Fractional variant Planck's over 2pi scaling for quantum kicked rotors without Cantori

Previous studies of quantum delta-kicked rotors have found momentum probability distributions with a typical width (localization length L) characterized by fractional variant Planck's over 2pi scaling; i.e., L approximately variant Planck's over 2pi;{2/3} in regimes and phase-space regions close to "golden-ratio" cantori. In contrast, in typical chaotic regimes, the scaling is integer, L approximately variant Planck's over 2pi;{-1}. Here we consider a generic variant of the kicked rotor, the random-pair-kicked particle, obtained by randomizing the phases every second kick; it has no Kol'mogorov-Arnol'd-Moser mixed-phase-space structures, such as golden-ratio cantori, at all. Our unexpected finding is that, over comparable phase-space regions, it also has fractional scaling, but L approximately variant Planck's over 2pi;{-2/3}. A semiclassical analysis indicates that the variant Planck's over 2pi;{2/3} scaling here is of quantum origin and is not a signature of classical cantori.     

混沌体系中寻找周期轨迹的方法

一个不可积混沌体系，由于扰动而遭到破坏时，存活的周期轨迹体现了体系的本质特征，是 体系的运动骨架.在一定程度上， 可以由周期轨迹来量子化不可积体系，这充分说明了 周期轨迹的重要性.而寻找周期轨迹，也就成为研究混沌体系动力学特性以及对混沌体系进 行量子化的关键问题.结合具体实例，给出了3种常用的寻找周期轨迹方法，并详细探讨了各 种方法的优缺点和适用范围. 关键词： 周期轨迹 数值方法 混沌     

Use of harmonic inversion techniques in the periodic orbit quantization of integrable systems

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained. Received 24 February 2000 and Received in final form 22 May 2000     

Periodic orbit quantization of chaotic maps by harmonic inversion

A method for the semiclassical quantization of chaotic maps is proposed, which is based on harmonic inversion. The power of the technique is demonstrated for the baker's map as a prototype example of a chaotic map.     

Applications of periodic orbit theory toN-particle systems

In recent years a number of new techniques have become available in nonequilibrium statistical mechanics, all derived from dynamical system theory, especially from the thermodynamic formalism of Ruelle. We focus here on periodic orbit theory, and we compare it with a novel approach proposed by Evans, Cohen, and Morriss, and developed further by Gallavotti and Cohen. We argue that the two approaches based on such theories are equivalent for systems of many particles if the underlying dynamics is similar to that of Anosov systems, and that such equivalence should remain in more general situations. We extend our previous explanation of irreversibility in the thermostatted Lorentz gas toN-particle diffusion and shearing systems.     

一种基于周期轨道统计的混沌信号不可预测性强弱的检测方法

论述了对混沌信号不可预测程度进行检测的基本原理,在此基础上提出了一种对混沌信号不可预测性强弱进行检测的方法.分别对硬件电路和数值仿真产生的混沌时间序列的周期轨道进行了统计分析,寻找热鞍周期轨道(SPO)并对其回归谱和回归频度等特性进行了研究.实验结果表明了该方法用于检测混沌信号不可预测性强弱的有效性. 关键词： 不可预测性 检测 回归 混沌     

Estimating generating partitions of chaotic systems by unstable periodic orbits

An outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1. It has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition. Here we present a general, dimension-independent, and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits. Our algorithm allows us to obtain the approximate location of the generating partition for the Ikeda-Hammel-Jones-Moloney map.     

Periodic orbit quantization of the anisotropic Kepler problem

The periodic orbit quantization on the anisotropic Kepler problem is tested. By computing the stability and action of some 2000 of the shortest periodic orbits, the eigenvalue spectrum of the anisotropic Kepler problem is calculated. The aim is to test the following claims for calculating the quantum spectrum of classically chaotic systems: (1) Curvature expansions of quantum mechanical zeta functions offer the best semiclassical estimates; (2) the real part of the cycle expansions of quantum mechanical zeta functions cut at appropriate cycle length offer the best estimates; (3) cycle expansions are superfluous; and (4) only a small subset of cycles (irreducible cycles) suffices for good estimates for the eigenvalues. No evidence is found to support any of the four claims.     

On the convergence of the magnus expansion for spin systems in periodic magnetic fields

An analytically solvable single-spin problem is employed to investigate the convergence of the Magnus expansion for spin systems in periodic magnetic fields. The slow decay of quasi-stationary magnetization under multiple-pulse spin-locking conditions is explained within the framework of a theory based on a time-independent effective hamiltonian.     

Geometric quantization of an OSp(1/2) coadjoint orbit

In this Letter, we show how the complete geometric quantization extends to specific supersymplectic supermanifolds. More precisely, we extend this procedure to OSp(1/2)-coadjoint orbits, which are graded extensions of elliptic Sp(2, )-coadjoint orbits. Our approach exploits results obtained in a previous work, where the notion of a super-Kähler supermanifold was defined, and the former orbits were shown to be nontrivial examples of such a notion. As their underlying Kähler manifolds, these supermanifolds carry a natural (super-Kähler) polarization, a crucial notion that was so far lacking. Geometric quantization leads here to a nontrivial representation of osp(1/2), which is realized in a space of square integrable holomorphic sections of a super-Hermitian complex line bundle sheaf-with-connection over the homogenous space OSp(1/2)/U(1).     

Stabilization of periodic and quasiperiodic motion in chaotic systems through changes in the system variables

This work deals with the suppression of chaos in dissipative systems that exhibit a transition from the coexistence of several periodic oscillations to deterministic chaos. The application of changes in the system variables is able to yield the prechaotic behaviour, that can be either quasiperiodic (two inconmensurate frequencies) or periodic (frequency locking), in the same way as for the original system. The performance of the method is shown by application to the two-dimensional Burgers map.     

Divergence of the chaotic layer width and strong acceleration of the spatial chaotic transport in periodic systems driven by an adiabatic ac force

We show for the first time that a weak perturbation in a Hamiltonian system may lead to an arbitrarily wide chaotic layer and fast chaotic transport. This generic effect occurs in any spatially periodic Hamiltonian system subject to a sufficiently slow ac force. We explain it and develop an explicit theory for the layer width, verified in simulations. Chaotic spatial transport as well as applications to the diffusion of particles on surfaces, threshold devices, and others are discussed.     

On the quantization of dissipative systems

A recent paper of Dekker on the quantization of dissipative systems is examined in some detail. It is argued that one can construct a large number of classical equivalent Hamiltonians for damped systems. These can be formally quantized according to Dirac's method, and the resulting equations are mathematically consistent, but yield different eigenfunctions for the same classical system. However, this procedure should be rejected on physical grounds. That is in quantum mechanics, unlike classical dynamics, the definition of the time derivative of a dynamical variable is unique, and is given by the commutator of the proper Hamiltonian (or the energy operator) and that variable. If the proper Hamiltonian is used for the quantization of a damped system, then the quantal equations are inconsistent for the cases where the rate of energy dissipation depends on the velocity of the particle. As an alternative approach to the quantal theory of dissipative phenomena, a generalization of the Hamilton-Jacobi formalism is considered, where the equation for the principle functionS, depends not only on the space and time derivatives ofS, but onS itself. This leads to a new class of damped systems in classical mechanics. The original Schrödinger method of quantization via the Hamilton-Jacobi equation has been applied to this class of dissipative systems, with the result that the wave equation in this case is a solution of a non-linear Schrödinger-Langevin equation. This formulation has no analogue in the Hamiltonian approach, since in the latter, the resulting wave equation is always linear.Supported in part by a grant from the National Research Council of Canada.     

外加正弦驱动力抑制一类分段光滑系统的混沌运动

针对含间隙碰撞振动的机械系统由于参数的改变，产生分叉和混沌运动，造成原稳定运行的系统产生异常振动时的故障消除方法展开研究.用外加正弦驱动力抑制混沌运动.提出了一种外加驱动力参数调节算法.该算法以原系统稳定运行时的可测状态量为希望目标，根据实际输出与希望目标的误差值，实时产生并逐步微调节至所需外加驱动力.通过对一个典型含间隙往复碰撞模型的数值仿真，检验了该方法的有效性. 关键词： 混沌抑制 碰撞振动 间隙 机械系统