A chaotic map with a flat segment can produce a noise-induced order

Matsumoto and Tsuda studied the effects of noise on chaos in a one-dimensional Belousov-Zhabotinsky (BZ) map and found noise-induced order, that is, an external noise destroys a chaotic behavior and produces some kind of order (periodicities). This phenomenon is very interesting in understanding the relation between chaos and natural phenomena. The present paper proposes a unimodal piecewise linear map which has a flat segment. It is shown numerically that the noise-induced order can be observed in this simple map in the same way as the BZ map. These numerical results clarify the mechanism of noise-induced order.     

Dynamics of oscillators with periodic dichotomous noise

The dynamics of bistable oscillators driven by periodic dichotomous noise is described. The stochastic differential equation governing the flow implies smooth trajectories between noise switching events. The dynamics of the two-branched map induced by this flow is a Markov process. Harmonic and quartic models of the bistable potential are studied in the overdamped limit. In the linear (harmonic) case the dynamics can be reduced to a stochastic one-dimensional map with two branches. The moments decay exponentially in this case, although the invariant measure may be multifractal. For strong damping, relaxation induces a cascade leading to a Cantor set and anomalous decay of the density in this case is modeled by a Markov chain. For the physically more realistic case of a quartic potential many additional features arise since the contraction factor is distance dependent. By tuning the barrier-height parameter in the quartic potential, noise-induced transition rates with the characteristics of intermittency are found.     

Noise-induced attractor annihilation in the delayed feedback logistic map

We study dynamics of the bistable logistic map with delayed feedback, under the influence of white Gaussian noise and periodic modulation applied to the variable. This system may serve as a model to describe population dynamics under finite resources in noisy environment with seasonal fluctuations. While a very small amount of noise has no effect on the global structure of the coexisting attractors in phase space, an intermediate noise totally eliminates one of the attractors. Slow periodic modulation enhances the attractor annihilation.     

Connection between noise-induced symmetry breaking and an information-decoding function for intracellular networks

The biological function of noise-induced symmetry breaking (NISB) is still unclear even though it may potentially occur in noisy intracellular systems. In this work, I demonstrate that information decoding from a noisy signal is a potential biological function of NISB by revealing that NISB naturally emerges from an optimal information-decoding dynamics and that several intracellular networks can be identified with the information-decoding dynamics. I also propose a mean first passage time profile as a way to experimentally identify NISB.     

Probability evolution method for exit location distribution

The exit problem in the framework of the large deviation theory has been a hot topic in the past few decades. The most probable escape path in the weak-noise limit has been clarified by the Freidlin–Wentzell action functional. However, noise in real physical systems cannot be arbitrarily small while noise with finite strength may induce nontrivial phenomena, such as noise-induced shift and noise-induced saddle-point avoidance. Traditional Monte Carlo simulation of noise-induced escape will take exponentially large time as noise approaches zero. The majority of the time is wasted on the uninteresting wandering around the attractors. In this paper, a new method is proposed to decrease the escape simulation time by an exponentially large factor by introducing a series of interfaces and by applying the reinjection on them. This method can be used to calculate the exit location distribution. It is verified by examining two classical examples and is compared with theoretical predictions. The results show that the method performs well for weak noise while may induce certain deviations for large noise. Finally, some possible ways to improve our method are discussed.     

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.     

Noise-induced stabilization of bumps in systems with long-range spatial coupling

The position of a localized region of active neurons (a “bump”) has been proposed to encode information for working memory, the head direction system, and feature selectivity in the visual system. Stationary bumps are ordinarily stable, but including spike frequency adaptation in the neural dynamics causes a stationary bump to become unstable to a moving bump through a supercritical pitchfork bifurcation in bump speed. Adding spatiotemporal noise to the network supporting the bump can cause the average speed of the bump to decrease to almost zero, reversing the effect of the adaptation and “restabilizing” the bump. This restabilizing occurs for noise levels lower than those required to break up the bump. The restabilizing can be understood by examining the effects of noise on the normal form of the pitchfork bifurcation where the variable involved in the bifurcation is bump speed. This noisy normal form can be further simplified to a persistent random walk in which the probability of changing direction is related to the noise level through an Arrhenius-type rate. The probability density function of position for the continuous-time version of this random walk satisfies the telegrapher’s equation, and the closed-form solution of this PDE allows us to find expressions for the mean and variance of the average speed of the particle (the bump) undergoing the random walk. This noise-induced stabilization is a novel example in which moderate amounts of noise have a beneficial effect on a system, specifically, stabilizing a spatiotemporal pattern.     

基于多重耦合映像格子的信号初值恢复研究

利用耦合映像格子恢复信号初值是信号处理研究中一个重要的问题.耦合映像格子具有混沌系统的初值敏感性，当初值受到噪声污染时将会影响到系统对其的恢复.提出了一种由多个一维耦合映像格子系统并列耦合而成的多重耦合映像格子系统，通过将多个一维系统耦合，使因受到噪声干扰而趋向于指数分离的混沌轨道相互靠近，以达到抑制噪声的目的.数值仿真表明，该系统具有较强的抗噪声能力和较高的鲁棒性.在耦合系数选取适当的情况下，即使初始信号受到噪声干扰，该多重耦合系统仍然能够很好地恢复信号初值的统计特性，且对单个初值的恢复情况及与初始信号 关键词： 耦合映像格子 恢复信号的统计特性 多重耦合     

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.     

Effect of noise on coupled chaotic systems

The effect of noise in inducing order on various chaotically evolving systems is reviewed, with special emphasis on systems consisting of coupled chaotic elements. In many situations it is observed that the uncoupled elements when driven by identical noise, show synchronization phenomena where chaotic trajectories exponentially converge towards a single noisy trajectory, independent of the initial conditions. In a random neural network, with infinite range coupling, chaos is suppressed due to noise and the system evolves towards a fixed point. Spatiotemporal stochastic resonance phenomenon has been observed in a square array of coupled threshold devices where a temporal characteristic of the system resonates at a given noise strength. In a chaotically evolving coupled map lattice with the logistic map as local dynamics and driven by identical noise at each site, we report that the number ofstructures (a structure is a group of neighbouring lattice sites for values of the variable follow which the certain predefined pattern) follows a power-law decay with the length of the structure. An interesting phenomenon, which we callstochastic coherence, is also reported in which the abundance and lifetimes of these structures show characteristic peaks at some intermediate noise strength.     

Lyapunov number for a noisy 2 n cycle

A stochastic one-dimensional map which produces a sequence of period doubling bifurcations is theoretically studied. We obtain analytic expressions, to a second-order approximation, of the local distribution function of fluctuating orbital points and the Lyapunov number for a noisy 2 ⁿ cycle. The expressions satisfy scaling laws and well agree with the results of numerical experiments when the external noise is weak. A scaling factor for the noise level is formulated in terms of the derivatives of a deterministic map. From it, the scaling factor is refined to be 6.6190 .... The Lyapunov number shows that, when the external noise is weaker than some extent, the noisy orbit is more stable rather than the deterministic one.     

Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations

There have been many contributions concerned with non-smooth dynamics. The purpose of this study is focused on the global stochastic dynamics of a kind of vibro-impact oscillator under the multiple harmonic and bounded noisy excitations. The well-known cell-to-cell mapping method is firstly developed to investigate the incursive fractal boundaries between the attracting domains of different random attractors, and a specific Poincaré map is then set up to explore the noise-contaminated dynamical transitions in the system. Lastly, the leading Lyapunov exponents and the surrogate tests are used to identify the noise-contaminated dynamics. It is shown that several random attractors will coexist in the phase space of the randomly driven system by adjusting the parameters’ values, and fractal boundaries may also arise between the attracting domains of different random attractors. Under the joint action of the harmonic excitation and the weak bounded noise excitation, the noisy period-doubling process, similar to a deterministic one, can appear in the Poincaré’s global cross-section by increasing the strength of the bounded noisy excitation. Moreover, the noisy periodic, the noisy chaotic, and the random-dominant dynamics are also distinguished from the noise-contaminated signals.     

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 reentrant transition of the stochastic Duffing oscillator

We derive the exact bifurcation diagram of the Duffing oscillator with parametric noise thanks to the analytical study of the associated Lyapunov exponent. When the fixed point is unstable for the underlying deterministic dynamics, we show that the system undergoes a noise-induced reentrant transition in a given range of parameters. The fixed point is stabilised when the amplitude of the noise belongs to a well-defined interval. Noisy oscillations are found outside that range, i.e., for both weaker and stronger noise.Received: 20 February 2004, Published online: 20 April 2004PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) - 05.45.-a Nonlinear dynamics and nonlinear dynamical systems     

Noise induced state transitions, intermittency, and universality in the noisy Kuramoto-Sivashinksy equation

Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.     

Firing rates of coupled noisy excitable elements

The dynamics of coupled excitable FitzHugh Nagumo systems under external noisy driving is studied. Different from most of previous work focusing on the noise-induced regularity in the framework of coherence resonance, here the average frequency （or firing rate） of coupled excitable elements is of much more concern. We find that （i） their frequencies first increase and then decrease with the increase of the coupling, and there is a clear crossover from a rush increase to a smooth increase with the increase of noise strength, and （ii） for nonidentical cases, all elements transit to an identical frequency simultaneously only after a certain coupling strength is achieved. These first-increase-thendecrease non-monotonic frequency behavior and isochronous frequency synchronization are believed to be two basic behaviors in coupled noisy excitable systems.     

Understanding synchronization induced by “common noise”

Noise-induced synchronization refers to the phenomenon where two uncoupled, independent nonlinear oscillators can achieve synchronization through a “common” noisy forcing. Here, “common” means identical. However, “common noise” is a construct which does not exist in practice. Noise by nature is unique and two noise signals cannot be exactly the same. How to justify and understand this central concept in noise-induced synchronization? What is the relation between noise-induced synchronization and the usual chaotic synchronization? Here we argue and demonstrate that noise-induced synchronization is closely related to generalized synchronization as characterized by the emergence of a functional relation between distinct dynamical systems through mutual interaction. We show that the same mechanism applies to the phenomenon of noise-induced (or chaos-induced) phase synchronization.     

格子复杂性和符号序列的细粒化

提出一种新的有限长一维符号序列的复杂性度量——格子复杂性，建立在Lempel Ziv复杂性和一维迭代映射系统的符号动力学基础上.同时提出了符号序列的细粒化方法，可与格子复杂性以及Lempel Ziv复杂性结合.新度量在细粒化指数较小时与Lempel Ziv复杂性基本一致，在细粒化指数增大时显示出截然不同的特性.以Logistic映射为对象的计算实验表明，格子复杂性对混沌区的边缘最敏感.最后还讨论了上述复杂性度量的其他一些重要性质. 关键词： 混沌 复杂性度量 格子复杂性 细粒化     

Pseudo-periodic surrogate test to sample time series in stochastic softening Duffing oscillator

Identification of typical noise-contaminated sample response is a hard task in a nonlinear system under stochastic background since irregularity of the sample response may come from measure noise, dynamical noise, or nonlinear effect, etc., and conventional dynamical methods are generally not useful. Here, the pseudo-periodic surrogate algorithm by Small is employed to test the sample time series in the softening Duffing oscillator under the Gaussian white noise excitation. The correlation dimensions of the noisy periodic and the noise-induced chaotic time series of the system are compared with those of their corresponding surrogate data respectively, the leading Lyapunov exponents by Rosenstein's algorithm are also presented for comparison.