反应扩散系统中螺旋波的失稳

文章以反应扩散系统为例,介绍了在可激发系统与振荡系统中螺旋波产生、发展、演化的一些基本性质及规律,并讨论了作者近年来对螺旋波的各种失稳途径、时空混沌的产生机理及螺旋波控制方面所做的实验与理论工作,重点讨论了两类螺旋波失稳现象：爱克豪斯失稳与多普勒失稳,两类失稳都使系统从有规律的螺旋波态变为时空混沌（缺陷湍流）态。     

A speculative study of 23-order fractional Laplacian modeling of turbulence: some thoughts and conjectures

This study makes the first attempt to use the 23-order fractional Laplacian modeling of Kolmogorov -53 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles. Nonlinear inertial interactions and molecular Brownian diffusivity are considered to be the bifractal mechanism behind multifractal scaling of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 23-order fractional Laplacian representation is also used to model nonlinear interactions of fluctuating velocity components, and then we conjecture a fractional Reynolds equation, underlying fractal spacetime structures of Levy 23 stable distribution and the Kolmogorov scaling at inertial scales. The new perspective of this study is that the fractional calculus is an effective approach to modeling the chaotic fractal phenomena induced by nonlinear interactions.     

Quantum turbulence: Theoretical and numerical problems

The term “quantum turbulence” (QT) unifies the wide class of phenomena where the chaotic set of one dimensional quantized vortex filaments (vortex tangles) appear in quantum fluids and greatly influence various physical features. Quantum turbulence displays itself differently depending on the physical situation, and ranges from quasi-classical turbulence in flowing fluids to a near equilibrium set of loops in phase transition. The statistical configurations of the vortex tangles are certainly different in, say, the cases of counterflowing helium and a rotating bulk, but in all the physical situations very similar theoretical and numerical problems arise. Furthermore, quite similar situations appear in other fields of physics, where a chaotic set of one dimensional topological defects, such as cosmic strings, or linear defects in solids, or lines of darkness in nonlinear light fields, appear in the system. There is an interpenetration of ideas and methods between these scientific topics which are far apart in other respects. The main purpose of this review is to bring together some of the most commonly discussed results on quantum turbulence, focusing on analytic and numerical studies. We set out a series of results on the general theory of quantum turbulence which aim to describe the properties of the chaotic vortex configuration, starting from vortex dynamics. In addition we insert a series of particular questions which are important both for the whole theory and for the various applications. We complete the article with a discussion of the hot topic, which is undoubtedly mainstream in this field, and which deals with the quasi-classical properties of quantum turbulence. We discuss this problem from the point of view of the theoretical results stated in the previous sections. We also included section, which is devoted to the experimental and numerical suggestions based on the discussed theoretical models.     

Suppress Winfree turbulence by local forcing excitable systems

The occurrence of Winfree turbulence is currently regarded as one of the principal mechanisms underlying cardiac fibrillation. We develop a local stimulation method that suppresses Winfree turbulence in three-dimensional excitable media. We find that Winfree turbulence can be effectively suppressed by locally injecting periodic signals to only a very small subset (around some surface region) of total space sites. Our method for the first time demonstrates the effectiveness of local low-amplitude periodic excitations in suppressing turbulence in 3D excitable media and has fundamental improvements in efficiency, convenience, and turbulence suppression speed compared with previous strategies. Therefore, it has great potential for developing into a practical low-amplitude defibrillation approach.     

Chaotic transients in spatially extended systems

Different transient-chaos related phenomena of spatiotemporal systems are reviewed. Special attention is paid to cases where spatiotemporal chaos appears in the form of chaotic transients only. The asymptotic state is then spatially regular. In systems of completely different origins, ranging from fluid dynamics to chemistry and biology, the average lifetimes of these spatiotemporal transients are found, however, to grow rapidly with the system size, often in an exponential fashion. For sufficiently large spatial extension, the lifetime might turn out to be larger than any physically realizable time. There is increasing numerical and experimental evidence that in many systems such transients mask the real attractors. Attractors may then not be relevant to certain types of spatiotemporal chaos, or turbulence. The observable dynamics is governed typically by a high-dimensional chaotic saddle. We review the origin of exponential scaling of the transient lifetime with the system size, and compare this with a similar scaling with system parameters known in low-dimensional problems. The effect of weak noise on such supertransients is discussed. Different crisis phenomena of spatiotemporal systems are presented and fractal properties of the chaotic saddles underlying high-dimensional supertransients are discussed. The recent discovery according to which turbulence in pipe flows is a very long lasting transient sheds new light on chaotic transients in other spatially extended systems.     

Theories and applications of second-order correlation of longitudinal velocity increments at three points in isotropic turbulence

In order to introduce new physics to traditional two-point correlations, we define the second-order correlation of longitudinal velocity increments at three points and obtain the analytical expressions in isotropic turbulence. By introducing the Kolmogorov 4/5 law, this three-point correlation explicitly contains velocity second- and third-order moments, which correspond to energy and energy transfer respectively. The combination of them then shows additional information of non-equilibrium turbulence by comparing to two-point correlations. Moreover, this three-point correlation shows the underlying inconsistency between numerical interpolation and three-point scaling law in numerical calculations, and inspires a preliminary model to correct this problem in isotropic turbulence.     

Direct numerical simulation of elastic turbulence and its mixing-enhancement effect in a straight channel flow

In this paper,we present a direct numerical simulation(DNS) of elastic turbulence of viscoelastic fluid at vanishingly low Reynolds number(Re = 1) in a three-dimensional straight channel flow for the first time,using the Giesekus constitutive model for the fluid.In order to generate and maintain the turbulent fluid motion in the straight channel,a sinusoidal force term is added to the momentum equation,and then the elastic turbulence is numerically realized with an initialized chaotic velocity field and a stretched conformation field.Statistical and structural characteristics of the elastic turbulence therein are analyzed based on the detailed information obtained from the DNS.The fluid mixing enhancement effect of elastic turbulence is also demonstrated for the potential applications of this phenomenon.     

Breakdown of universality in quantum chaotic transport: the two-phase dynamical fluid model

We investigate the transport properties of open quantum chaotic systems in the semiclassical limit. We show how the transmission spectrum, the conductance fluctuations, and their correlations are influenced by the underlying chaotic classical dynamics, and result from the separation of the quantum phase space into a stochastic and a deterministic phase. Consequently, sample-to-sample conductance fluctuations lose their universality, while the persistence of a finite stochastic phase protects the universality of conductance fluctuations under variation of a quantum parameter.     

Chaotic free-space laser communication over a turbulent channel

The dynamics of errors caused by atmospheric turbulence in a self-synchronizing chaos-based communication system that stably transmits information over an approximately 5 km free-space laser link is studied experimentally. Binary information is transmitted using a chaotic sequence of short-term pulses as a carrier. The information signal slightly shifts the chaotic time position of each pulse depending on the information bit. We report the results of an experimental analysis of the atmospheric turbulence in the channel and the impact of turbulence on the bit-error-rate performance of this chaos-based communication system.     

Statistics of defect-mediated turbulence influenced by noise

The influence of white noise on defect-mediated turbulence which is modeled by the complex Ginzburg-Landau equation is investigated. We show that the dynamics of defects in the noise-driven spatiotemporal chaos can be described by a simple statistical model. The noise enhances significantly the ability of the turbulent background to advocate new defects with a constant rate, and at the same time it increases the vanishing of defects in the system by introducing an additional annihilation rate that is proportional to the number of defects. A universal probability distribution function is derived for the number of defect pairs.     

Stochastic versus chaotic dynamics in a deterministic system

We analyze a dynamical system whose time evolution depends on an externally controlled model parameter. We observe that the introduction of state-dependent perturbations induces a variety of phenomena which can have either a chaotic or stochastic nature. We analyze the sensitivity of the dynamics and the underlying attractors to the strength, frequency, and time correlations of the external perturbations.     

Don't bleach chaotic data

A common first step in time series signal analysis involves digitally filtering the data to remove linear correlations. The residual data is spectrally white (it is "bleached"), but in principle retains the nonlinear structure of the original time series. It is well known that simple linear autocorrelation can give rise to spurious results in algorithms for estimating nonlinear invariants, such as fractal dimension and Lyapunov exponents. In theory, bleached data avoids these pitfalls. But in practice, bleaching obscures the underlying deterministic structure of a low-dimensional chaotic process. This appears to be a property of the chaos itself, since nonchaotic data are not similarly affected. The adverse effects of bleaching are demonstrated in a series of numerical experiments on known chaotic data. Some theoretical aspects are also discussed.     

Transverse instability of line defects of period-2 spiral waves

Spiral waves that arise in period-2 oscillatory media extended in space generically bear "line defects" along which the local kinetics exhibits a period-1 oscillation. Locally, these defect structures can be viewed as a front separating two period-2 oscillatory domains oscillating 2pi out of phase. Here we show that their shape can become sinusoidal with a transverse instability as in bistable fronts. This instability eventually leads to a line-defect filled spatiotemporal chaotic state having erratic proliferations, annihilations, and regenerations of line defects. The same sequence of phenomena is observed in a model reaction-diffusion system as well as in an experimental system.     

Spatiotemporal dynamics in the coherence collapsed regime of semiconductor lasers with optical feedback

This paper presents a spatiotemporal characterization of the dynamics of a single-mode semiconductor laser with optical feedback. I use the two-dimensional representation of a time-delayed system (where the delay time plays the role of a space variable) to represent the time evolution of the output intensity and the phase delay in the external cavity. For low feedback levels the laser output is generally periodic or quasiperiodic and with the 2D representation I obtain quasiperiodic patterns. For higher feedback levels the coherence collapsed regime arises, and in the 2D patterns the quasiperiodic structures break and "defects" appear. In this regime the patterns present features that resemble those of an extended spatiotemporally chaotic system. The 2D representation allows the recognition of two distinct types of transition to coherence collapse. As the feedback intensity grows the number of defects increases and the patterns become increasingly chaotic. As the delay time increases the number of defects in the patterns do not increase and there is a signature of the previous quasiperiodic structure that remains. The nature of the two transitions is understood by examining the behavior of various chaotic indicators (the field autocorrelation function, the Lyapunov spectrum, the fractal dimension, and the metric entropy) when the feedback intensity and the delay time vary. (c) 1997 American Institute of Physics.     

An electronic technique for measuring phase space dimension from chaotic time series

An instrument for determining the number of independent variables involved in chaotic dynamical systems is described. The elaborated technique allows to obtain this measure in real time from a single observable. It has been used for the quantitative characterization of strange attractors underlying chaotic oscillations in anumber of nonlinear electronic circuits. This technique is found to be fairly insensitive to stochastic noise inherent in real experimental systems.     

Shell models of magnetohydrodynamic turbulence

Shell models of hydrodynamic turbulence originated in the seventies. Their main aim was to describe the statistics of homogeneous and isotropic turbulence in spectral space, using a simple set of ordinary differential equations. In the eighties, shell models of magnetohydrodynamic (MHD) turbulence emerged based on the same principles as their hydrodynamic counter-part but also incorporating interactions between magnetic and velocity fields. In recent years, significant improvements have been made such as the inclusion of non-local interactions and appropriate definitions for helicities. Though shell models cannot account for the spatial complexity of MHD turbulence, their dynamics are not over simplified and do reflect those of real MHD turbulence including intermittency or chaotic reversals of large-scale modes. Furthermore, these models use realistic values for dimensionless parameters (high kinetic and magnetic Reynolds numbers, low or high magnetic Prandtl number) allowing extended inertial range and accurate dissipation rate. Using modern computers it is difficult to attain an inertial range of three decades with direct numerical simulations, whereas eight are possible using shell models.     

On the dynamical mechanism of cross-over from chaotic to turbulent states

The Portevin-Le Chatelier effect is one of the few examples of organization of defects. Here the spatio-temporal dynamics emerges from the cooperative behavior of the constituent defects, namely dislocations and point defects. Recent dynamical approach to the study of experimental time series reports an intriguing cross-over phenomenon from a low dimensional chaotic to an infinite dimensional scale invariant power-law regime of stress drops in experiments on CuAl single crystals and AlMg polycrystals, as a function of strain rate. We show that an extension of a dynamical model due to Ananthakrishna and coworkers for the Portevin-Le Chatelier effect reproduces this cross-over. At low and medium strain rates, the model shows chaos with the structure of the attractor resembling the reconstructed experimental attractor. At high strain rates, the model exhibits a power-law statistics for the magnitudes and durations of the stress drops as in experiments. Concomitantly, the largest Lyapunov exponent is zero. In this regime, there is a finite density of null exponents which itself follows a power law. This feature is similar to the Lyapunov spectrum of a shell model for turbulence. The marginal nature of this state is visualized through slow manifold approach.     

Field correlations of annular beams in extremely strong turbulence

The field correlations of annular beams are formulated when the atmosphere assumes extremely strong turbulence. Thicker and larger ring sized annular beams are found to exhibit larger absolute field correlations. For the same transverse distance at the receiver plane, annular beams attain larger field correlations if the transverse distance starts from the receiver origin. Comparisons of the annular beam absolute field correlations in extremely strong turbulence with the no turbulence results show that the absolute field correlation variations follow similar trends, except that the magnitudes of the absolute field correlations are much smaller in extremely strong turbulence and the annular fields become decorrelated at very short transverse distances. When the inner scale of turbulence becomes smaller, the absolute field correlations of the annular beams in extremely strong turbulence become smaller.     

Direct numerical simulation of the turbulent MHD channel flow at low magnetic Reynolds number for electric correlation characteristics

Direct numerical simulation (DNS) of incompressible magnetohydrodynamic (MHD) turbulent channel flow has been performed under the low magnetic Reynolds number assumption.The velocity-electric field and electric-electric field correlations were studied in the present work for different magnetic field orientations.The Kenjeres-Hanjalic (K-H) model was validated with the DNS data in a term by term manner.The numerical results showed that the K-H model makes good predictions for most components of the velocity-electric field correlations.The mechanisms of turbulence suppression were also analyzed for different magnetic field orientations utilizing the DNS data and the K-H model.The results revealed that the dissipative MHD source term is responsible for the turbulence suppression for the case of streamwise and spanwise magnetic orientation,while the Lorentz force which speeds up the near-wall fluid and decreases the production term is responsible for the turbulence suppression for the case of the wall normal magnetic orientation.     

A numerical simulation of the transition to turbulence in a two-dimensional flow

A numerical simulation of the transition to turbulence is performed using a finite element method. The unsteady Navier-Stokes equations are discretized using a standard Galerkin approximation and a loading strategy for increasing the Reynolds number. The numerical results are then analysed at different Reynolds numbers showing a transition from a steady-state solution to a weakly chaotic one. Phase space diagrams are presented showing the presence of strange attractors. The dimension and Lyapunov's exponents of these attractors are computed and compared with existing results in the literature.