具有线性乘积白噪声的随机非自治吊桥方程的随机指数吸引子

本文考虑具有线性乘积白噪声的随机非自治吊桥方程长时间行为.首先,建立了所研究共圈系统的适定性;第二步,研究了该系统随机吸引子的存在性;第三步,当随机系数趋于0时,得到了随机吸引子的上半连续性;第四步,通过``迭代''法证明了随机吸引子在高正则空间中的正则性;最后,给出了该系统随机指数吸引子的存在性,同时得到了吸引子的有限分形维数.     

Fractals in the open quantum kicked top model

Fractals are one of the most important features of the classically chaotic systems. We analyze the fractal phenomena in a quantum chaos system in terms of its fidelity and dynamical localization properties in the paper. We show that, even in the open and dissipative quantum kicked top model, the fidelity displays fractal fluctuations if the underlying dynamics is in the classically chaotic regime. Moreover, the fluctuations of the inverse participation ratio which characterize the dynamical localization behavior also exhibit fractality. The relations between the fractal dimensions and the decoherence rates are explored.     

Scaling law of diffusivity generated by a noisy telegraph signal with fractal intermittency

In many complex systems the non-linear cooperative dynamics determine the emergence of self-organized, metastable, structures that are associated with a birth–death process of cooperation. This is found to be described by a renewal point process, i.e., a sequence of crucial birth–death events corresponding to transitions among states that are faster than the typical long-life time of the metastable states. Metastable states are highly correlated, but the occurrence of crucial events is typically associated with a fast memory drop, which is the reason for the renewal condition. Consequently, these complex systems display a power-law decay and, thus, a long-range or scale-free behavior, in both time correlations and distribution of inter-event times, i.e., fractal intermittency.The emergence of fractal intermittency is then a signature of complexity. However, the scaling features of complex systems are, in general, affected by the presence of added white or short-term noise. This has been found also for fractal intermittency.In this work, after a brief review on metastability and noise in complex systems, we discuss the emerging paradigm of Temporal Complexity. Then, we propose a model of noisy fractal intermittency, where noise is interpreted as a renewal Poisson process with event rate r_p. We show that the presence of Poisson noise causes the emergence of a normal diffusion scaling in the long-time range of diffusion generated by a telegraph signal driven by noisy fractal intermittency. We analytically derive the scaling law of the long-time normal diffusivity coefficient. We find the surprising result that this long-time normal diffusivity depends not only on the Poisson event rate, but also on the parameters of the complex component of the signal: the power exponent μ of the inter-event time distribution, denoted as complexity index, and the time scale T needed to reach the asymptotic power-law behavior marking the emergence of complexity. In particular, in the range μ < 3, we find the counter-intuitive result that normal diffusivity increases as the Poisson rate decreases.Starting from the diffusivity scaling law here derived, we propose a novel scaling analysis of complex signals being able to estimate both the complexity index μ and the Poisson noise rate r_p.     

Characterization of welding defects by fractal analysis of ultrasonic signals

In this work we apply tools developed for the study of fractal properties of time series to the problem of classifying defects in welding joints probed by ultrasonic techniques. We employ the fractal tools in a preprocessing step, producing curves with a considerably smaller number of points than in the original signals. These curves are then used in the classification step, which is realized by applying an extension of the Karhunen–Loève linear transformation. We show that our approach leads to small error rates, comparable with those obtained by using more time-consuming methods based on non-linear classifiers.     

Nonlinear correlations in the hydrophobicity and average flexibility along the glycolytic enzymes sequences

Nonlinear methods widely used for time series analysis were applied to glycolytic enzyme sequences to derive information concerning the correlation of hydrophobicity and average flexibility along their chains. The 20 sequences of different types of the 10 human glycolytic enzymes were considered as spatial series and were analyzed by spectral analysis, detrended fluctuations analysis and Hurst coefficient calculation. The results agreed that there are both short range and long range correlations of hydrophobicity and average flexibility within investigated sequences, the short range correlations being stronger and indicating that local interactions are the most important for the protein folding. This correlation is also reflected by the fractal nature of the structures of investigated proteins.     

Asymptotically Exactly Solvable Models of Processes in Stochastically Homogeneous Disordered Lattice Media

We construct and study asymptotically exactly solvable models (including new models) of evolution processes in random stationary stochastically homogeneous lattice media. For the first time, we present an effective method for obtaining long-time asymptotic expansions for spatial fluctuations of the propagator in many models previously studied as well as in new models.     

Frequency and dependence of long range temporal correlations in human hippocampal energy fluctuations

Spontaneous energy fluctuations in human hippocampal EEG show prominent amplitude and temporal variability. Here we show hippocampal energy fluctuations often exhibit long‐range temporal correlations with power‐law scaling. In most cases this scaling behavior persisted on time scales in excess of 10 minutes, the maximum duration we could detect with our recording durations. During these epochs we find that the energy fluctuations exhibit long‐range correlations over a broad frequency range (0.5–100 Hz) with greater persistence of the correlations in the lower frequency bands (0.5–30 Hz) than the higher (30–100 Hz). The correlation in hippocampal energy fluctuations is characterized by a bias for energy fluctuations to be followed by similar magnitude fluctuations over all energy scales, i.e. large fluctuations begets large fluctuations and small begets small. © 2005 Wiley Periodicals, Inc. Complexity 10: 35–45, 2005     

Group generalized inverses of M-matrices associated with periodic and nonperiodic jacobi matrices

We consider two types of irreducible stochastic matrices —tridiagonal matrices and periodic jacobi matrices —which can be viewed as transition matrices of interval and circular random walks,respcetively. For both types of matrices we give a formula for the group inverse of the associated singular M-matrix. We discuss both the sign patterns and the relative sizes of the entries in these group inverses and apply our results to give qualitative information about random walks on an interval and on a circle.     

Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics

In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.

     

On fractal nature of groundwater level fluctuations due to rainfall process

Hourly resolution time series of groundwater level fluctuations are analyzed after removing the seasonal cycle. It is found that fluctuations of groundwater levels have fractal scaling and a persistent behavior. We show also that groundwater level fluctuations exhibit non-Gaussian heavy tailed probability distribution that is well fitted by the Lévy stable distribution. Implications of the present results on the groundwater system modeling as a fractional Lévy motion and the connection with the anomalous diffusion inside the soil are discussed.     

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.     

Short-term prediction method of wind speed series based on fractal interpolation

In order to improve the prediction performance of the wind speed series, the rescaled range analysis is used to analyze the fractal characteristics of the wind speed series. An improved fractal interpolation prediction method is proposed to predict the wind speed series whose Hurst exponents are close to 1. An optimization function which is composed of the interpolation error and the constraint items of the vertical scaling factors in the fractal interpolation iterated function system is designed. The chaos optimization algorithm is used to optimize the function to resolve the optimal vertical scaling factors. According to the self-similarity characteristic and the scale invariance, the fractal extrapolate interpolation prediction can be performed by extending the fractal characteristic from internal interval to external interval. Simulation results show that the fractal interpolation prediction method can get better prediction result than others for the wind speed series with the fractal characteristic, and the prediction performance of the proposed method can be improved further because the fractal characteristic of its iterated function system is similar to that of the predicted wind speed series.     

IS THE NILE OUTFLOW FRACTAL? HURST'S ANALYSIS REVISITED

In a now classic study, Hurst [1951, 1955] found significant long-term correlations among fluctuations in Nile River outflows and described these correlations in terms of power laws. Mandelbrot's theory of random fractals [1982] later provided an axiomatic framework for Hurst's work. More recently, Bak, Tang and Weisenfeld's [1987] theory of “self-organized criticality” predicted that the fluctuations in Nile River outflows should follow power laws such as those observed by Hurst. In reexamining Hurst's data, we found small but significant deviations from these power laws, and in particular, evidence for a natural time scale of 32–128 years in global climate dynamics, possibly driven by ocean dynamics.     

Fractal response of physiological signals to stress conditions,environmental changes,and neurodegenerative diseases

In the past two decades the biomedical community has witnessed several applications of nonlinear system theory to the analysis of biomedical time series and the development of nonlinear dynamic models. The development of this area of medicine can best be described as nonlinear and fractal physiology. These studies have been intended to develop more reliable methodologies for understanding how biological systems respond to peculiar altered conditions induced by internal stress, environment stress, and/or disease. Herein, we summarize the theory and some of our results showing the fractal dependency on different conditions of physiological signals such as inter‐breath intervals, heart inter‐beat intervals, and human stride intervals. © 2007 Wiley Periodicals, Inc. Complexity 12: 12–17, 2007     

The distance-decay function of geographical gravity model: Power law or exponential law?

The distance-decay function of the geographical gravity model is originally an inverse power law, which suggests a scaling process in spatial interaction. However, the distance exponent of the model cannot be reasonably explained with the ideas from Euclidean geometry. This results in a dimension dilemma in geographical analysis. Consequently, a negative exponential function was used to replace the inverse power function to serve for a distance-decay function. But a new puzzle arose that the exponential-based gravity model goes against the first law of geography. This paper is devoted for solving these kinds of problems by mathematical reasoning and empirical analysis. New findings are as follows. First, the distance exponent of the gravity model is demonstrated to be a fractal dimension using the geometric measure relation. Second, the similarities and differences between the gravity models and spatial interaction models are revealed using allometric relations. Third, a four-parameter gravity model possesses a symmetrical expression, and we need dual gravity models to describe spatial flows. The observational data of China's cities and regions (29 elements indicative of 841 data points) in 2010 are employed to verify the theoretical inferences. A conclusion can be reached that the geographical gravity model based on power-law decay is more suitable for analyzing large, complex, and scale-free regional and urban systems. This study lends further support to the suggestion that the underlying rationale of fractal structure is entropy maximization. Moreover, it suggests that many dimensional dilemmas of spatial modeling can be solved using the concepts from fractal geometry.     

Heat kernel fluctuations for a resistance form with non-uniform volume growth

In this article, we consider the problem of estimating the heatkernel on measure-metric spaces equipped with a resistance form.Such spaces admit a corresponding resistance metric that reflectsthe conductivity properties of the set. In this situation, ithas been proved that when there is uniform polynomial volumegrowth with respect to the resistance metric the behaviour ofthe on-diagonal part of the heat kernel is completely determinedby this rate of volume growth. However, recent results haveshown that for certain random fractal sets, there are globaland local (point-wise) fluctuations in the volume as r 0 andso these uniform results do not apply. Motivated by these examples,we present global and local on-diagonal heat kernel estimateswhen the volume growth is not uniform, and demonstrate thatwhen the volume fluctuations are non-trivial, there will benon-trivial fluctuations of the same order (up to exponents)in the short-time heat kernel asymptotics. We also provide boundsfor the off-diagonal part of the heat kernel. These resultsapply to deterministic and random self-similar fractals, andmetric space dendrites (the topological analogues of graph trees).     

Fractal dimension of a random invariant set

In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations.     

发展方程长时间计算的稳定性与收敛性

本文是[5]的继续,作者在[5]中从稳定性与收敛性之间的关系入手,研究了半线性发展方程数值法的长时间误差估计,那里要求离散化方程当tn=nτ充分大后在某种较严格的意义下稳定,这限制了它的应用范围,本文引进γ-相容的概念,证明由γ－相容和较弱意义下的稳定性可以推出 收敛性,特别对齐次线性和初值问题的差分格式得到了无穷时域上的Lax等价定理,本文2引进γ－相容的概念,并就齐次线性初值问题的差分格式,给出Lax等价定理对无穷时域的推广,3讨论非线性初值问题的数值逼近,对全离散逼近,我们给出两类由长时间稳定（严格稳定）＋γ－相容（相容）无穷时域上收敛性的定量,由于γ－相容性对偏微分方程（同时含有时间和人间变量）通常比对常微分方程组（只含有时间变量）更难检验,为此,我们特别针对常微分方程组数值解给出第三个收敛性定理,我们知道偏微分方程（组）半离散化就是常微分方程组,若能证明半离散解的长时间收敛性,我们就可以利用这一定理证明全离散解的长时间收敛性了,最后在3,作为应用我们指出,可以利用本文方法证明[2]和[4]的结果。并改进[1]的结果。γ