Glassy behavior of light

We study the nonlinear dynamics of a multimode random laser using the methods of statistical physics of disordered systems. A replica-symmetry breaking phase transition is predicted as a function of the pump intensity. We thus show that light propagating in a random nonlinear medium displays glassy behavior; i.e., the photon gas has a multitude of metastable states and a nonvanishing complexity, corresponding to mode-locking processes in random lasers. The present work reveals the existence of new physical phenomena, and demonstrates how nonlinear optics and random lasers can be a benchmark for the modern theory of complex systems and glasses.     

Synchronization from disordered driving forces in arrays of coupled oscillators

The effects of disorder in external forces on the dynamical behavior of coupled nonlinear oscillator networks are studied. When driven synchronously, i.e., all driving forces have the same phase, the networks display chaotic dynamics. We show that random phases in the driving forces result in regular, periodic network behavior. Intermediate phase disorder can produce network synchrony. Specifically, there is an optimal amount of phase disorder, which can induce the highest level of synchrony. These results demonstrate that the spatiotemporal structure of external influences can control chaos and lead to synchronization in nonlinear systems.     

The nonlinear decay of complex signals in dissipative media

The solution of Burgers' equation with random initial conditions is often said to describe "Burgers turbulence." The Burgers equation describes two fundamental effects characteristic of any turbulence-the nonlinear transfer of energy over the spectrum and the dissipation of energy in the small-scale components. Strong interaction between coherent harmonics, associated with the nondispersive nature of the dynamics, leads to the appearance of local self-similar structure. In Burgers' equation, continuous random initial fields are transformed into sequences of regions with regular behavior, with random locations of the shocks separating them. Moreover, the statistical properties of such random fields are also self-similar. It is already known that the merging of the shocks leads to an increase of the external scale of the turbulence, and because of this the energy of a random signal ("noise") decreases more slowly than the energy of simple signals. Here we show that similar behavior takes place for complex regular signals with fractal structure in the coordinate or in the wave-number space. In all these cases, the law of increase of the external scale is determined by the behavior of the structure function of the integral of the initial field-i.e., the structure function of the initial action. (c) 1995 American Institute of Physics.     

Analytic approximations of statistical quantities and response of noisy oscillators

Nonlinear oscillators have been utilized in many contexts because they encompass a large class of phenomena. For a reduced phase oscillator model with weak noise forcing that is necessarily multiplicative, we provide analytic formulas for the stationary statistical quantities of the random period. This is an important quantity which we term ‘response’ (i.e., the spike times, instantaneous frequency in neuroscience, the cycle time in chemical reactions, etc.) that is often analytically intractable in noisy oscillator systems. The analytic formulas are accurate in the weak noise limit so that one does not have to numerically solve a time-varying Fokker-Planck equation. The steady-state and dynamic responses are also analyzed with deterministic forcing. A second order analytic formula is derived for the steady-state response, whereas the dynamic response with time-varying forcing is more complicated. We focus on the specific case where the forcing is sinusoidal and accurately capture the frequency response with an analytic approximation that is obtained with a rescaling of the equation. By utilizing various techniques in the weak noise regime, this work leads to a better understanding of how the random period of nonlinear oscillators are affected by multiplicative noise and external forcing. Comparisons of the asymptotic formulas with a full oscillator system confirm the qualitative accurateness of the theory.     

表面催化反应模型中关联噪声诱导非平衡相变

建立含有关联噪声的双分子-单分子(DM)表面催化反应延迟反馈模型,该模型能同时显示一级和二级非平衡动力学相变,即在一级和二级非平衡动力学相变之间的反应窗口展现.讨论双分子在DM延迟反馈模型中两种吸附机制,即局域和随机吸附模型.研究结果表明:1)外部噪声及两噪声关联性致使反应窗口的宽度收缩;2)内部噪声对非平衡动力学相变行为的影响依赖两噪声关联性,即当两噪声负关联,内部噪声致使反应窗口的宽度变宽;而当两噪声正关联时,内部噪声致使反应窗口的宽度收缩;3)关联噪声致使反应窗口变化对DM模型中一级和二级非平衡动力学相变研究具有重要的科学意义.     

A comparison between transitions induced by random and periodic fluctuations

It is shown that a certain class of nonlinear systems possesses a unique and stable stationary state when subjected to periodic dichotomous modulations of an external parameter. This result enables us to define a probability density for the system and to characterize its shape and support. We compare this probability density with the one obtained in the case that the external parameter fluctuates randomly like a Markovian dichotomous noise and discuss various fluctuation-induced transition phenomena. The effects of these two types of fluctuations are quite dissimilar: the random fluctuations give rise to a richer behavior. The results are applied to the Freedericksz transition in nematic liquid crystals.Fellow of the University of Texas at Austin.     

Duffing系统随机相位抑制混沌与随机共振并存现象的机理研究

针对随机相位作用的Duffing混沌系统, 研究了随机相位强度变化时系统混沌动力学的演化行为及伴随的随机共振现象. 结合Lyapunov指数、庞加莱截面、相图、时间历程图、功率谱等工具, 发现当噪声强度增大时, 系统存在从混沌状态转化为有序状态的过程, 即存在噪声抑制混沌的现象, 且在这一过程中, 系统亦存在随机共振现象, 而且随机共振曲线上最优的噪声强度恰为噪声抑制混沌的参数临界点. 通过含随机相位周期力的平均效应分析并结合系统的分岔图, 探讨了噪声对混沌运动演化的作用机理, 解释了在此过程中随机共振的形成机理, 论证了噪声抑制混沌与随机共振的相互关系.     

Effect of common noise on phase synchronization in coupled chaotic oscillators

We report a general phenomenon concerning the effect of noise on phase synchronization in coupled chaotic oscillators: the average phase-synchronization time exhibits a nonmonotonic behavior with the noise amplitude. In particular, we find that the time exhibits a local minimum for relatively small noise amplitude but a local maximum for stronger noise. We provide numerical results, experimental evidence from coupled chaotic circuits, and a heuristic argument to establish the generality of this phenomenon.     

How Disorder Originates and Grows Inside Order

We present a study of disorder origination and growth inside an ordered phase processes induced by the presence of multiplicative noise within mean-field approximation. Our research is based on the study of solutions of the nonlinear self-consistent Fokker-Planck equation for a stochastic spatially extended model of a chemical reaction. We carried out numerical simulation of the probability distribution density dynamics and statistical characteristics of the system under study for varying noise intensity values and system parameter values corresponding to a spatially inhomogeneous ordered state in a deterministic case. Physical interpretation of the results obtained that determines the scenario of noise-induced order–disorder transition is given. Mean-field results are compared with numerical simulations of the evolution of the model under study. We find that beginning from some value of external noise intensity the “embryo” of disorder appears inside the ordered phase. Its lifetime is finite, and it increases with growth of noise intensity. At some second noise intensity value the ordered and disordered phases begin to alternate repeatedly and almost periodically. The frequency of intermittency grows with the increasing of noise intensity. Ordered and disordered phase intermittency affects the process of spatial pattern formation as a consequent change of spatial inhomogeneity configurations.

     

Mechanisms of stochastic phase locking

Periodically driven nonlinear oscillators can exhibit a form of phase locking in which a well-defined feature of the motion occurs near a preferred phase of the stimulus, but a random number of stimulus cycles are skipped between its occurrences. This feature may be an action potential, or another crossing by a state variable of some specific value. This behavior can also occur when no apparent external periodic forcing is present. The phase preference is then measured with respect to a time scale internal to the system. Models of these behaviors are briefly reviewed, and new mechanisms are presented that involve the coupling of noise to the equations of motion. Our study investigates such stochastic phase locking near bifurcations commonly present in models of biological oscillators: (1) a supercritical and (2) a subcritical Hopf bifurcation, and, under autonomous conditions, near (3) a saddle-node bifurcation, and (4) chaotic behavior. Our results complement previous studies of aperiodic phase locking in which noise perturbs deterministic phase-locked motion. In our study however, we emphasize how noise can induce a stochastic phase-locked motion that does not have a similar deterministic counterpart. Although our study focuses on models of excitable and bursting neurons, our results are applicable to other oscillators, such as those discussed in the respiratory and cardiac literatures. (c) 1995 American Institute of Physics.     

"Green" noise in quasistationary stochastic systems

We consider the behavior of stochastic systems driven by noise with a zero value of spectral density at zero frequency ("green" noise). For this purpose we propose the version of the Krylov-Bogoliubov averaging method to study the systems which are not stationary in the case of an external white noise. We use the ergodicity of a nonlinear random function in the method, and obtain equations for any approximation of the theory. In particular, it is shown in the first approximation that there is an effective potential to describe the averaged motion of the system. We consider a phase-locked loop as an example and show that metastable states are possible. The lifetime of these states essentially increases if the form of a green noise spectrum becomes sharper in the low-frequency region. The high stability of the system driven by green noise is confirmed by numerical simulation. It is important that the theoretical result obtained by the averaging method and the one obtained in the simulation coincide with sufficient accuracy. In conclusion, we discuss some of the unsolved green noise problems. (c) 2001 American Institute of Physics.     

Anticipating the response of excitable systems driven by random forcing

We study the regime of anticipated synchronization in unidirectionally coupled model neurons subject to a common external aperiodic forcing that makes their behavior unpredictable. We show numerically and by analog hardware electronic circuits that, under appropriate coupling conditions, the pulses fired by the slave neuron anticipate (i.e., predict) the pulses fired by the master neuron. This anticipated synchronization occurs even when the common external forcing is white noise.     

Phase-Measuring Algorithms to Suppress Spatially Nonuniform Phase Modulation in a Two-Beam Interferometer

Phase modulation of presently used phase-shifting interferometers is assumed to be spatially uniform across the observing aperture. However, calibration errors or the configuration of an interferometer can cause a spatial nonuniformity in the phase modulation. Spatial nonuniformity causes a significant error in the measured phase when the phase modulator has nonlinear sensitivity. An even-order nonlinearity in the phase modulation in particular contributes to the errors. Lowest-order errors can be suppressed by adding a new symmetry to the sampling functions of the phase-shifting algorithm, however the algorithm suffers from large random noise. The random noise is shown to be decreased substantially by applying one more sampled frame to the algorithm. We derive new seven-sample and eight-sample algorithms that can compensate for a nonuniform phase shift and has much less random noise than the previous algorithm we proposed.     

Random-phase solitons in nonlinear periodic lattices

We predict the existence of random phase solitons in nonlinear periodic lattices. These solitons exist when the nonlinear response time is much longer than the characteristic time of random phase fluctuations. The intensity profiles, power spectra, and statistical (coherence) properties of these stationary waves conform to the periodicity of the lattice. The general phenomenon of such solitons is analyzed in the context of nonlinear photonic lattices.     

Asymptotic probability density of nonlinear phase noise

The asymptotic probability density of nonlinear phase noise, often called the Gordon-Mollenauer effect, is derived analytically when the number of fiber spans is large. Nonlinear phase noise is the summation of infinitely many independently distributed noncentral chi2 random variables with two degrees of freedom. The mean and the standard deviation of those random variables are both proportional to the square of the reciprocal of all odd natural numbers. Nonlinear phase noise can also be accurately modeled as the summation of a noncentral chi2 random variable with two degrees of freedom and a Gaussian random variable.     

Random oscillator with general Gaussian noise

We study the long time behaviour of a nonlinear oscillator subject to a random multiplicative noise with a spectral density (or power-spectrum) that decays as a power law at high frequencies. When the dissipation is negligible, physical observables, such as the amplitude, the velocity and the energy of the oscillator grow as power-laws with time. We calculate the associated scaling exponents and we show that their values depend on the asymptotic behaviour of the external potential and on the high frequencies of the noise. Our results are generalized to include dissipative effects and additive noise.     

Observation of a noise-induced phase transition in a parametric oscillator

We have observed a phase transition induced by external noise, using a parametric oscillator driven by a random current, additionally supplied to the sinusoidal pumping current.     

Autocorrelation functions of nonlinear Fokker-Planck equations

We compute autocorrelation functions from nonlinear Fokker-Planck equations that describe nonlinear families of Markov diffusion processes and illustrate this approach for the Plastino-Plastino Fokker-Planck equation related to the Tsallis entropy.Received: 30 October 2003, Published online: 15 March 2004PACS: 05.20.-y Classical statistical mechanics - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion