General mechanism for the meandering instability of rivulets of Newtonian fluids

A rivulet flowing down an inclined plane often does not follow a straight path, but starts to meander spontaneously. Here we show that this instability is the result of two key ingredients: fluid inertia and anisotropy of the friction between rivulet and substrate. Meandering only occurs if the motion normal to the instantaneous flow direction is more difficult than parallel to it. We give a quantitative criterion for the onset of meandering and confirm it by comparing to the flow of a rivulet between two glass plates which are wetted completely. Above the threshold, the rivulet follows an irregular pattern with a typical wavelength of a few cm.     

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.     

Motion of spiral tip driven by local forcing in excitable media

We study the motion of a spiral wave controlled by a local periodic forcing imposed on a region around the spiral tip in an excitable medium. Three types of trajectories of spiral tip are observed： the epicycloid-like meandering, the resonant drift, and the hypocycloid-like meandering. The frequency of the spiral is sensitive to the local periodic forcing. The dependency of spiral frequency on the amplitude and size of local periodic forcing are presented. In addition, we show how the drift speed and direction are adjusted by the amplitude and phase of local periodic forcing, which is consistent with a theoretical analysis based on the weak deformation approximation.     

Effect of phase noise on parametric instabilities

We report an experimental study on the effect of an external phase noise on the parametric amplification of surface waves. We observe that both the instability growth rate and the wave amplitude above the instability onset are decreased in the presence of noise. We show that all the results can be understood with a deterministic amplitude equation for the wave in which the effect of noise is just to change the forcing term. All the data for the growth rate (respectively the wave amplitude), obtained for different forcing amplitudes and different intensities of the noise, can be collapsed on a single curve using this renormalized forcing in the presence of noise.     

Synchronized Firing in Coupled Inhomogeneous Excitable Neurons

We study the firing synchronization behavior of the inhomogeneous excitable media. Phase synchronization of neuron firings is observed with increasing the coupling, while the phases of neurons are different (out-of-phase synchronization). We found the synchronization of bursts can be greatly enhanced by applying an external forcing (in-phase synchronization). The external forcing can be either a periodic or just homogeneous thermal noise. The mechanism responsible for this enhancement is discussed.     

Synchronized Firing in Coupled Inhomogeneous Excitable Neurons

We study the firing synchronization behavior of the inhomogeneous excitable media. Phase synchronizationof neuron firings is observed with increasing the coupling, while the phases of neurons are different (out-of-phase synchronization). We found the synchronization of bursts can be greatly enhanced by applying an external forcing (in-phasesynchronization). The external forcing can be either a periodic or just homogeneous thermal noise. The mechanismresponsible for this enhancement is discussed.PACS numbers: 05.45.-a, 87.10. e     

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.     

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.     

Control of scroll wave turbulence in a three-dimensional reaction-diffusion system with gradient

In this paper, we summarize our recent experimental and theoretical works on observation and control of scroll wave (SW) turbulence. The experiments were conducted in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system with chemical concentration gradients in one dimension. A spatially homogeneous external forcing was used in the experiments as a control; it was realized by illuminating white light on the light sensitive reaction medium. We observed that, in the oscillatory regime of the system, SW can appear automatically in the gradient system, which will be led to spatiotemporal chaos under certain conditions. A suitable periodic forcing may stabilize inherent turbulence of SW. The mechanism of the transition to SW turbulence is due to the phase twist of SW in the presence of chemical gradients, while modulating the phase twist with a proper periodic forcing can delay this transition. Using the FitzHugh-Nagumo model with an external periodic forcing, we confirmed the control mechanism with numerical simulation. Moreover, we also show in the simulation that adding temporal external noise to the system may have the same control effect. During this process, we observed a new state called "intermittent turbulence," which may undergo a transition into a new type of SW collapse when the noise intensity is further increased. The intermittent state and the collapse could be explained by a random process.     

随机相位扰动抑制激发介质中漂移的螺旋波

研究了一类二维变量描述的激发系统中漂移螺旋波的抑制问题.通过在整个系统中局部注入带随机相位的电信号，如在系统256×256格点的边界或中心区域中选取4×4或者5×5格点区域施加一个带随机相位的外部激励电信号，在系统内部产生一个持续的靶波信号，实现靶波对螺旋波的动态竞争.数值计算表明：该方法对于Barkley模型中螺旋波有很强的抑制作用，与简单的局部周期信号驱动比较，具有暂态过程比较短的特点，而且对于时空噪声具有一定的抗干扰性.在一定的噪声范围内，即使系统出现不均匀性，也可以观测到靶波，新出现的靶波对螺旋波有抑制作用. 关键词： 螺旋波 靶波 Barkley模型 随机相位     

线性过阻尼分数阶Langevin方程的共振行为

通过将广义Langevin方程中的系统内噪声建模为分数阶高斯噪声,推导出分数阶Langevin方程, 其分数阶导数项阶数由系统内噪声的Hurst指数所确定.讨论了处于强噪声环境下的线性过阻尼分数阶 Langevin方程在周期信号激励下的共振行为,利用Shapiro-Loginov公式和Laplace变换, 推导了系统响应的一、二阶稳态矩和稳态响应振幅、方差的解析表达式.分析表明,适当参数下, 系统稳态响应振幅和方差随噪声的某些特征参数、周期激励信号的频率及系统部分参数的变化出现了 广义的随机共振现象.     

Proton dynamics in hydrogen-bonded systems

We discuss a simplified version of an ice lattice which consists of an alternating sequence of heavy and light masses. The light masses (protons) are each subject to a bistable potential caused by the heavy masses (oxygens). The protons interact with one another, as do the heavy ions. The interactions between the protons and the oxygens modulate the bistable proton potential. This system is known to exhibit kink and antikink solutions associated with mobile ionic defects accompanied by a lattice distortion. We show that at finite temperatures and in the presence of a constant external field on the protons, the defect velocity is a nonmonotonic function of the temperature, reflecting an interesting interplay of thermal effects (noise) and the constant deterministic external forcing in this nonlinear system. We discuss extensions of the model to higher dimensions, and present preliminary results for the proton motion in such networks.     

Brownian motors and stochastic resonance

We study the transport properties for a walker on a ratchet potential. The walker consists of two particles coupled by a bistable potential that allow the interchange of the order of the particles while moving through a one-dimensional asymmetric periodic ratchet potential. We consider the stochastic dynamics of the walker on a ratchet with an external periodic forcing, in the overdamped case. The coupling of the two particles corresponds to a single effective particle, describing the internal degree of freedom, in a bistable potential. This double-well potential is subjected to both a periodic forcing and noise and therefore is able to provide a realization of the phenomenon of stochastic resonance. The main result is that there is an optimal amount of noise where the amplitude of the periodic response of the system is maximum, a signal of stochastic resonance, and that precisely for this optimal noise, the average velocity of the walker is maximal, implying a strong link between stochastic resonance and the ratchet effect.     

Stochastic Perturbations to Dynamical Systems: A Response Theory Approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.     

Response of axisymmetric separated flow to its spatially localized perturbation

The flow past an axisymmetric body with laminar boundary-layer separation in a low-velocity air stream has been studied. The hot-wire technique was employed to identify the variation of velocity field induced by a local stationary perturbation of separation region at the stern of the experimental model. A large-scale influence upon the near-wall flow due to a cylinder roughness element provided on the model surface was observed. The obtained data substantiate the possibility of controlling the laminar boundary-layer separation on an axisymmetric body using a local external forcing.     

Coherence and stochastic resonance in threshold crossing detectors with delayed feedback

We consider the dynamical behavior of threshold systems driven by external periodic and stochastic signals and internal delayed feedback. Specifically, the effect of positive delayed feedback on the sensitivity of a threshold crossing detector (TCD) to periodic forcing embedded in noise is investigated. The system has an intrinsic ability to oscillate in the presence of positive feedback. We first show conditions under which such reverberatory behavior is enhanced by noise, which is a form of coherence resonance (CR) for this system. Further, for input signals that are subthreshold in the absence of feedback, the open-loop stochastic resonance (SR) characteristic can be sharply enhanced by positive delayed feedback. This enhancement is shown to depend on the stimulus period, and is maximal when this period is matched to an integer multiple of the delay. Reverberatory oscillations, which are particularly prominent after the offset of periodic forcing, are shown to be eliminated by a summing network of such TCDs with local delayed feedback. Theoretical analysis of the crossing rate dynamics qualitatively accounts for the existence of CR and the resonant behavior of the SR effect as a function of delay and forcing frequency.     

Chaotic dynamics of a bouncing ball

A detailed study of a mapping on a two-dimensional manifold is made. The mapping describes a system subject to periodic forcing, in particular an imperfectly elastic ball bouncing on a vibrating platform. Quasiperiodic motion on a one-dimensional manifold is proven, and observed numerically, at low forcing, while at higher forcing Smale horseshoes are present. We examine the evolution of the attracting set with changing parameter. Spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises. A new type of chaos, in which a trajectory alternates between two distinct chaotic regions, is described and explained in terms of manifold collisions. Throughout we are concerned to examine the behaviour of Lyapunov exponents. Typical behaviour of Lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed. At higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given.     

Rivulet flow on the outer surface of an inclined cylinder

A study of the three-dimensional flow of a liquid film (rivulet) over the external part of an inclined cylinder was conducted for liquids with various physical properties. Patterns of the flow regimes were constructed. Good agreement is observed between the experimental data on the thickness and wall friction with the calculation with an asymptotic model in the case of a waveless rivulet. A comparison of the evolution of natural waves on rivulets with the theory of waves of maximal growth shows good agreement for small Re numbers. During the experiments, the wave characteristics of excited waves on a rivulet were investigated. The thickness, amplitude, frequency, and phase velocity of the waves over a wide range of variable parameters are given. Phase velocity integrated functions of the amplitude are constructed for various liquids. The friction on the cylinder wall is measured in the presence of natural and excited waves. The effects of wave regimes on the average values and RMS (root-mean-square) friction pulsations are studied.     

Role of the driving frequency in a randomly perturbed Hodgkin-Huxley neuron with suprathreshold forcing

The present paper examines the influence of the forcing frequency on the response of a randomly perturbed Hodgkin-Huxley system in the realm of suprathreshold amplitudes. Our results show that, in the presence of noise, the choice of driving frequency can seriously affect the precision of the external information transmission. At the same level of noise the precision can either decrease or increase depending on the driving frequency. We demonstrate that the destructive influence of noise on the interspike interval can be effectively reduced. That is, with driving signals in certain frequency ranges, the system can show stable periodic spiking even for relatively large noise intensities. Here, the most accurate transmission of an external signal occurs. Outside these frequency ranges, noise of the same intensity destroys the regularity of the spike trains by suppressing the generation of some spikes. On the other hand, we show that noise can have a reconstructive role for certain driving frequencies. Here, increasing noise intensity enhances the coherence of the neuronal response.     

Collective effects induced by diversity in extended systems

We show that diversity, in the form of quenched noise, can have a constructive effect in the dynamics of extended systems. We first consider a bistable φ⁴ model composed by many coupled units and show that the global response to an external periodic forcing is enhanced under the presence of the right amount of diversity (measured as the dispersion in one of the parameters defining the model). As a second example, we consider a system of active-rotators and show that while they are at rest in the homogeneous case, the disorder introduced by the diversity suffices to trigger the appearance of common firings or pulses. Both effects require very simple ingredients and we expect the results presented here to be of interest in similar models.