Modification of instability processes by multiplicative noises

We study two dynamical systems submitted to white and Gaussian random noise acting multiplicatively. The first system is an imperfect pitchfork bifurcation with a noisy departure from onset. The second system is a pitchfork bifurcation in which the noise acts multiplicatively on the non-linear term of lowest order. In both cases noise suppresses some solutions that exist in the deterministic regime. Besides, for the first system, the imperfectness of the bifurcation reduces the regime of on-off intermittency. For the second system, the unstable mode can achieve a jump of finite amplitude at instability but without hysteresis. We finally identify a generic property that is verified by the stationary probability density function of the dynamical variable when a control parameter is varied.     

New patterns of activity in a pair of interacting excitatory-inhibitory neural fields

In this Letter, we study stationary bump solutions in a pair of interacting excitatory-inhibitory (E-I) neural fields in one dimension. We demonstrate the existence of localized bump solutions of persistent activity that can be maintained by the pair of interacting layers when a stationary bump is not supported by either layer in isolation--a scenario which may be relevant as a mechanism for the persistent activity associated with working memory in the prefrontal cortex and may explain why bumps are not seen in in vitro slice preparations. Furthermore, we describe a new type of stationary bump solution arising from a pitchfork bifurcation which produces a stationary bump in each layer with a spatial offset that increases with the bifurcation parameter.     

Escape statistics for systems driven by dichotomous noise. II. The imperfect pitchfork bifurcation as a case study

We use the general results for the escape probabilities and mean exit times obtained in an accompanying paper to analyze in detail a nonlinear system presenting an imperfect (subcritical) pitchfork bifurcation. We redraw the bifurcation diagram to show the effect of the noise. To avoid spurious results we introduce the concept ofextinction level as the minimum possible value for the system, and discuss its effect on the bifurcation diagram.     

On the low-frequency limit of the Schönfeld pulse 1/f law

On the basis of propositions of the common fluctuation theory, peculiarities of small fluctuations in real physical systems with limited sizes are analyzed. It is established that small fluctuations should necessarily be divided into two types of fluctuations: “small” and “very small”. It is shown that the damping process of “small” fluctuations has relaxation character, while the damping process of “very small” fluctuations is of random character, i.e., it represents a random rectangular signal. The probability density of “very small” fluctuations is shown to be Gaussian. The agreement of the obtained results with experimental data acquired from semiconductor-based devices is analyzed. A relation between the generation–recombination noise and phonon number fluctuations in semiconductors is studied. On the basis of this consideration it is shown that the Schönfeld pulse spectrum preserves its well-known 1/f form only in the range of intermediate frequencies; at lower frequencies the spectrum gets saturated. An expression for the low-frequency limit of Schönfeld pulse 1/f law is obtained.     

Coloured-noise-induced transitions: Exact results for external dichotomous Markovian noise

A general method to calculate explicitly the stationary probability of nonlinear systems subjected to a special case of coloured noise is presented. For a simple model system the “phase diagram” for the various noise-induced transitions is determined.     

Comment on asymptotic properties of coupled Langevin equations

The asymptotic behavior of coupled Langevin equations in the limit of weak noise is studied by general normal form techniques, in the vicinity of a pitchfork bifurcation. The non-Gaussian behavior of the critical variable is established. The conditional probability of the noncritical variable around the center manifold is determined. It is shown that in certain cases the distribution of this later variable may be non-Gaussian.On leave of absence from Facultad de Ciencias Fisicas Y Mathemàticas, Universidad de Chile, Santiago, Chile     

The red queen's walk

We study a new type of walk with memory which might serve as a toy model for the behavior one must adopt to avoid exhaustion of resources and attraction of parasites and predators. The walk takes place on a regular square lattice with periodic boundary conditions. Although the walk is completely deterministic, it mimics a “true” self-avoiding walk, i.e. a random walk with weak autocorrelation. This shows that the Red Queen effect can lead to aperiodic behavior. In addition to the case of single walkers in a flat landscape we also study the cases of hilly landscapes and of several walkers performing simultaneous walks.     

Stochastic bifurcation for a tumor–immune system with symmetric Lévy noise

In this paper, we investigate stochastic bifurcation for a tumor–immune system in the presence of a symmetric non-Gaussian Lévy noise. Stationary probability density functions will be numerically obtained to define stochastic bifurcation via the criteria of its qualitative change, and bifurcation diagram at parameter plane is presented to illustrate the bifurcation analysis versus noise intensity and stability index. The effects of both noise intensity and stability index on the average tumor population are also analyzed by simulation calculation. We find that stochastic dynamics induced by Gaussian and non-Gaussian Lévy noises are quite different.     

A modified pitchfork bifurcation as a model of the TI–TII transition in liquid helium counterflow with external noise

An amended pitchfork bifurcation is introduced to model recent experiments by Griswold and Tough on superfluid turbulence in liquid helium counterflow subject to strong external noise. We adopt the generalized white noise limit of Blankenship and Papanicolaou to take a short-correlation-time limit of the nonlinear noise which enters into the model, and we implement this limit by means of the wideband perturbation expansion. Novel boundary conditions are applied to the resultant diffusion process in order to obtain behavior in qualitative agreement with the observations at low vortex line density. We are able to account for the sharp peak in probability observed experimentally at a small positive line density. The drift and diffusion of our diffusion process may be estimated experimentally; we describe how to do this.     

Deterministically generated noise amplification in the period-doubling bifurcations of an additive-pulse mode-locked laser

We examine the noise behavior of an additive-pulse mode-locked (APM) laser as it undergoes a period-one/period-two pitchfork bifurcation. We find evidence of noise amplification, “wiggly” bifurcation phenomena, and frequency shifting of the signal within the noise background. The total amount of frequency tuning is about 5.5% of the fundamental frequency.     

Spatial instabilities in reaction random walks with direction-independent kinetics

We study spatial instabilities in reacting and diffusing systems, where diffusion is modeled by a persistent random walk instead of the usual Brownian motion. Perturbations in these reaction walk systems propagate with finite speed, whereas in reaction-diffusion systems localized disturbances affect every part instantly, albeit with heavy damping. We present evolution equations for reaction random walks whose kinetics do not depend on the particles' direction of motion. The homogeneous steady state of such systems can undergo two types of transport-driven instabilities. One type of bifurcation gives rise to stationary spatial patterns and corresponds to the Turing instability in reaction-diffusion systems. The other type occurs in the ballistic regime and leads to oscillatory spatial patterns; it has no analog in reaction-diffusion systems. The conditions for these bifurcations are derived and applied to two model systems. We also analyze the stability properties of one-variable systems and find that small wavelength perturbations decay in an oscillatory manner.     

非线性动力系统分岔点邻域内随机共振的特性

研究了叉形分岔系统和FitzHugh Nagumo(FHN)细胞模型两种非线性动力系统分岔点邻域内 随机共振的特性.研究结果表明：这两种系统在分岔发生时具有由一个吸引子变为两个吸引 子或者由两个吸引子变为一个吸引子共同的分岔特性，即在分岔点的邻域内, 系统在分岔点 的两侧有分岔前吸引子和分岔后吸引子存在，在噪声的作用下，系统的运动除了像传统随机 共振的机理那样在分岔点一侧共存的吸引子之间跃迁，还在分岔点两侧三个吸引子（分岔前 一个吸引子和分岔后两个吸引子）之间跃迁，并且这种跃迁单独诱发了随机共振 ；在两种 跃迁都发生的情况下, 在其分岔点的邻域内，由第二种跃迁诱发的随机共振在引起第一种跃 迁噪声的强度很大的范围内变化仍可维持, 而第一种跃迁诱发的随机共振在引起第二种跃迁 噪声的强度很小的范围内变化即迅速消失. 关键词： 随机共振 吸引子 分岔点 跃迁     

一类五次方振子系统的叉形分叉及振动共振研究

研究了一类具有分数阶导数阻尼的五次方振子系统中的叉形分叉及振动共振现象. 基于快慢变量分离法, 消去系统中的高频激励成分, 得到关于慢变量的等效系统, 根据等效系统中稳态平衡点的变化情况研究了系统的叉形分叉现象. 结果表明: 高频信号幅值的递增变化会引起亚临界叉形分叉, 高频信号频率和分数阶导数阻尼阶数的递增变化都会引起超临界叉形分叉; 振动共振和叉形分叉是关联的, 当叉形分叉发生时, 振动共振曲线会出现两个峰值, 否则只会出现一个峰值. 通过解析结果和数值模拟结果的对比, 验证了解析分析的正确性. 关键词： 亚临界叉形分叉 超临界叉形分叉 分数阶导数阻尼 振动共振     

Classical random walk with memory versus quantum walk on a one-dimensional infinite chain

Quantum walk is a very useful tool for building quantum algorithms due to the faster spreading of probability distributions as compared to a classical random walk. Comparing the spreading of the probability distributions of a quantum walk with that of a mnemonic classical random walk on a one-dimensional infinite chain, we find that the classical random walk could have a faster spreading than that of the quantum walk conditioned on a finite number of walking steps. Quantum walk surpasses classical random walk with memory in spreading speed when the number of steps is large enough. However, in such a situation, quantum walk would seriously suffer from decoherence. Therefore, classical walk with memory may have some advantages in practical applications.     

Noise Filtering Strategies in Adaptive Biochemical Signaling Networks

Two distinct mechanisms for filtering noise in an input signal are identified in a class of adaptive sensory networks. We find that the high-frequency noise is filtered by the output degradation process through time-averaging; while the low-frequency noise is damped by adaptation through negative feedback. Both filtering processes themselves introduce intrinsic noises, which are found to be unfiltered and can thus amount to a significant internal noise floor even without signaling. These results are applied to E. coli chemotaxis. We show unambiguously that the molecular mechanism for the Berg-Purcell time-averaging scheme is the dephosphorylation of the response regulator CheY-P, not the receptor adaptation process as previously suggested. The high-frequency noise due to the stochastic ligand binding-unbinding events and the random ligand molecule diffusion is averaged by the CheY-P dephosphorylation process to a negligible level in E. coli. We identify a previously unstudied noise source caused by the random motion of the cell in a ligand gradient. We show that this random walk induced signal noise has a divergent low-frequency component, which is only rendered finite by the receptor adaptation process. For gradients within the E. coli sensing range, this dominant external noise can be comparable to the significant intrinsic noise in the system. The dependence of the response and its fluctuations on the key time scales of the system are studied systematically. We show that the chemotaxis pathway may have evolved to optimize gradient sensing, strong response, and noise control in different time scales.     

Additive global noise delays Turing bifurcations

We apply a stochastic center manifold method to the calculation of noise-induced phase transitions in the stochastic Swift-Hohenberg equation. This analysis is applied to the reduced mode equations that result from Fourier decomposition of the field variable and of the temporal noise. The method shows a pitchfork bifurcation at lower perturbation order, but reveals a novel additive-noise-induced postponement of the Turing bifurcation at higher order. Good agreement is found between the theory and the numerics for both the reduced and the full system. The results are generalizable to a broad class of nonlinear spatial systems.     

Local bifurcation analysis of a four-dimensional hyperchaotic system

Local bifurcation phenomena in a four-dlmensional continuous hyperchaotic system, which has rich and complex dynamical behaviours, are analysed. The local bifurcations of the system are investigated by utilizing the bifurcation theory and the centre manifold theorem, and thus the conditions of the existence of pitchfork bifurcation and Hopf bifurcation are derived in detail. Numerical simulations are presented to verify the theoretical analysis, and they show some interesting dynamics, including stable periodic orbits emerging from the new fixed points generated by pitchfork bifurcation, coexistence of a stable limit cycle and a chaotic attractor, as well as chaos within quite a wide parameter region.     

The exact probability distribution of a two-dimensional random walk

A calculation is made of the exact probability distribution of the two-dimensional displacement of a particle at timet that starts at the origin, moves in straight-line paths at constant speed, and changes its direction after exponentially distributed time intervals, where the lengths of the straight-line paths and the turn angles are independent, the angles being uniformly distributed. This random walk is the simplest model for the locomotion of microorganisms on surfaces. Its weak convergence to a Wiener process is also shown.     

Quantum walk under coherence non-generating channels

We investigate the probability distribution of the quantum walk under coherence non-generating channels. We definea model called generalized classical walk with memory. Under certain conditions, generalized classical random walk withmemory can degrade into classical random walk and classical random walk with memory. Based on its various spreadingspeed, the model may be a useful tool for building algorithms. Furthermore, the model may be useful for measuring thequantumness of quantum walk. The probability distributions of quantum walks are generalized classical random walkswith memory under a class of coherence non-generating channels. Therefore, we can simulate classical random walkand classical random walk with memory by coherence non-generating channels. Also, we find that for another class ofcoherence non-generating channels, the probability distributions are influenced by the coherence in the initial state of thecoin. Nevertheless, the influence degrades as the number of steps increases. Our results could be helpful to explore therelationship between coherence and quantum walk.     

Diffusion with restrictions

A nonlinear diffusion equation is derived by taking into account hopping rates depending on the occupation of next neighbouring sites. There appears additional repulsive and attractive forces leading to a changed local mobility. The stationary and the time dependent behaviour of the system are studied based upon the master equation approach. Different to conventional diffusion it results in a time dependent bump the position of which increases with time described by an anomalous diffusion exponent. The fractal dimension of this random walk is exclusively determined by the space dimension. The applicability of the model to describe glasses is discussed.