Discrete dynamics and metastability: Mean first passage times and escape rates

The problem of escape from a domain of attraction is applied to the case of discrete dynamical systems possessing stable and unstable fixed points. In the presence of noise, the otherwise stable fixed point of a nonlinear map becomes metastable, due to noise-induced hopping events, which eventually pass the unstable fixed point. Exact integral equations for the moments of the first passage time variable are derived, as well as an upper bound for the first moment. In the limit of weak noise, the integral equation for the first moment, i.e., the mean first passage time (MFPT), is treated, both numerically and analytically. The exponential leading part of the MFPT is given by the ratio of the noise-induced invariant probability at the stable fixed point and unstable fixed point, respectively. The evaluation of the prefactor is more subtle: It is characterized by a jump at the exit boundaries, which is the result of a discontinuous boundary layer function obeying an inhomogeneous integral equation. The jump at the boundary is shown to be always less than one-half of the maximum value of the MFPT. On the basis of a clear-cut separation of time scales, the MFPT is related to the escape rate to leave the domain of attraction and other transport coefficients, such as the diffusion coefficient. Alternatively, the rate can also be obtained if one evaluates the current-carrying flux that results if particles are continuously injected into the domain of attraction and captured beyond the exit boundaries. The two methods are shown to yield identical results for the escape rate of the weak noise result for the MFPT, respectively. As a byproduct of this study, we obtain general analytic expressions for the invariant probability of noisy maps with a small amount of nonlinearity.     

Noise-induced intermittency in the quasiperiodic regime of Rayleigh-Bénard convection

Thermal measurements on a converting dilute³He-superfluid⁴He solution in the quasiperiodic regime show a transition from a mode-locked periodic state to chaotic time dependence via intermittency. The type of intermittency is discussed in the context of standard models of the phenomenon. In a region just below the onset of intermittency, injection of external multiplicative noise with noise amplitude above a certain threshold level induces the chaotic state. This noise-induced transition can be understood to be due to perturbations of a system with a barely stable attractor; the noise causes the system to escape the weakly attracting periodic points. We present a numerical simulation of a 1D map with external noise which explains some aspects of the noise-induced behavior, and a 2D map which has certain features of the intermittency.     

小世界神经元网络随机共振现象:混合突触和部分时滞的影响

实际神经元网络中,信息传递时电突触和化学突触同时存在,并且有些神经元间的时滞很小可以忽略.本文构建了带有不同类型突触耦合的小世界网络,研究部分时滞、混合突触及噪声对随机共振的影响.结果表明:兴奋性和抑制性突触的比例影响共振的产生;在抑制性突触为主的网络里,几乎不产生随机共振.系统最佳噪声强度和化学突触比例大致呈线性递增关系;特别是在以化学耦合为主的混合突触网络里,仅当兴奋性突触与抑制性突触比例约为4:1时,噪声才可诱导网络产生共振行为.在此比例下,引入部分时滞发现时滞可诱导网络产生随机多共振,且随网络中时滞边比例的增加,系统响应强度达到最优水平的时滞取值区间逐渐变窄;同时发现,网络中含有的化学突触越多,部分时滞诱导产生的多共振行为越强.此外,当时滞为系统固有周期的整数倍时,时滞越大共振所对应的噪声区域越广;并且网络中时滞边越多,越容易促使噪声和时滞诱导其产生明显的共振行为.     

Fluctuational escape from a quasi-hyperbolic attractor in the Lorenz system

Noise-induced escape from the basin of attraction of a quasi-hyperbolic chaotic attractor in the Lorenz system is considered. The investigation is carried out in terms of the theory of large fluctuations by experimentally analyzing the escape prehistory. The optimal escape trajectory is shown to be unique and determined by the saddle-point manifolds of the Lorenz system. We established that the escape process consists of three stages and that noise plays a fundamentally different role at each of these stages. The dynamics of fluctuational escape from a quasi-hyperbolic attractor is shown to differ fundamentally from the dynamics of escape from a nonhyperbolic attractor considered previously [1]. We discuss the possibility of analytically describing large noise-induced deviations from a quasi-hyperbolic chaotic attractor and outline the range of outstanding problems in this field.     

Stochastic dynamics of the cubic map: A study of noise-induced transition phenomena

The effects of finite-amplitude, additive noise on the dynamics generated by a one-dimensional, two-parameter cubic map are considered. The underlying deterministic system exhibits bistability and hysteresis, and noise-induced processes associated with these phenomena are studied. If a bounded noise source is applied to this system, trajectories may be confined to a finite region. Mechanisms are given for the merging transitions between different parts of this region and the eventual escape from it as the noise level is increased. The noisy dynamics is also represented by an integral evolution operator, with an equilibrium density function with finite support. The operator's spectrum is determined as a function of map parameters and noise amplitude. Such noisy one-dimensional maps can provide models for the study of noise-induced phenomena described by stochastic differential equations.     

Hénon映射的噪声诱导间歇现象及其临界值估算研究

本文研究了Hénon映射在噪声诱导下发生的间歇现象.通过数值模拟和全局分析手段,揭示了噪声诱导间歇现象的机理.基于随机敏感度函数法,通过检测噪声作用下周期吸引子的置信椭圆与混沌鞍的碰撞情况,给出了诱发间歇现象的噪声强度临界值的估算方法.结果表明,Hénon映射中噪声诱导间歇现象是由随机周期吸引子和混沌鞍不稳定流形的相互作用引发,随机敏感度函数的方法可以较好地估算发生间歇现象的噪声强度临界值.     

Required width of exit to avoid the faster-is-slower effect in highly competitive evacuation

A group of competitive people escaping through an exit could lead to the formation of a deadlock, which significantly increases the evacuation time. Such a phenomenon is called the faster-is-slower effect(FIS) and it has been experimentally verified in different systems of particles flowing through an opening. In this paper, the numerical simulation based on discrete element method(DEM) is adopted to study a group of highly competitive people through an exit of varying widths. The FIS effect is observed for a narrow exit whilst it is not observed for the exit wide enough to accommodate two people through it side-by-side. Experimental validation of such a phenomenon with humans is difficult due to ethical issues. The mouse is a kind of self-driven and soft-body creature and it exhibits selfish behaviour under stressed conditions.Particles flowing through an opening in different systems, such as pedestrian flow, animal flow, silo flow, etc. have similar characteristics. Therefore, experimental study is conducted by driving mice to escape through an exit of different widths at varying levels of stimulus. The escape time through a narrow exit(i.e., 2 cm) increases obviously with the increase of stimulus level but it is quite opposite to a wider exit(i.e., 4 cm). The FIS effect is avoided for an exit wide enough to accommodate two mice passing through it side-by-side. The study illustrates that FIF effect could be effectively prevented for an exit when its width is twice the size of particles.     

Escaping from nonhyperbolic chaotic attractors

The noise-induced escape process from a nonhyperbolic chaotic attractor is of physical and fundamental importance. We address this problem by uncovering the general mechanism of escape in the relevant low noise limit using the Hamiltonian theory of large fluctuations and by establishing the crucial role of the primary homoclinic tangency closest to the basin boundary in the dynamical process. In order to demonstrate that, we provide an unambiguous solution of the variational equations from the Hamiltonian theory. Our results are substantiated with the help of physical and dynamical paradigms, such as the Hénon and the Ikeda maps. It is further pointed out that our findings should be valid for driven flow systems and for experimental data.     

Negative mobility and multiple current reversals induced by colored thermal fluctuation in an asymmetric periodic potential

The phenomena of negative mobility (NM) and multiple current reversals (MCR) are investigated numerically in an asymmetric periodic potential with a Gaussian colored noise under the influence of a periodic driving and a constant bias. Two cases have been considered: the case of noise-induced normal transport and the case of noise-induced anomalous transport. The results indicate: (1) within tailored parameter regimes, a robust and wide range of NM can be obtained for a fixed regime of correlation time; (2) nonzero correlation time can induce and diminish MCR; (3) the asymmetry can induce and significantly facilitate the anomalous transport of inertial Brownian particle.     

Stabilization by dissipation and stochastic resonant activation in quantum metastable systems

In this tutorial paper we present a comprehensive review of the escape dynamics from quantum metastable states in dissipative systems and related noise-induced effects. We analyze the role of dissipation and driving in the escape process from quantum metastable states with and without an external driving force, starting from a nonequilibrium initial condition. We use the Caldeira–Leggett model and a non-perturbative theoretical technique within the Feynman–Vernon influence functional approach in strong dissipation regime. In the absence of driving, we find that the escape time from the metastable region has a nonmonotonic behavior versus the system-bath coupling and the temperature, producing a stabilizing effect in the quantum metastable system. In the presence of an external driving, the escape time from the metastable region has a nonmonotonic behavior as a function of the frequency of the driving, the thermal-bath coupling and the temperature. The quantum noise enhanced stability phenomenon is observed in both systems investigated. Finally, we analyze the resonantly activated escape from a quantum metastable state in the spin-boson model. We find quantum stochastic resonant activation, that is the presence of a minimum in the escape time as a function of the driving frequency. Background and introductory material has been added in the first three sections of the paper to make this tutorial review reasonably self-contained and readable for graduate students and non-specialists from related areas.     

Resonant activation in a stochastic Hodgkin-Huxley model: Interplay between noise and suprathreshold driving effects

The paper considers an excitable Hodgkin-Huxley system subjected to a strong periodic forcing in the presence of random noise. The influence of the forcing frequency on the response of the system is examined in the realm of suprathreshold amplitudes. Our results confirm that the presence of noise has a detrimental effect on the neuronal response. Fluctuations can induce significant delays in the detection of an external signal. We demonstrate, however, that this negative influence may be minimized by a resonant activation effect: Both the mean escape time and its standard deviation exhibit a minimum as functions of the forcing frequency. The destructive influence of noise on the interspike interval can also be reduced. With driving signals in a certain frequency range, the system can show stable periodic spiking even for relatively large noise intensities. Outside this frequency range, noise of similar intensity destroys the regularity of the spike trains by suppressing the generation of some of the spikes.     

Could one single dichotomous noise cause resonant activation for exit time over potential barrier？

This paper studies the mean first passage time （or exit time, or escape time） over the non-fluctuating potential harrier for a system driven only by a dichotomous noise. It finds that the dichotomous noise can make the particles escape over the potential barrier, in some circumstances; but in other circumstances, it can not. In the case that the particles escape over the potential harrier, a resonant activation phenomenon for the mean first passage time over the potential barrier is obtained.     

关联高斯与非高斯噪声激励的FHN神经元系统的稳态分析

本文主要研究了关联乘性非高斯噪声和加性高斯白噪声共同激励的FHN(Fitz Hugh-Nagumo)神经元系统.利用路径积分法和统一色噪声近似,推导出该系统的定态概率密度函数表达式.通过研究发现,乘性噪声强度D、加性噪声强度Q、噪声自关联时间τ以及互关联系数λ均可以诱导系统产生非平衡相变现象,而非高斯参数q却不可以诱导系统产生非平衡相变现象.此外,我们还发现参数D和λ的增大有利于神经元系统从激发态向静息态转换,Q和τ的增大有利于神经元系统从静息态向激发态转换,q的增大会使得神经元系统停留在静息态的概率增加.     

Noise-Induced Phase Transition: Zero-Dimensional Brownian Particles Varying between Ergodicity and Nonergodicity

We study in phase space a zero-dimensional system of Brownian particles which move in a periodic potential and subject to an internal time derivative Ornstein-Uhlenbeck noise. To resolve the Fokker-Planck equation in such a case, we propose an approximate analytical method. The theoretical predictions exhibit a second order noise-induced nonequilibrium phase transition, which is confirmed by numerical simulation results. The phase transition brings the system from an ergodicity to a nonergodicity phase as the potential barrier height decreases.     

Noise-induced phase synchronization and synchronization transitions in chaotic oscillators

Whether common noise can induce complete synchronization in chaotic systems has been a topic of great relevance and long-standing controversy. We first clarify the mechanism of this phenomenon and show that the existence of a significant contraction region, where nearby trajectories converge, plays a decisive role. Second, we demonstrate that, more generally, common noise can induce phase synchronization in nonidentical chaotic systems. Such a noise-induced synchronization and synchronization transitions are of special significance for understanding neuron encoding in neurobiology.     

Escape rate of Brownian particles from a metastable potential well under time derivative Ornstein-Uhlenbeck noise

We investigate the escape rate of Brownian particles that move in a cubic metastablepotential subjected to an internal time derivative Ornstein-Uhlenbeck noise (DOUN). Thisnoise can induce the ballistic diffusion of force-free Brownian particles. Some newfeatures are found. The escape rate for DOUN shows qualitative different dependence onpotential well width compared with OUN which induces normal diffusion. As the potentialbarrier height decreases, the escape rate of DOUN deviates from Arrhenius law considerablyearlier than that of Ornstein-Uhlenbeck noise (OUN). The Brownian particles escape fasterunder DOUN than that under OUN. A quasi-periodic oscillation occurs in transient state. Asolvable case is presented to demonstrate the significant cancellation behavior in thebarrier region that governs most of these phenomena. The physical mechanism of thefindings can be clarified by the noise features. These characteristics should be commonfor internal noises that induce superdiffusion, especially the ballistic diffusion.     

Additive noise in noise-induced nonequilibrium transitions

We study different nonlinear systems which possess noise-induced nonequlibrium transitions and shed light on the role of additive noise in these effects. We find that the influence of additive noise can be very nontrivial: it can induce first- and second-order phase transitions, can change properties of on-off intermittency, or stabilize oscillations. For the Swift-Hohenberg coupling, that is a paradigm in the study of pattern formation, we show that additive noise can cause the formation of ordered spatial patterns in distributed systems. We show also the effect of doubly stochastic resonance, which differs from stochastic resonance, because the influence of noise is twofold: multiplicative noise and coupling induce a bistability of a system, and additive noise changes a response of this noise-induced structure to the periodic driving. Despite the close similarity, we point out several important distinctions between conventional stochastic resonance and doubly stochastic resonance. Finally, we discuss open questions and possible experimental implementations. (c) 2001 American Institute of Physics.     

Pseudo-regular oscillations induced by external noise

An examination of the effect of noise on a general system at a saddle-node bifurcation has revealed that, in the limit of weak noise, the probability density of the time to pass through the saddle-node has a universal shape, the specific kinetics of the particular system serving only to set the time scale. This probability density is displayed and its salient features are explicated. In the case of a saddle-node bifurcation leading to relaxation oscillations, this analysis leads to the prediction of the existence of noise-induced oscillations which appear much less random than might at first be expected. The period of these oscillations has a well-defined, nonzero most probable value, the inverse of which is a noise-induced frequency. This frequency can be detected as a peak in power spectra from numerical simulations of such a system. This is the first case of the prediction and detection of a noise-induced frequency of which the authors are aware.     

Lifetime of the superconductive state in short and long Josephson junctions

We study the transient statistical properties of short and long Josephson junctions under the influence of thermal and correlated fluctuations. In particular, we investigate the lifetime of the superconductive metastable state finding the presence of noise induced phenomena. For short Josephson junctions we investigate the lifetime as a function both of the frequency of the current driving signal and the noise intensity and we find how these noise-induced effects are modified by the presence of a correlated noise source. For long Josephson junctions we integrate numerically the sine-Gordon equation calculating the lifetime as a function of the length of the junction both for inhomogeneous and homogeneous bias current distributions. We obtain a non-monotonic behavior of the lifetime as a function of the frequency of the current driving signal and the correlation time of the noise. Moreover we find two maxima in the non-monotonic behaviour of the mean escape time as a function of the correlated noise intensity.