It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependance on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and round-off error) always exist in practice. Thus, due to the SDIC, long-term accurate prediction of chaotic dynamic systems is practically impossible, and therefore numerical simulations of chaos are only mixtures of "true" solution with physical meanings and numerical noises without physical meanings. However, it is traditionally believed that statistic computations based on such kind of "mixtures" of numerical simulations of chaotic dynamic systems are acceptable. In this paper, using the so-called "clean numerical simulation" (CNS) whose numerical noises might be much smaller even than micro-level physical uncertainty and thus are negligible, we gain accurate prediction of a chaotic dynamic system in a long enough interval of time. Then, based on these reliable simulations, the influence of numerical noises on statistic computations is investigated. It is found that the influence of numerical noises is negligible when statistic results are time-independent. Unfortunately, when a chaotic dynamic system is far from equilibrium state so that its statistics are time-dependent, numerical noises have a great influence even on statistic computations. It suggests that even the direct numerical simulations (DNS) might not give reliable statistic computations for non-equilibrium dynamic systems with the SDIC.
We illustrate that, like the truncation error, the round-off error has a significant influence on the reliability of numerical simulations of chaotic dynamic systems. Due to the butterfly-effect, all numerical approaches in double precision cannot give a reliable long-term simulation of chaotic dynamic systems. So, in order to avoid man-made uncertainty of numerical simulations of chaos, we had to greatly decrease both of the truncation and round-off error to a small enough level, plus a verification of solution reliability by means of an additional computation using even smaller truncation an…
It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependence on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and round-off error) always exist in practice. Thus, due to the SDIC, long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e. the Clean Numerical Simulation (CNS), is briefly described, together with its applications to a few Hamiltonian chaotic systems. With negligible numerical noises, t…
The statistical properties of wave chaotic systems of varying dimensionalities and realizations have been studied extensively. These systems are commonly characterized by the statistics of the eigenmode-spacings and the statistics of the eigenfunctions. Here, we propose photonic crystal (PC) defect waveguide graphs as a new physical setting for chaotic graph studies. Photonic crystal waveguides possess a dispersion relation for the propagating modes which is engineerable. Graphs constructed by joining these waveguides possess junctions and bends with distinct scattering properties. We present …
Dynamical localization of chaotic eigenstates in the mixed-type systems: spectral statistics in a billiard system after separation of regular and chaotic eigenstates
The aim of this paper is to compare different sources of stochasticity in the solar system. More precisely we study the importance of the long term influence of asteroids on the chaotic dynamics of the solar system. We show that the effects of asteroids on planets is similar to a white noise process, when those effects are considered on a time scale much larger than the correlation time $τ_φ\simeq10^{4}$ yr of asteroid trajectories. We compute the time scale $τ_{e}$ after which the effects of the stochastic evolution of the asteroids lead to a loss of information for the initial conditions of …
An assumption of smooth response to small parameter changes, of statistics or long-time averages of a chaotic system, is generally made in the field of sensitivity analysis, and the parametric derivatives of statistical quantities are critically used in science and engineering. In this paper, we propose a numerical procedure to assess the differentiability of statistics with respect to parameters in chaotic systems. We numerically show that the existence of the derivative depends on the Lebesgue-integrability of a certain density gradient function, which we define as the derivative of logarith…
Parametric Correlations of Phase Shifts and Statistics of Time Delays in Quantum Chaotic Scattering: Crossover between Unitary and Orthogonal Symmetries
This article presents a study of the electromagnetic response of a chaotic reverberation chamber (RC) in the presence of losses. By means of simulations and of experiments, the fluctuations in the maxima of the field obtained in a conventional mode-stirred RC are compared with those in a chaotic RC in the neighborhood of the Lowest Useable Frequency (LUF). The present work illustrates that the universal spectral and spatial statistical properties of chaotic RCs allow to meet more adequately the criteria required by the Standard IEC 61000-4-21 to perform tests of electromagnetic compatibility.
Distributions of eigenmodes are widely concerned in both bounded and open systems. In the realm of chaos, counting resonances can characterize the underlying dynamics (regular vs. chaotic), and is often instrumental to identify classical-to-quantum correspondence. Here, we study, both theoretically and experimentally, the statistics of chaotic resonances in an optical microcavity with a mixed phase space of both regular and chaotic dynamics. Information on the number of chaotic modes is extracted by counting regular modes, which couple to the former via dynamical tunneling. The experimental da…
Using the Random Matrix Theory approach we derive explicit distributions of the real and imaginary parts for off-diagonal entries of the Wigner reaction matrix $\mathbf{K}$ for wave chaotic scattering in systems with and without time-reversal invariance, in the presence of an arbitrary uniform absorption.
The Random Coupling Model (RCM) is a statistical approach for studying the scattering properties of linear wave chaotic systems in the semi-classical regime. Its success has been experimentally verified in various over-moded wave settings, including both microwave and acoustic systems. It is of great interest to extend its use to nonlinear systems. This paper studies the impact of a nonlinear port on the measured statistical electromagnetic properties of a ray-chaotic complex enclosure in the short wavelength limit. A Vector Network Analyzer is upgraded with a high power option which enables c…
We derive a trace formula that expresses the level density of chaotic many-body systems as a smooth term plus a sum over contributions associated to solutions of the nonlinear Schrödinger (or Gross-Pitaevski) equation. Our formula applies to bosonic systems with discretised positions, such as the Bose-Hubbard model, in the semiclassical limit as well as in the limit where the number of particles is taken to infinity. We use the trace formula to investigate the spectral statistics of these systems, by studying interference between solutions of the nonlinear Schrödinger equation. We show that in…
Comment on "Shadowability of Statistical Averages in Chaotic Systems" by Y.C. Lai, Z. Liu, G.W. Wei, and C.H. Lai, Phys. Rev. Lett. 89, 184101
Lai { \it等} \cite{Lai:2002}研究具有稳定周期轨道和混沌鞍共存的系统。他们声称，通过添加高斯白噪声（GN），平均值在一定的噪声阈值以上会随着代数标度律而变化，并认为这会导致平均值的阴影能力崩溃。在这里，我们表明，（i）阴影没有很好地定义，即使它是，没有故障发生。我们澄清误解（ii）GN的阈值。我们进一步指出（iii）对噪声平均值的影响的误解。最后，我们表明，（iv）缺乏适当的阈值可以导致任何（无意义的）标度指数。
