随机扰动下时滞复杂动力网络的一致性

研究了随机扰动下一般时滞复杂动力网络的一致性问题,此复杂动力网络不仅具有随机扰动而且时变时滞同时出现在耦合项和节点系统中,所以这样的网络更具有一般性.基于随机Lyapunov稳定性理论、线性反馈控制理论和线性矩阵不等式,从理论上提出了此网络各个节点与孤立系统达到时滞无关和时滞相关一致性的充分条件.最后的数值模拟验证了理论结果的正确性和有效性.     

Linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations

Abstract

Necessary and sufficient conditions for the linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations are given. Stochastic integrating factor is introduced to solve the linear jump- diffusion stochastic differential equations. Closed form solutions to certain linearizable jump-diffusion stochastic differential equations are obtained.     

A stochastic model of tweet diffusion on the Twitter network

We introduce a stochastic model which describes the information diffusion on the network of a popular web service, Twitter. Restricting our attention to the daily diffusion from a popular account, we model the information diffusion as a random multiplicative process. We justify our model by directly observing the statistics of the multiplicative factors in the Twitter data.     

Diffusion in stochastic sandpiles

We study diffusion of particles in large-scale simulations of one-dimensional stochastic sandpiles, in both the restricted and unrestricted versions. The results indicate that the diffusion constant scales in the same manner as the activity density, so that it represents an alternative definition of an order parameter. The critical behavior of the unrestricted sandpile is very similar to that of its restricted counterpart, including the fact that a data collapse of the order parameter as a function of the particle density is possible, but with a narrow scaling region. We also develop a series expansion, in inverse powers of the density, for the collective diffusion coefficient in a variant of the stochastic sandpile in which the toppling rate at a site with n particles is n(n-1), and compare the theoretical prediction with simulation results.     

Optimal weights decoding of M-ary suprathreshold stochastic resonance in stochastic pooling network

Suprathreshold stochastic resonance (SSR) is a noise enhancing signal processing phenomenon, occurring in a parallel array of nonlinear elements. In this paper, we investigate the optimal decoding scheme of SSR in stochastic pooling network with a quantization evolution of the output signal-to-quantization-noise ratio (SQNR) by studying the optimal weights and the optimal thresholds. Firstly, we introduce an effective method of weights decoding which makes the better SSR effects than the Wiener linear decoding and is defined as the optimal weights decoding. Moreover, in order to find the optimal thresholds, we select three common forms of thresholds in [0,1] interval the uniform thresholds, the random thresholds and the group thresholds. The result indicates that the group thresholds make a better SSR effect than uniform thresholds, but worse than the random thresholds. Therefore, the random thresholds are the optimal thresholds setting in the M-ary stochastic pooling network. Finally, we discuss the influences of number of elements N and threshold level M on SSR, and find that changing the number of the comparators N in stochastic pooling network is more easier to enhance the performance of SSR than changing the values of M. These works as a complement to the optimal quantification theory will be helpful for the study of optimal thresholds in stochastic pooling network.     

The stochastic Burgers Equation

We study Burgers Equation perturbed by a white noise in space and time. We prove the existence of solutions by showing that the Cole-Hopf transformation is meaningful also in the stochastic case. The problem is thus reduced to the anaylsis of a linear equation with multiplicativehalf white noise. An explicit solution of the latter is constructed through a generalized Feynman-Kac formula. Typical properties of the trajectories are then discussed. A technical result, concerning the regularizing effect of the convolution with the heat kernel, is proved for stochastic integrals.     

Approximate solution methods for linear stochastic difference equations: I. Moments

The cumulant expansion for linear stochastic differential equations is extended to the case of linear stochastic difference equations. We consider a vector difference equation, which contains a deterministic matrix A₀ and a random perturbation matrix A₁(t). The expansion proceeds in powers of ατ_c, where τ_c is the correlation time of the fluctuations in A₁(t) and α a measure for their strength. Compared to the differential case, additional cumulants occur in the expansion. Moreover one has to distinguish between a nonsingular and a singular A₀. We also discuss a limiting situation in which the stochastic difference equation can be replaced by a stochastic differential equation. The derivation is not restricted to the case where in the limit the stochastic parameters in the difference equation are replaced by white noise.     

过阻尼分数阶Langevin方程及其随机共振

通过对广义Langevin方程阻尼核函数的适当选取,在过阻尼的情形下, 推导出分数阶Langevin方程.给合反常扩散理论和分数阶导数的记忆性, 讨论了分数阶Langevin方程的物理意义,进而得出分数阶Langevin方程产生随机共振的内在机理.数值模拟表明,在一定的阶数范围内,分数阶Langevin方程可以产生随机共振, 并且分数阶下的信噪比增益好于整数阶情形.     

On a transformed stochastic equation

A linear stochastic equation is considered. As a result of the transformation used in the theory of integral equations for improving the convergence of successive approximations, transformed stochastic equations are obtained. The latter are exact and are equivalent to the original equation. By solving the transformed stochastic equations by the method of small perturbations the conditions are derived for the applicability of the approximate Keller equations for a value of the field averaged over the ensemble, which satisfies the original stochastic equation. As an application, the applicability boundaries of the Dyson equations are estimated in the Foldy and Burre approximation. In the first case it is assumed that the medium consists of Rayleigh scatters, while in the second case it is assumed that the fluctuations of the permeability of the medium are small-scale ones. If the medium is bounded and has the form of a sphere, the applicability condition of the Dyson equations impose an upper constraint on the radius of the sphere which nevertheless may take values that exceed the extinction length.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 1, pp. 66–72, January, 1972.     

Quantum stochastic processes III

We construct an example of a quantum stochastic process with a non-zero, linear, time-independent source, for a massive scalar Boson field in four space-time dimensions. Also we study in detail a similar process with only a single degree of freedom.Work supported by A.F.O.S.R. contract no. F44620-67-C-0029     

Finite temperature field from stochastic mechanics

Stochastic mechanics of Nelson when generalized to positive temperature for a scalar field, gives rise to a stochastic field which appears to be a hybrid of euclidean and minkowskian field if the usual value of the diffusion parameter is taken. The stochastic process associated with it is a gaussian non-Markov process. The thermal expectations of this stochastic field fails to satisfy the KMS periodic condition. If the diffusion parameter is allowed to continue analytically to a purely imaginary value, the resulting field can be identified with the usual finite temperature quantum field in minkowskian space-time. The relation of this field with that of thermo field dynamics is discussed.     

A crisis of a stochastic web

In a kicked rotor subjected to a piecewise-continuous force field, it is observed that a stochastic web and the chaotic diffusion on it suddenly change to transients when an adjustable parameter drives the dissipation. This phenomenon appears to be a new crisis type, which occurs in systems where conservative dynamics may be converted to a dissipative one with a contraction rate showing linear time dependence. It is analytically and numerically shown that, in the crisis, the lifetime dependence obeys universal scaling law suggested by Grebogy, Ott, and Yorke [Phys. Rev. Lett. 57, 1284 (1986)], and the scaling exponent takes a special value, 1, due to the dissipation characteristics. Additionally presented is another power law that describes, from another viewpoint, the transition of a conservative stochastic web (which is a fat fractal) to a non-attracting thin fractal (the strange repeller).Received: 13 December 2003, Published online: 9 March 2004PACS: 05.45.Ac Low-dimensional chaos     

Estimating parameters in stochastic systems: A variational Bayesian approach

This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.     

Kernel-based regression of drift and diffusion coefficients of stochastic processes

To improve the estimation of drift and diffusion coefficients of stochastic processes in case of a limited amount of usable data due to e.g. non-stationarity of natural systems we suggest to use kernel-based instead of histogram-based regression. We propose a method for bandwidth selection and compare it to a widely used cross-validation method. Kernel-based regression reveals an enhanced ability to estimate drift and diffusion especially for a small amount of data. This allows one to improve resolvability of changes in complex dynamical systems as evidenced by an exemplary analysis of electroencephalographic data recorded from a human epileptic brain.     

动态随机最短路径算法研究

静态最短路径问题已经得到很好解决, 然而现实中的网络大多具有动态性和随机性. 网络弧和节点的状态及耗费不仅具有不确定性且相互关联, 弧和节点的耗费都服从一定的概率分布, 因此把最短路径问题看作是一个动态随机优化问题更具有一般性. 文中分析了网络弧和节点的动态随机特性及其相互关系, 定义了动态随机最短路径; 给出了动态随机最短路径优化数学模型, 提出了一种动态随机最短路径遗传算法; 针对网络的拓扑特性设计了高效合理的遗传算子. 实验结果表明, 文中提出的模型和算法能有效地解决动态随机最短路径问题, 可以运用到交通、通信等网络的网络流随机优化问题中.     

Michaelis-Menten mechanism for single-enzyme and multi-enzyme system under stochastic noise and spatial diffusion

The investigation of enzymatic reaction under stochastic effect and spatial effect is an interesting problem. By virtue of Monte Carlo simulation, the stochastic dynamic of enzyme and the related Michaelis-Menten mechanism with stochastic internal noise and spatial diffusion are explored in this article. (i) For the single-enzyme system, two cases, including the fast phosphorylation case [X. S. Xie, et al., J. Phys. Chem. B 109 (2005) 19068] and slow phosphorylation case [X. S. Xie, et al., Nat. Chem. Biol. 2 (2006) 87] are considered. It is found the micro enzymatic velocity rate shows a rough hyperbolic dependence on the substrate concentration, hence obeys the Michaelis-Menten law qualitatively. In addition, our result reveals that diffusion rate can adjust the Michaelis-Menten curve; especially, it is shown that increasing diffusion rate enhances the micro enzyme rate. (ii) For the multi-enzyme system, a typical example, i.e., MAPK signaling pathway is used. We apply the Michaelis-Menten mechanism to the MAPK cascade and give a simple comparison for the signaling ability between the Michaelis-Menten mechanism and the single collision mechanism [J. W. Locasale et al., PLOS Comput. Biol. 4 (2008) e1000099].     

On the derivation of the Schrödinger equation from stochastic mechanics

It is shown that the existing formulations of stochastic mechanics are not equivalent to the Schrödinger equation, as had previously been believed. It is argued that this is a reflection of fundamental inadequacies in the physical foundations of stochastic mechanics.Some relatively minor difficulties with the demonstration of equivalence are already known for the special case in which the nodal surface separates the manifold of the diffusion into disjoint components.^(1,11) The problems described in this paper are much more general and quite unrelated.