Cantori for the stadium billiard

Although almost all orbits in the stadium billiard are "chaotic," there are many regular orbits as well-the ordered periodic orbits and cantori. The symmetries of the stadium are exploited to find maximizing and saddle periodic orbits. Cantori are maximizing quasiperiodic orbits; they have caustics. Transport in the stadium should be impeded by cantori, just as in any twist map; this is particularly important for those orbits that are nearly glancing.     

Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories. The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method. The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at small spacings because of the C_4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas distribution of random matrix theory. Received 12 February 2001 and Received in final form 17 April 2001     

Test of Trace Formulas for Spectra of Superconducting Microwave Billiards

Experimental tests of various trace formulas, which in general relate the density of states for a given quantum mechanical system to the properties of the periodic orbits of its classical counterpart, for spectra of superconducting microwave billiards of varying chaoticity are reviewed by way of examples. For a two-dimensional Bunimovich stadium billiard the application of Gutzwiller's trace formula is shown to yield correctly locations and strengths of the peaks in the Fourier transformed quantum spectrum in terms of the shortest unstable classical periodic orbits. Furthermore, in two-dimensional billiards of the Limaçon family the transition from regular to chaotic dynamics is studied in terms of a recently derived general trace formula by Ullmo, Grinberg and Tomsovic. Finally, some salient features of wave dynamical chaos in a fully chaotic three-dimensional Sinai microwave billiard are discussed. Here the reconstruction of the spectrum is not as straightforward as in the two-dimensional cases and a modified trace formula as suggested by Balian and Duplantier will have eventually to be applied.     

Quantum spectra and classical periodic orbit in the cubic billiard

Quantum billiards have attracted much interest in many fields. People have made a lot of researches on the two-dimensional (2D) billiard systems. Contrary to the 2D billiard, due to the complication of its classical periodic orbits, no one has studied the correspondence between the quantum spectra and the classical orbits of the three-dimensional (3D) billiards. Taking the cubic billiard as an example, using the periodic orbit theory, we find the periodic orbit of the cubic billiard and study the correspondence between the quantum spectra and the length of the classical orbits in 3D system. The Fourier transformed spectrum of this system has allowed direct comparison between peaks in such plot and the length of the periodic orbits, which verifies the correctness of the periodic orbit theory. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.     

Experimental test of a trace formula for a chaotic three-dimensional microwave cavity

We have measured resonance spectra in a superconducting microwave cavity with the shape of a three-dimensional generalized Bunimovich stadium billiard and analyzed their spectral fluctuation properties. The experimental length spectrum exhibits contributions from periodic orbits of nongeneric modes and from unstable periodic orbits of the underlying classical system. It is well reproduced by our theoretical calculations based on the trace formula derived by Balian and Duplantier for chaotic electromagnetic cavities.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

Quantifying intermittency in the open drivebelt billiard

A "drivebelt" stadium billiard with boundary consisting of circular arcs of differing radius connected by their common tangents shares many properties with the conventional "straight" stadium, including hyperbolicity and mixing, as well as intermittency due to marginally unstable periodic orbits (MUPOs). Interestingly, the roles of the straight and curved sides are reversed. Here, we discuss intermittent properties of the chaotic trajectories from the point of view of escape through a hole in the billiard, giving the exact leading order coefficient lim(t→∞)tP(t) of the survival probability P(t) which is algebraic for fixed hole size. However, in the natural scaling limit of small hole size inversely proportional to time, the decay remains exponential. The big distinction between the straight and drivebelt stadia is that in the drivebelt case, there are multiple families of MUPOs leading to qualitatively new effects. A further difference is that most marginal periodic orbits in this system are oblique to the boundary, thus permitting applications that utilise total internal reflection such as microlasers.     

Dual polygonal billiards and necklace dynamics

We study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics. Our main result is that for a certain class of tables, called the quasi-rational polygons, the dual billiard orbits are bounded. This implies that for the subset of rational tables (i.e. polygons with rational vertices) the dual billiard orbits are periodic.Partially supported by NSF Grant DMS 88-02643     

Big islands in dispersing billiard-like potentials

We derive a rigorous estimate of the size of islands (in both phase space and parameter space) appearing in smooth Hamiltonian approximations of scattering billiards. The derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials. Thus, universality classes for the local behavior are found. Moreover, for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter ε→0, islands of polynomial size in ε appear. This suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling, for example, scattering molecules, may be of physically noticeable effect.     

Spectra and Wavefunctions in a Ray-Splitting Sinai Microwave Billiard and their Semiclassical Interpretation

Experimental results on spectra and wave functions of a ray-splitting microwave billiard are presented. The billiard is formed by a flat rectangular microwave cavity with a quarter-circle insert made of teflon in one of the corners. Using the Gutzwiller trace formula, the contribution of the periodic orbits of the billiard to the density of states are determined. The wave functions, many of them showing scars associated with periodic orbits, are interpreted in terms of the semiclassical Green function.     

The growth rate for the number of singular and periodic orbits for a polygonal billiard

For any simply connected polygon in the plane, the number of billiard orbits which begin and end at a vertex grows subexponentially with respect to the length or to the number of reflections. This implies that the numbers of isolated periodic orbits and of families of parallel periodic orbits do grow subexponentially. The main technical device is a calculation showing that the topological entropy of the Poincaré map for the billiard flow is equal to zero.Supported in part by NSF Grant #DMS-8414400     

矩形弹子球中的量子波包分析(英文)

利用波包分析量子力学体系的动力学行为在研究经典和量子的对应关系方面越来越成为一个非常重要的方法.利用高斯波包分析方法,我们计算了矩形弹子球体系的自关联函数,自关联函数的峰和经典周期轨道的周期符合的很好,这表明经典周期轨道的周期可以通过含时的量子波包方法产生.我们还讨论了矩形弹子球的波包回归和波包的部分回归,计算结果表明在每一个回归时间,波包出现精确的回归.对于动量为零的波包,初始位置在弹子球内部的特殊对称点处,出现一些时间比较短的附加的回归.     

二维Sinai台球系统的量子混沌研究

研究了二维Sinai台球系统的经典与量子的对应关系,运用定态展开法和Gutzwiller的周期轨道理论对Sinai台球系统的态密度经傅里叶变换得到的量子长度谱进行分析,并把量子长度谱中峰的位置与其所对应的经典体系的周期轨道长度做对比,发现两者之间存在很好的对应关系.观察到了一些量子态局域在短周期轨道附近形成量子scarred态或量子superscarred态.还研究了同心与非同心Sinai台球系统的能级最近邻间距分布,发现同心Sinai台球系统是近可积的,非同心Sinai台球系统在θ=3π/8下,随两中心间距离的增加,能级最近邻间距分布将由近可积向维格那分布过渡.     

Wavefunction statistics using scar states

We describe the statistics of chaotic wavefunctions near periodic orbits using a basis of states which optimise the effect of scarring. These states reflect the underlying structure of stable and unstable manifolds in phase space and provide a natural means of characterising scarring effects in individual wavefunctions as well as their collective statistical properties. In particular, these states may be used to find scarring in regions of the spectrum normally associated with antiscarring and suggest a characterisation of templates for scarred wavefunctions which vary over the spectrum. The results are applied to quantum maps and billiard systems.     

Dynamical zeta functions for Artin''s billiard and the Venkov-Zograf factorization formula

Dynamical zeta functions are expected to relate the Schrödinger operator's spectrum to the periodic orbits of the corresponding fully chaotic Hamiltonian system. The relationship is exact in the case of surfaces of constant negative curvature. The recently found factorization of the Selberg zeta function for the modular surface is known to correspond to a decomposition of the Schrödinger operator's eigenfunctions into two sets obeying different boundary condition on Artin's billiard. Here we express zeta functions for Artin's billiard in terms of generalized transfer operators, providing thereby a new dynamical proof of the above interpretation of the factorization formula. This dynamical proof is then extended to the Artin-Venkov-Zograf formula for finite coverings of the modular surface.     

矩形弹子球中的量子谱和经典周期轨道

利用SU(2)相干态的表示,我们构造了二维矩形弹子球中与经典周期轨道对应的波函数.经典周期轨道和量子波函数之间的关系可以通过物理图像清晰的表示出来.另外,利用周期轨道理论,我们计算了二维矩形弹子球体系的量子谱的傅立叶变换ρ(L).变换谱|ρN(L)|2对L图像中的峰可以和粒子在二维矩形腔中运动的经典轨迹的长度相比较.量子谱中的每一条峰正好对应一条经典周期轨道的长度,表明量子力学和经典力学的对应关系.     

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     

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     

Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

Globally enumerating unstable periodic orbits for observed data using symbolic dynamics

The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an observed time series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from observed time series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.