Theory of oscillations in average crisis-induced transient lifetimes

Analytical and numerical study of the roughly periodic oscillations emerging on the background of the well-known power law governing the scaling of the average lifetimes of crisis induced chaotic transients is presented. The explicit formula giving the amplitude of "normal" oscillations in terms of the eigenvalues of unstable orbits involved in the crisis is obtained using a simple geometrical model. We also discuss the commonly encountered situation when normal oscillations appear together with "anomalous" ones caused by the fractal structure of basins of attraction.     

Application of the diffraction trace formula to the three-disk scattering system

The diffraction trace formula derived previously and the spectral determinant are tested on the open three-disk scattering system. The system contains a generic and exponentially growing number of diffraction periodic orbits. In spite of this it is shown that even the scattering resonances with large imaginary part can be reproduced semiclassically. The nontrivial interplay of the diffraction periodic orbits with the usual geometrical orbits produces the fine structure of the complicated spectrum of scattering resonances, which are beyond the resolution of the conventional periodic orbit theory.     

The dynamical feature of transition of a Hamiltonian system to a dissipative system

The mechanism of generation and annihilation of attractors during transition from a Hamiltonian system to a dissipative system is studied numerically using the dissipative standard map. The transient process related to the formation of attracting basins of periodic attractors is studied by discussing the evolution of the KAM tori of the standard map. The result shows that as damping increases, attractors are mainly generated from elliptic orbits of the Hamiltonian system and annihilated by colliding with unstable periodic orbits originating from the corresponding hyperbolic orbits of the Hamiltonian system. The transient process also exhibits the general feature of bifurcation.     

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.     

Linear stability of periodic orbits in lagrangian systems

The orbits of lagrangian systems are given by paths with stationary action. We derive a formula for the linear stability of periodic orbits in discrete-time one-degree-of-freedom lagrangian systems, in terms of the second variation of the action about the orbit in the space of periodic paths.     

二维可积系统的求迹公式和周期轨道的量子化

从Berry–Tabor求迹公式出发,导出了二维可积系统周期轨道作用量的半经典量子化条件.利用此量子化条件,考虑周期轨道满足的周期条件,得到了二维无关联四次振子系统周期轨道作用量的半经典量子化条件,并给出了半经典能级公式.对能级与周期轨道的对应关系做了分析.     

周期轨道的量子化与量子能谱中的长程关联

用二维可积系统的半经典量子化方案和二维无关联振子系统的量子能级与周期轨道之间的对应关系,讨论了一组量子能级之间具有长程关联的内在机制,在二维无关联振子系统中,发现了具有相同拓扑M(M1,M2)的周期轨道相对应的量子能级之间存在着长程关联,并以二维4次无关联振子系统为例做了具体说明.     

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.     

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.     

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.     

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.     

Exact solution of the one-dimensional deterministic fixed-energy sandpile

By reason of the strongly nonergodic dynamical behavior, universality properties of deterministic fixed-energy sandpiles are still an open and debated issue. We investigate the one-dimensional model, whose microscopical dynamics can be solved exactly, and provide a deeper understanding of the origin of the nonergodicity. By means of exact arguments, we prove the occurrence of orbits of well-defined periods and their dependence on the conserved energy density. Further statistical estimates of the size of the attraction's basins of the different periodic orbits lead to a complete characterization of the activity vs energy density phase diagram in the limit of large system's size.     

Quasiclassical Wavefunctions

A convergent quasiclassical formula for the wavefunctions of a closed quantum or wave system is obtained. This is expressed entirely in terms of classical orbits. The result is at the same level as earlier results expressing the spectrum as a finite resurgent sum over composite periodic orbits.     

Prevalence of unstable attractors in networks of pulse-coupled oscillators

We present and analyze the first example of a dynamical system that naturally exhibits attracting periodic orbits that are unstable. These unstable attractors occur in networks of pulse-coupled oscillators, and become prevalent with increasing network size for a wide range of parameters. They are enclosed by basins of attraction of other attractors but are remote from their own basin volume such that arbitrarily small noise leads to a switching among attractors.     

Cycling chaotic attractors in two models for dynamics with invariant subspaces

Nonergodic attractors can robustly appear in symmetric systems as structurally stable cycles between saddle-type invariant sets. These saddles may be chaotic giving rise to "cycling chaos." The robustness of such attractors appears by virtue of the fact that the connections are robust within some invariant subspace. We consider two previously studied examples and examine these in detail for a number of effects: (i) presence of internal symmetries within the chaotic saddles, (ii) phase-resetting, where only a limited set of connecting trajectories between saddles are possible, and (iii) multistability of periodic orbits near bifurcation to cycling attractors. The first model consists of three cyclically coupled Lorenz equations and was investigated first by Dellnitz et al. [Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1243-1247 (1995)]. We show that one can find a "false phase-resetting" effect here due to the presence of a skew product structure for the dynamics in an invariant subspace; we verify this by considering a more general bi-directional coupling. The presence of internal symmetries of the chaotic saddles means that the set of connections can never be clean in this system, that is, there will always be transversely repelling orbits within the saddles that are transversely attracting on average. Nonetheless we argue that "anomalous connections" are rare. The second model we consider is an approximate return mapping near the stable manifold of a saddle in a cycling attractor from a magnetoconvection problem previously investigated by two of the authors. Near resonance, we show that the model genuinely is phase-resetting, and there are indeed stable periodic orbits of arbitrarily long period close to resonance, as previously conjectured. We examine the set of nearby periodic orbits in both parameter and phase space and show that their structure appears to be much more complicated than previously suspected. In particular, the basins of attraction of the periodic orbits appear to be pseudo-riddled in the terminology of Lai [Physica D 150, 1-13 (2001)].     

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.     

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.     

Multiple coexisting attractors,Basin boundaries and basic sets

Orbits initialized exactly on a basin boundary remain on that boundary and tend to a subset on the boundary. The largest ergodic such sets are called basic sets. In this paper we develop a numerical technique which restricts orbits to the boundary. We call these numerically obtained orbits “straddle orbits”. By following straddle orbits we can obtain all the basic sets on a basin boundary. Furthermore, we show that knowledge of the basic sets provides essential information on the structure of the boundaries. The straddle orbit method is illustrated by two systems as examples. The first system is a damped driven pendulum which has two basins of attraction separated by a fractal basin boundary. In this case the basic set is chaotic and appears to resemble the product of two Cantor sets. The second system is a high-dimensional system (five phase space dimensions), namely, two coupled driven Van der Pol oscillators. Two parameter sets are examined for this system. In one of these cases the basin boundaries are not fractal, but there are several attractors and the basins are tangled in a complicated way. In this case all the basic sets are found to be unstable periodic orbits. It is then shown that using the numerically obtained knowledge of the basic sets, one can untangle the topology of the basin boundaries in the five-dimensional phase space. In the case of the other parameter set, we find that the basin boundary is fractal and contains at least two basic sets one of which is chaotic and the other quasiperiodic.     

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     

Quantum Energy Levels and Classical Periodic Orbits: Discreteness and Statistical Duality

It is shown that both universal and non-universal correlations must exist between classical periodic orbits in order that Gutzwiller's semiclassical trace formula is consistent with a real, discrete quantum energy spectrum. Formulae for the two-point correlations are derived. The universal correlations are consistent with those conjectured by Argaman et al. (1993). Likewise, both universal and non-universal correlations must exist between quantum energy levels in order that the trace formula be consistent with the fact that periodic orbit actions are real and discrete. In this case, the two-point correlations implied are consistent with random matrix theory and previous semiclassical calculations. These ideas are illustrated with reference to the primes and the Riemann zeros.