝热不变量     

利用阵不变量的浅海宽带声源距离深度估计方法

现有阵不变量方法未能确定声源深度,且通常对远距离声源的测距误差较大。针对这一问题,提出一种利用阵不变量的宽带声源距离深度估计方法。该方法先对宽带接收信号进行频域波束形成和傅里叶逆变换，转换到波束-时间域以实现模态分离，波束-时间域各峰值位置的连线形成波束时间迁移曲线,曲线形状由阵不变量参数决定且与声源距离相关,曲线上能量分布由各阶模态激励决定且与声源深度有关;再沿波束时间迁移曲线提取波束时间强度并进行匹配处理,最终实现声源距离深度估计。SWellEx-96实验数据处理结果表明,本文所提方法的声源距离和深度估计平均相对误差分别为3.9%和3.4%,而常规阵不变量方法的平均相对测距误差为5.1%,验证了所提方法的优越性能。     

广义经典力学系统对称性的摄动与绝热不变量

在高维增广相空间中研究广义经典力学系统的精确不变量和绝热不变量.建立了该空间中系 统的对称性与不变量的关系；基于力学系统受到小干扰力作用的高阶绝热不变量的概念，给出了系统的高阶绝热不变量的形式及存在条件，并建立了绝热不变量与对称变换之间的对应关系；最后，举例说明结果的应用. 关键词： 对称性 摄动 不变量 广义经典力学系统 增广相空间     

Birkhoff系统的一类新型绝热不变量

研究Birkhoff系统对称性的摄动与绝热不变量.给出了未受扰Birkhoff系统的Lie对称性导致的Hojman型精确不变量.基于力学系统的高阶绝热不变量的定义，研究在小扰动作用下Birkhoff系统Lie对称性的摄动，得到了系统的一类新型绝热不变量.举例说明结果的应用. 关键词： Birkhoff系统 Lie对称性 摄动 绝热不变量     

Extracting the invariant model from the feedback paths of digital hearing aids

Feedback whistling is a severe problem with hearing aids. A typical acoustical feedback path represents a wave propagation path from the receiver to the microphone and includes many complicated effects among which some are invariant or nearly invariant for all users and in all acoustical environments given a specific type of hearing aids. Based on this observation, a feedback path model that consists of an invariant model and a variant model is proposed. A common-acoustical-pole and zero model-based approach and an iterative least-square search-based approach are used to extract the invariant model from a set of impulse responses of the feedback paths. A hybrid approach combining the two methods is also proposed. The general properties of the three methods are studied using artificial datasets, and the methods are cross-validated using the measured feedback paths. The results show that the proposed hybrid method gives the best overall performance, and the extracted invariant model is effective in modeling the feedback path.     

Meanders and reconnection-collision sequences in the standard nontwist map

New global periodic orbit collision and separatrix reconnection scenarios exhibited by the standard nontwist map are described in detail, including exact methods for determining reconnection thresholds, methods that are implemented numerically. Results are compared to a parameter space breakup diagram for shearless invariant curves. The existence of meanders, invariant tori that are not graphs, is demonstrated numerically for both odd and even period reconnection for certain regions in parameter space. Implications for transport are discussed.     

Renormalization-group analysis for the transition to chaos in Hamiltonian systems

We study the stability of Hamiltonian systems in classical mechanics with two degrees of freedom by renormalization-group methods. One of the key mechanisms of the transition to chaos is the break-up of invariant tori, which plays an essential role in the large scale and long-term behavior. The aim is to determine the threshold of break-up of invariant tori and its mechanism. The idea is to construct a renormalization transformation as a canonical change of coordinates, which deals with the dominant resonances leading to qualitative changes in the dynamics. Numerical results show that this transformation is an efficient tool for the determination of the threshold of the break-up of invariant tori for Hamiltonian systems with two degrees of freedom. The analysis of this transformation indicates that the break-up of invariant tori is a universal mechanism. The properties of invariant tori are described by the renormalization flow. A trivial attractive set of the renormalization transformation characterizes the Hamiltonians that have a smooth invariant torus. The set of Hamiltonians that have a non-smooth invariant torus is a fractal surface. This critical surface is the stable manifold of a single strange set encompassing all irrational frequencies. This hyperbolic strange set characterizes the Hamiltonians that have an invariant torus at the threshold of the break-up. From the critical strange set, one can deduce the critical properties of the tori (self-similarity, universality classes).     

Investigation of relativistic bosons in the presence of two-dimensional time-dependent harmonic interaction

The Duffin–Kemmer–Petiau (DKP) equation has been exactly solved for the spin-one particle in the presence of time-dependent harmonic potential in a two dimensional space using the Lewis–Riesenfeld dynamical invariant and unitary transform methods. The dynamical invariant has been constructed and its eigen functions have been obtained. The total wave function as well as the evolution operator have been derived.     

Exponentially Long Stability Times for a Nonlinear Lattice in the Thermodynamic Limit

In this paper, we construct an adiabatic invariant for a large 1–d lattice of particles, which is the so called Klein Gordon lattice. The time evolution of such a quantity is bounded by a stretched exponential as the perturbation parameters tend to zero. At variance with the results available in the literature, our result holds uniformly in the thermodynamic limit. The proof consists of two steps: first, one uses techniques of Hamiltonian perturbation theory to construct a formal adiabatic invariant; second, one uses probabilistic methods to show that, with large probability, the adiabatic invariant is approximately constant. As a corollary, we can give a bound from below to the relaxation time for the considered system, through estimates on the autocorrelation of the adiabatic invariant.     

Invariants and numerical methods for ODEs

The preservation of some invariants is important during the numerical integration of ODEs. In some cases, failure to maintain certain invariants leads to physically impossible solutions, in other cases to instability. Some authors report that the long term macroscopic characteristics of a solution are better represented if certain invariants are maintained. This paper considers two types of invariants, point invariants which are functions along a particular trajectory, and differential invariants which are invariant relations between neighboring trajectories, such as symplectic invariants. Conventional numerical methods usually introduce (h^p+1) errors in an invariant at each numerical step and these will accumulate over the integration interval. Two approaches to preventing this accumulation are (1) to find methods that maintain the invariants within round-off error, or (2) to find methods that maintain another invariant which is no more than (h^q) different from the invariant satisfied by the ODE. These approaches will be illustrated with some examples. The major objective of this paper is to consider the potential of general methods (that is, methods that do not depend on the specific differential equation) to maintain the invariants.     

三种方法求解外场驱动含时谐振子系统的异同

讨论了外场驱动含量谐振子系统的三种解法的异同，发现利用相干态平均方法与量子不变量理论所得结果一致，而利用海森伯运动方程所求结果与上述两种结果相关一个相位因子。     

Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant

We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. We determine high-probability control-actuation sets by computing regions of uncertainty, almost invariant sets, and Lagrangian coherent structures. The combination of geometric and probabilistic methods allows us to design regions of control, which provide an increase in loitering time while minimizing the amount of control actuation. We show how the loitering time in almost invariant sets scales exponentially with respect to the control actuation, causing an exponential increase in loitering times with only small changes in actuation force. The result is that the control actuation makes almost invariant sets more invariant.     

Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation

We study a natural construction of an invariant measure for the 2-dimensional periodic focusing nonlinear Schrödinger equation, with the critical cubic nonlinearity. We find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased, but that the natural construction, successful for the 1-dimensional case and for the 2-dimensional defocusing case, cannot produce an invariant measure. Our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of Berlin and Kac.     

Degenerate Elliptic Resonances

Quasi-periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations. The parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations. Here we prove that, for a class of elliptic tori, a resummation algorithm can be devised and proved to be convergent, thus extending to such lower-dimensional invariant tori the methods employed to prove convergence of the Lindstedt series either for the maximal (i.e. KAM) tori or for the hyperbolic lower-dimensional invariant tori.