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.     

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.     

A linear code for the sawtooth and cat maps

The sawtooth maps are a one-parameter set of piecewise linear area preserving maps on the torus. For positive integer values of the parameter K they are automorphisms of the torus, known as the cat maps. We present a symbolic dynamics for these maps in which the symbols are integers. This code is related to a practical problem of the stabilisation of a system which is perturbed by impulses. The code is linear in the sense that an orbit and its code are linearly related, so it is not difficult to obtain a good approximation to one from the other in practice. A stationary stochastic process for generating the code is given explicitly. The theory uses Green function methods, which are also used to study ordered periodic orbits and cantori. The problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed.     

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.     

Chaotic diffusion and 1/f-noise of particles in two-dimensional solids

We investigate the classical nonlinear dynamics of a particle moving conservatively in a two-dimensional periodic potential. The particle exhibits diffusive motion in the absence of random forces. In a broad range of energies above the potential barrier, the diffusion process is anomalously accelerated and associated with 1/f-noise in the power spectrum of velocity fluctuations. The analysis of Poincaré surfaces of section and the distribution of free paths indicate that the phenomenon is caused by a trapping of orbits in a self-similar hierarchy of nested cantori. We describe a statistical theory for this mechanism in terms of a renewal process and a random walk on a hierarchical lattice.Work supported by Deutsche Forschungsgemeinschaft     

Phase space scars and quantum billiards

A semiclassical expression is derived for the spectral Wigner function of ergodic billiards in terms of a sum over contributions from classical periodic orbits. It represents a generalization of a similar formula by Berry, which does not immediately apply to billiard systems. These results are a natural generalization of Gutzwiller's trace formula for the density of states. Our theory clarifies the origin of scars in the eigenfunctions of billiard systems. However, in its present form, it is unable to predict what states will be dominated by individual periodic orbits. Finally, we compare some of the predictions of our theory with numerical results from the stadium. Within the limitations of numerical resolution, we find agreement between the two.     

Diffusion and scaling in escapes from two-degrees-of-freedom Hamiltonian systems

This paper summarizes an investigation of the statistical properties of orbits escaping from three different two-degrees-of-freedom Hamiltonian systems which exhibit global stochasticity. Each time-independent H=H(0)+ varepsilon H('), with H(0) an integrable Hamiltonian and varepsilon H(') a nonintegrable correction, not necessarily small. Despite possessing very different symmetries, ensembles of orbits in all three potentials exhibit similar behavior. For varepsilon below a critical varepsilon (0), escapes are impossible energetically. For somewhat higher values, escape is allowed energetically but still many orbits never escape. The escape probability P computed for an arbitrary orbit ensemble decays toward zero exponentially. At or near a critical value varepsilon (1)> varepsilon (0) there is a rather abrupt qualitative change in behavior. Above varepsilon (1), P typically exhibits (1) an initial rapid evolution toward a nonzero P(0)( varepsilon ), the value of which is independent of the detailed choice of initial conditions, followed by (2) a much slower subsequent decay toward zero which, in at least one case, is well fit by a power law P(t) proportional, variant t(-&mgr;), with &mgr; approximately 0.35-0.40. In all three cases, P(0) and the time T required to converge toward P(0) scale as powers of varepsilon - varepsilon (1), i.e., P(0) proportional, variant ( varepsilon - varepsilon (1))(alpha) and T proportional, variant ( varepsilon - varepsilon (1))(beta), and T also scales in the linear size r of the region sampled for initial conditions, i.e., T proportional, variant r(-delta). To within statistical uncertainties, the best fit values of the critical exponents alpha, beta, and delta appear to be the same for all three potentials, namely alpha approximately 0.5, beta approximately 0.4, and delta approximately 0.1, and satisfy alpha-beta-delta approximately 0. The transitional behavior observed near varepsilon (1) is attributed to the breakdown of some especially significant KAM tori or cantori. The power law behavior at late times is interpreted as reflecting intrinsic diffusion of chaotic orbits through cantori surrounding islands of regular orbits. (c) 1999 American Institute of Physics.     

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

The three-body problem can be traced back to Newton in 1687,but it is still an open question today.Note that only a few periodic orbits of three-body systems were found in 300 years after Newton mentioned this famous problem.Although triple systems are common in astronomy,practically all observed periodic triple systems are hierarchical(similar to the Sun,Earth and Moon).It has traditionally been believed that non-hierarchical triple systems would be unstable and thus should disintegrate into a stable binary system and a single star,and consequently stable periodic orbits of non-hierarchical triple systems have been expected to be rather scarce.However,we report here one family of 135445 periodic orbits of non-hierarchical triple systems with unequal masses;13315 among them are stable.Compared with the narrow mass range(only 10-5)in which stable"Figure-eight"periodic orbits of three-body systems exist,our newly found stable periodic orbits have fairly large mass region.We find that many of these numerically found stable non-hierarchical periodic orbits have mass ratios close to those of hierarchical triple systems that have been measured with astronomical observations.This implies that these stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses quite possibly can be observed in practice.Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses.Note that our approach has general meaning:in a similar way,every known family of periodic orbits of three-body systems with two or three equal masses can be used as a starting point to generate thousands of new periodic orbits of triple systems with distinctly unequal masses.     

On the Periodic Orbits of the Static,Spherically Symmetric Einstein-Yang-Mills Equations

In this paper we analyze the existence of the periodic orbits of the static, spherically symmetric Einstein–Yang–Mills Equations by using the qualitative theory of the ordinary differential equation. We prove that there are no periodic orbits restricted to some invariant set of codimension 1. Furthermore if there is a periodic orbit out of this invariant set, then there must be other periodic orbits, which are symmetric to the first one. We also have results on the non–existence of periodic orbits when the cosmological constant is negative.     

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.     

Quantum transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.     

Stability transformation: a tool to solve nonlinear problems

We present an analysis of the properties as well as the diverse applications and extensions of the method of stabilisation transformation. This method was originally invented to detect unstable periodic orbits in chaotic dynamical systems. Its working principle is to change the stability characteristics of the periodic orbits by applying an appropriate global transformation of the dynamical system. The theoretical foundations and the associated algorithms for the numerical implementation of the method are discussed. This includes a geometrical classification of the periodic orbits according to their behaviour when the stabilisation transformations are applied. Several refinements concerning the implementation of the method in order to increase the numerical efficiency allow the detection of complete sets of unstable periodic orbits in a large class of dynamical systems. The selective detection of unstable periodic orbits according to certain stability properties and the extension of the method to time series are discussed. Unstable periodic orbits in continuous-time dynamical systems are detected via introduction of appropriate Poincaré surfaces of section. Applications are given for a number of examples including the classical Hamiltonian systems of the hydrogen and helium atom, respectively, in electromagnetic fields. The universal potential of the method is demonstrated by extensions to several other nonlinear problems that can be traced back to the detection of fixed points. Examples include the integration of nonlinear partial differential equations and the numerical determination of Markov-partitions of one-parametric maps.     

AN ANALYTICAL AND NUMERICAL STUDY OF A MODIFIED VAN DER POL OSCILLATOR

A three-dimensional system of differential equations that models an electronic oscillator is considered. The equations allow a variety of periodic orbits that originate from a degenerate Hopf bifurcation, which is analytically studied. Numerical results are presented that show the existence of saddle-node cusps of periodic orbits, as well as period-doubling bifurcations, that result in the coexistence of multiple “canard” orbits if one of the parameters is small. The presence of chaotic attractors is also detected.     

二维均匀耦合映象格子中的时空周期图案

目的——构造二维均匀耦合映象格子中的时空周期图案;方法——通过一维耦合映象格子模型的相空间中已知低空间周期轨道,直接构造二维均匀耦合映象格子模型中一系列空间周期轨道,而不必求解其模型方程,并对构造轨道的稳定性进行分析;结果——L²×L²雅可比矩阵可化简为几个2×2矩阵组成的对角矩阵;结论——所构造轨道的稳定性不可能比原来轨道的稳定性高. 关键词： 耦合映象格子 时空周期图案 雅可比矩阵     

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.     

Experimental versus numerical eigenvalues of a Bunimovich stadium billiard: a comparison

We compare the statistical properties of eigenvalue sequences for a gamma=1 Bunimovich stadium billiard. The eigenvalues have been obtained in two ways: one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator (real system), and the other set is calculated numerically (ideal system). We show influence of mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system. We also discuss the influence of a family of marginally stable orbits, the bouncing ball orbits, in two microwave stadium billiards with different geometrical dimensions.     

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     

Double lunar swing-by orbits revisited

The double lunar swing-by orbits are a special kind of orbits in the Earth-Moon system.These orbits repeatedly pass through the vicinity of the Moon and change their shapes due to the Moon’s gravity.In the synodic frame of the circular restricted three-body problem consisting of the Earth and the Moon,these orbits are periodic,with two close approaches to the Moon in every orbit period.In this paper,these orbits are revisited.It is found that these orbits belong to the symmetric horseshoe periodic families which bifurcate from the planar Lyapunov family around the collinear libration point L3.Usually,the double lunar swing-by orbits have k=i+j loops,where i is the number of the inner loops and j is the number of outer loops.The genealogy of these orbits with different i and j is studied in this paper.That is,how these double lunar swing-by orbits are organized in the symmetric horseshoe periodic families is explored.In addition,the 2n lunar swing-by orbits(n≥2)with 2n close approaches to the Moon in one orbit period are also studied.     

量子能谱中的长程关联

从可积系统求迹公式出发,运用Einstein-Brillouin-Keller(EBK)量子化条件,导出了二维无关联振子系统周期轨道作用量量子化条件,由此发现了量子能级与周期轨道之间的对应关系.这种对应关系表明,如果两条能级对应的周期轨道的拓扑相同,这两条能级对回归函数的贡献相干.回归谱中的一个峰是量子能谱中一组与具有相同拓扑的周期轨道相对应的能级之间相干的结果,这一组能级间存在着长程关联.     

Properties of Some Chaotic Billiards with Time-Dependent Boundaries

A dispersing billiard (Lorentz gas) and focusing billiards (in the form of a stadium) with time-dependent boundaries are considered. The problem of a particle acceleration in such billiards is studied. For the Lorentz gas two cases of the time-dependence are investigated: stochastic perturbations of the boundary and its periodic oscillations. Two types of focusing billiards with periodically forced boundaries are explored: stadium with strong chaotic properties and a near-rectangle stadium. It is shown that in all cases billiard particles can reach unbounded velocities. Average velocities of the particle ensemble as functions of time and the number of collisions are obtained.