Dynamical System Approach to a Coupled Dispersionless System: Localized and Periodic Traveling Waves

We investigate the dynamical behavior of a coupled dispersionless system describing a current-conducting string with infinite length within a magnetic field. Thus, following a dynamical system approach, we unwrap typical miscellaneous traveling waves including localized and periodic ones. Studying the relative stabilities of such structures through their energy densities, we find that under some boundary conditions, localized waves moving in positive directions are more stable than periodic waves which in contrast stand for the most stable traveling waves in another boundary condition situation.     

Oscillatory convection in binary mixtures: Thermodiffusion,solutal buoyancy,and advection

The role of thermodiffusive generation of concentration fluctuations via the Soret effect, their contribution to the buoyancy forces that drive convection, the advective mixing effect of the latter, and the diffusive homogenisation are compared and elucidated for oscillatory convection. Numerically obtained solutions of the field equations in the form of spatially extended relaxed traveling waves, of standing waves, and of the transient growth of standing waves and their transition to traveling waves are discussed as well as spatially localized convective states of traveling waves that are surrounded by the quiescent fluid.     

Deltons, peakons and other traveling-wave solutions of a Camassa-Holm hierarchy

In this letter, we study an integrable Camassa-Holm hierarchy whose high-frequency limit is the Camassa-Holm equation. Phase plane analysis is employed to investigate bounded traveling wave solutions. An important feature is that there exists a singular line on the phase plane. By considering the properties of the equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. Those paths in phase planes which represented bounded solutions are discussed one-by-one. Besides solitary, peaked and periodic waves, the equations are shown to admit a new type of traveling waves, which concentrate all their energy in one point, and we name them deltons as they can be expressed as some constant multiplied by a delta function. There also exists a type of traveling waves we name periodic deltons, which concentrate their energy in periodic points. The explicit expressions for them and all the other traveling waves are given.     

Directional motion, current reversals and mass separation in a symmetrical periodic potential

The transport of a symmetric periodic potential driven by a static bias and correlated noises is investigated for both the over-damped case and the under-damped case. By both theoretical approximation and numerical simulations, we study steady current of an over-damped Brownian particle moving in the potential. It is shown that the symmetric periodic potential driven by a static bias and the correlated noises is simultaneously able to exhibit directional transport, a single current reversal, as well as a double current reversal. For the under-damped case, we examine the dynamic at various inertial strengths by direct simulations of the stochastic differential equations. We specially focus on the influence of inertial term in the particle dynamics for the noise induced, directed current. Different directions of the steady current is found for different masses of the particles, thus an efficient scheme to separate the Brownian particles according to their mass is suggested.     

Current reversal and mass separation of inertial Brownian motors in a two-noise ratchet

Transport of an inertial Brownian motor moving in an asymmetric periodic potential driven by an external force and correlated noises is investigated. Using the numerical algorithm, the asymptotic mean velocity (AMV) for characterizing directed transport is obtained. The effects of the external driving force f and the correlation λ between the two noises on the AMV are discussed. The results manifest: (1) the external driving force and the correlation between the two noises can lead to the phenomena like current reversal. (2) The competitions among the external driving force and the correlation between the two noises are necessary for current reversal, i.e., fλ>0. (3) Different directions of transport are found for different masses of the Brownian particles under the condition fλ>0. Therefore a theoretical study is suggested for separating Brownian particles according to their different masses in the ratchet system.     

Additive Temporal Coloured Noise Induced Eckhaus Instability in Complex Ginzburg-Landau Equation System

The effect of additive coloured noises, which are correlated in time, on one-dimensional travelling waves in the complex Ginzburg-Landau equation is studied by numerical simulations. We found that a small coloured noise with temporal correlation could considerably influence the stability of one-dimensional wave trains. There exists an optimal temporal correlation of noise where travelling waves are the most vulnerable. To elucidate the phenomena, we statistically calculated the convective velocities Vg of the wave packets, and found that the coloured noise with an appropriate temporal correlation can decrease Vg, making the system convectively more unstable.     

Two Stochastic Resonances Induced by Two Different Multiplicative Telegraphic Noises for an Electric System

In this paper, an electric system with two dichotomous resistors is investigated. It is shown that this system can display two stochastic resonances, which are the amplitude of the periodic response as the functions of the two dichotomous resistors strengthes respectively. In the limits of Gaussian white noise and shot white noise (i.e., the two noises are both Gaussian white noise or shot white noise), no phenomena of resonance appear. By further study, we find that when the system is with three or more multiplicative telegraphic noises, there are three or more stochastic resonances.     

Stochastic resonance, reverse-resonance, and resonant activation induced by a multi-state noise

In this paper, we investigate the periodic response for a linear system driven by a multiplicative multi-state noise (which is composed of the multiplication of two dichotomous noises) to an input temporal oscillatory signal, and the escape of Brownian particles over the fluctuating potential barrier for a system with a piece-wise linear potential and driven by an additive multi-state noise (which is also composed of the multiplication of two dichotomous noises). For the first system, we get the stochastic resonance phenomenon for the amplitude of the periodic response vs. the two dichotomous noise strengths, and the phenomenon of reverse-resonance for the amplitude of the periodic response vs. k, which represents the asymmetry degree of the dichotomous noises. For the second system, we obtain the resonant activation phenomenon, for which the mean first passage time of the Brownian particles over the fluctuating potential barrier shows a minimum as the function of the transition rates of the multi-state noise.     

Dynamics of defects and traveling waves in an interfacial finger pattern

We present the results of an experimental study of one-dimensional traveling finger patterns dominated by source and sink defects. The system studied is a driven fluid–air interface, and the traveling-wave state arises via a subcritical bifurcation. We analyze space–time images of the patterns to obtain the amplitude and local wave number of the left- and right-going traveling waves near the defects. We determine the width of the defects, and find that the width of the sinks does not vary significantly with distance above the bifurcation over the range of our experiments, while the source width increases as the bifurcation is approached from above. We also observe both localized depressions or enhancements in wave amplitude and periodic modulations of the pattern amplitude and wave number.     

Surface loving and surface avoiding modes

We theoretically study the propagation of sound waves in GaAs/AlAs superlattices focusing on periodic modes in the vicinity of the band gaps. Based on analytical and numerical calculations, we show that these modes are the product of a quickly oscillating function times a slowly varying envelope function. We carefully study the phase of the envelope function compared to the surface of a semi-infinite superlattice. Especially, the dephasing of the superlattice compared to its surface is a key parameter. We exhibit two kind of modes: Surface Avoiding and Surface Loving Modes whose envelope functions have their minima and respectively maxima in the vicinity of the surface. We finally consider the observability of such modes. While Surface Avoiding Modes have experimentally been observed [Phys. Rev. Lett. 97, 1224301 (2006)], we show that Surface Loving Modes are likely to be observable and we discuss the achievement of such experiments. The proposed approach could be easily transposed to other types of wave propagation in unidimensional semi-infinite periodic structures as photonic Bragg mirror.     

Nonlinear trans-resonant waves,vortices and patterns: From microresonators to the early Universe

Perturbed wave equations are considered. Approximate general solutions of these equations are constructed, which describe wave phenomena in different physical and chemical systems. Analogies between surface waves, nonlinear and atom optics, field theories and acoustics of the early Universe can be seen in the similarities between the general solutions that govern each system. With the help of the general solutions and boundary conditions and/or resonant conditions we have derived the basic highly nonlinear ordinary differential equation or the basic algebraic equation for traveling waves. Then, approximate analytic resonant solutions are constructed, which describe the trans-resonant transformation of harmonic waves into traveling shock-, jet-, or mushroom-like waves. The mushroom-like waves can evolve into cloud-like and vortex-like structures. The motion and oscillations of these waves and structures can be very complex. Under parametric excitation these waves can vary their velocity, stop, and change the direction of their motion. Different dynamic patterns are yielded by these resonant traveling waves in the x-t and x-y planes. They simulate many patterns observed in liquid layers, optical systems, superconductors, Bose-Einstein condensates, micro- and electron resonators. The harmonic excitation may be compressed and transformed inside the resonant band into traveling or standing particle-like waves. The area of application of these solutions and results may possibly vary from the generation of nuclear particles, acoustical turbulence, and catastrophic seismic waves to the formation of galaxies and the Universe. In particular, the formation of galaxies and galaxy clusters may be connected with nonlinear and resonant phenomena in the early Universe. (c) 2001 American Institute of Physics.     

噪声交叉关联强度的时间周期调制对线性过阻尼系统的随机共振的影响

针对由加性、乘性噪声和周期信号共同作用的线性过阻尼系统, 在噪声交叉关联强度受到时间周期调制的情况下,利用随机平均法推导了系统响应的信噪比的解析表达式. 研究发现这类系统比噪声间互不相关或噪声交叉关联强度为常数的线性系统具有更丰富的动力学特性, 系统响应的信噪比随交叉关联调制频率的变化出现周期振荡型随机共振, 噪声的交叉关联参数导致随机共振现象的多样化.噪声交叉关联强度的时间周期调制的引入有利于提高对微弱周期信号检测的灵敏度和实现对周期信号的频率估计. 关键词： 随机共振 周期振荡型共振 噪声交叉关联强度 信噪比     

Synchronized firing of FitzHugh-Nagumo neurons by noise

We investigate the influence of noise on synchronization between the spiking activities of neurons with external impulsive forces. We first analyze the dependence of the synchronized firing on the amplitude and the angular frequency of the impulsive force in the noise-free system. Three cases (regular spiking, traveling wave, and chaotic spiking) with low synchronized firing are chosen to study effects due to noise. In each case we find that small noise can be a promoter of synchronization phenomena in neural activities, by choosing an appropriate noise intensity acting on some of the neurons.     

二值噪声激励下欠阻尼周期势系统的随机共振

研究了二值噪声和周期信号共同激励下欠阻尼周期势系统的随机共振. 利用随机能量法计算了系统的平均输入能量和平均输出信号的振幅和相位差, 讨论了二值噪声对随机共振的影响. 发现随着噪声强度的增大, 平均输入能量曲线存在一个极小值和一个极大值, 系统出现先抑制后共振的现象; 同时, 系统信噪比曲线随噪声强度的增加出现单峰现象, 说明系统存在随机共振现象.     

Constructive influence of noise-flatness in correlation ratchets

The influence of noise-flatness on overdamped motion of Brownian particles in a 1D periodic system with a simple sawtooth potential subjected to both unbiased thermal noise and three-level telegraph noise is considered. The exact formula for the stationary probability flux (current) is presented. The phenomenon of multiple current reversals and some topological properties of the hypersurface of zero current in the parameter space of noises are investigated and illustrated by phase diagrams. The conditions for the existence of four current reversals versus the switching rate of nonequilibrium noise are given. An alternative interpretation of the results in terms of cross-correlation between two dichotomous noises is presented.     

Chemical waves in electric field-modelling

Experimental results on pulse wave propagation in a one-dimensional reactor, including effects of an imposed electric field, are reviewed. Then results of the modelling of the experiments by means of a reaction-diffusion model with SH kinetics are presented. Permanent concentration perturbation on the left-hand side of the system produces periodic generation of pulse waves propagating through the system. Periodic perturbation gives both simple and complex periodic pulses; chaotic (aperiodic) pulse waves were observed for certain types of perturbations. Modelling of pulse stopping, splitting and reversal by the imposed electric field is then discussed.     

Effects of the Low Frequencies of Noise on On–Off Intermittency

A bifurcating system subject to multiplicative noise can exhibit on–off intermittency close to the instability threshold. For a canonical system, we discuss the dependence of this intermittency on the Power Spectrum Density (PSD) of the noise. Our study is based on the calculation of the Probability Density Function (PDF) of the unstable variable. We derive analytical results for some particular types of noises and interpret them in the framework of on-off intermittency. Besides, we perform a cumulant expansion (N. G. van Kampen, 24, 171 (1976).) for a random noise with arbitrary power spectrum density and we show that the intermittent regime is controlled by the ratio between the departure from the threshold and the value of the PSD of the noise at zero frequency. Our results are in agreement with numerical simulations performed with two types of random perturbations: colored Gaussian noise and deterministic fluctuations of a chaotic variable. Extensions of this study to another, more complex, system are presented and the underlying mechanisms are discussed. PACS Number: 05.40.-a, 05.45.-a, 91.25.-r     

周期矩形信号作用下时滞非对称单稳系统 的随机共振

研究了周期矩形信号对时滞非对称单稳系统随机共振的影响,系统中加入的噪声均为Gauss白噪声.得到了信噪比的解析表达式,通过分析信噪比曲线发现系统存在随机共振现象.数值结果还表明乘性与加性噪声强度对信噪比的影响是不同的,在SNR-D参数平面上共振与抑制共存.在信噪比随着时滞量变化的曲线图上发现,当系统的非对称性|r|取值很大或者乘性与加性噪声强度比D/α小于1时,参数平面上的随机共振现象会消失.