Model-based detector and extraction of weak signal frequencies from chaotic data

Detecting a weak signal from chaotic time series is of general interest in science and engineering. In this work we introduce and investigate a signal detection algorithm for which chaos theory, nonlinear dynamical reconstruction techniques, neural networks, and time-frequency analysis are put together in a synergistic manner. By applying the scheme to numerical simulation and different experimental measurement data sets (Henon map, chaotic circuit, and NH(3) laser data sets), we demonstrate that weak signals hidden beneath the noise floor can be detected by using a model-based detector. Particularly, the signal frequencies can be extracted accurately in the time-frequency space. By comparing the model-based method with the standard denoising wavelet technique as well as supervised principal components analysis detector, we further show that the nonlinear dynamics and neural network-based approach performs better in extracting frequencies of weak signals hidden in chaotic time series.     

Neural networks and chaos: construction, evaluation of chaotic networks, and prediction of chaos with multilayer feedforward networks

Many research works deal with chaotic neural networks for various fields of application. Unfortunately, up to now, these networks are usually claimed to be chaotic without any mathematical proof. The purpose of this paper is to establish, based on a rigorous theoretical framework, an equivalence between chaotic iterations according to Devaney and a particular class of neural networks. On the one hand, we show how to build such a network, on the other hand, we provide a method to check if a neural network is a chaotic one. Finally, the ability of classical feedforward multilayer perceptrons to learn sets of data obtained from a dynamical system is regarded. Various boolean functions are iterated on finite states. Iterations of some of them are proven to be chaotic as it is defined by Devaney. In that context, important differences occur in the training process, establishing with various neural networks that chaotic behaviors are far more difficult to learn.     

多变量时间序列最大李雅普诺夫指数的计算

根据单变量时间序列计算最大Lyapunov指数的算法思想，本文提出了一种于多变量时间序列最大Lyapunov指数计算的方法.针对原有算法需要使用重构相空间的特点，推广算法给出了多变量时间序列相空间重构参数的选择方法，并采用多变量重构相空间进行最大Lyapunov指数计算.经耦合Rssler系统产生多变量时间序列的仿真计算，验证了该算法的有效性，推广算法的计算结果表明多变量时间序列的计算结果优于单变量的结果，且更加接近理论计算结果. 关键词： 多变量时间序列 相空间重构 Lyapunov指数     

Detecting synchronizations in an asymmetric vocal fold model from time series data

A nonlinear modeling approach is presented for the reconstruction of the synchronization structure in an asymmetric two-mass model from time series data. The asymmetric two-mass model describes a variety of normal and pathological human voices associated with synchronous and desynchronous oscillations of the two asymmetric vocal folds. Our technique recovers the synchronization diagram, which yields the regimes of synchronization as well as desynchronization, which are dependent upon the asymmetry parameter and the subglottal pressure. This allows the prediction of the regime of pathological phonation associated with desynchronization of the vocal folds from a few sets of recorded time series. It is shown that the modeling is quite effective when the time series data are chaotic and if they are taken from a regime of desynchronization. We discuss the applicability of the present approach as a diagnostic tool for voice pathologies.     

Chaotic synchronization: a nonlinear predictive filtering approach

The synchronization of chaotic systems is a difficult task due to their sensitive dependence on the initial conditions. Perfect synchronization is almost impossible when noise is present in the system. One of the well known stochastic filtering algorithms that is used to synchronize chaotic systems in the presence of noise is the extended Kalman filter (EKF). However, for highly nonlinear systems, the approximation error introduced by the EKF has been shown to be relatively high. In this paper, a nonlinear predictive filter (NPF) is proposed for synchronizing chaotic systems. In this scheme, it is not required to approximate the underlying nonlinearity and hence there is no need to compute the Jacobian of the chaotic system. Numerical simulations are carried out to compare the performances of the NPF and EKF algorithms for synchronizing different sets of chaotic systems and/or maps. The well known Lorenz and Mackey-Glass systems as well as Ikeda map are used for numerical evaluation of the performance. Results clearly show that the NPF based approach is superior to the EKF based approach in terms of the normalized mean square error (NMSE), total NMSE, and the time taken for synchronization (measured in terms of the normalized instantaneous square error) for all the systems and/or maps considered.     

The geometry of chaotic dynamics — a complex network perspective

Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently ε-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using ε-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that ε-recurrence networks exhibit an important link between dynamical systems and graph theory.     

Clustering data by inhomogeneous chaotic map lattices

A new approach to clustering, based on the physical properties of inhomogeneous coupled chaotic maps, is presented. A chaotic map is assigned to each data point and short range couplings are introduced. The stationary regime of the system corresponds to a macroscopic attractor independent of the initial conditions. The mutual information between pairs of maps serves to partition the data set in clusters, without prior assumptions about the structure of the underlying distribution of the data. Experiments on simulated and real data sets show the effectiveness of the proposed algorithm.     

基于在线小波支持向量回归的混沌时间序列预测

混沌时间序列预测是非线性动力学研究中一个十分重要的问题，支持向量回归方法为其提供了一种有效的解决思路.通过分析新样本加入训练集后支持向量集的变化情况，建立了一种混沌时间序列预测的支持向量回归算法，具备了在线学习的特点.同时，针对混沌信号提出了一种满足小波框架的小波核函数，它不但能以较高的精度逼近任意函数，而且适合于混沌信号的局部分析，提高了支持向量回归的泛化能力.最后就Mackey-Glass混沌时间序列在线预测问题进行了大量仿真.结果表明，本文算法与现有的算法相比具有训练时间短、预测精度高等特点，有一定 关键词： 混沌时间序列 支持向量回归 在线学习 小波核     

Detecting anomalous phase synchronization from time series

Modeling approaches are presented for detecting an anomalous route to phase synchronization from time series of two interacting nonlinear oscillators. The anomalous transition is characterized by an enlargement of the mean frequency difference between the oscillators with an initial increase in the coupling strength. Although such a structure is common in a large class of coupled nonisochronous oscillators, prediction of the anomalous transition is nontrivial for experimental systems, whose dynamical properties are unknown. Two approaches are examined; one is a phase equational modeling of coupled limit cycle oscillators and the other is a nonlinear predictive modeling of coupled chaotic oscillators. Application to prototypical models such as two interacting predator-prey systems in both limit cycle and chaotic regimes demonstrates the capability of detecting the anomalous structure from only a few sets of time series. Experimental data from two coupled Chua circuits shows its applicability to real experimental system.     

Transition to chaos in the conservative four-wave parametric interactions

In this work, we analyze the transition from regular to chaotic states in the parametric four-wave interactions. The temporal evolution describing the coupling of two sets of three-waves with quadratic nonlinearity is considered. This system is shown to undergo a chaotic transition via the separatrix chaos scenario, where a soliton-like solution (separatrix) that is found for the integrable (perfect matched) case becomes irregular as a small mismatch is turned on. As the mismatch is increased the separatrix chaotic layer spreads along the phase space, eventually engrossing most part of it. This scenario is typical of low-dimensional Hamiltonian systems.     

常微分方程系统中内部激变现象的研究

应用广义胞映射图论方法研究常微分方程系统的激变.揭示了边界激变是由于混沌吸引子与 在其吸引域边界上的周期鞍碰撞产生的,在这种情况下,当系统参数通过激变临界值时,混 沌吸引子连同它的吸引域突然消失,在相空间原混沌吸引子的位置上留下了一个混沌鞍.研 究混沌吸引子大小(尺寸和形状)的突然变化,即内部激变.发现这种混沌吸引子大小的突然 变化是由于混沌吸引子与在其吸引域内部的混沌鞍碰撞产生的,这个混沌鞍是相空间非吸引 的不变集,代表内部激变后混沌吸引子新增的一部分.同时研究了这个混沌鞍的形成与演化. 给出了对永久自循环胞集和瞬态自循环胞集进行局部细化的方法. 关键词： 广义胞映射 有向图 激变 混沌鞍     

From chaos to disorder: Statistics of the eigenfunctions of microwave cavities

We study the statistics of the experimental eigenfunctions of chaotic and disordered microwave billiards in terms of the moments of their spatial distributions, such as the inverse participation ratio (IPR) and density-density auto-correlation. A path from chaos to disorder is described in terms of increasing IPR. In the chaotic, ballistic limit, the data correspond well with universal results from random matrix theory. Deviations from universal distributions are observed due to disorder induced localization, and for the weakly disordered case the data are well-described by including finite conductance and mean free path contributions in the framework of nonlinear sigma models of supersymmetry.     

基于模糊边界模块化神经网络的混沌时间序列预测

提出一种模糊边界模块化神经网络（FBMNN）的混沌时间序列预测方法,该方法先对混沌时间序列观测点重构的相空间进行模块化划分,划分点的选取由遗传算法自动寻优.然后定义一个模糊隶属度函数,在划分边界一侧按照一定的模糊隶属度设定模糊边界带,通过模糊化处理,解决了各模块划分点附近预测结果的跳跃问题.最后每一模块,及其模糊边界的样本点都对应一个递归神经网络进行训练,通过预测合成模块输出结果.该方法对三个混沌时间序列基准数据集Mackey-Glass,Lorenz,Henon进行实验,结果表明该方法有效地提高了混沌时间序列预测效果. 关键词： 模糊边界 模块化神经网络 混沌时间序列 预测     

A method of estimating the noise level in a chaotic time series

An attempt is made in this study to estimate the noise level present in a chaotic time series. This is achieved by employing a linear least-squares method that is based on the correlation integral form obtained by Diks in 1999. The effectiveness of the method is demonstrated using five artificial chaotic time series, the Henon map, the Lorenz equation, the Duffing equation, the Rossler equation and the Chua's circuit whose dynamical characteristics are known a priori. Different levels of noise are added to the artificial chaotic time series and the estimated results indicate good performance of the proposed method. Finally, the proposed method is applied to estimate the noise level present in some real world data sets.     

Origin of transient and intermittent dynamics in spatiotemporal chaotic systems

Nonattracting chaotic sets (chaotic saddles) are shown to be responsible for transient and intermittent dynamics in an extended system exemplified by a nonlinear regularized long-wave equation, relevant to plasma and fluid studies. As the driver amplitude is increased, the system undergoes a transition from quasiperiodicity to temporal chaos, then to spatiotemporal chaos. The resulting intermittent time series of spatiotemporal chaos displays random switching between laminar and bursty phases. We identify temporally and spatiotemporally chaotic saddles which are responsible for the laminar and bursty phases, respectively. Prior to the transition to spatiotemporal chaos, a spatiotemporally chaotic saddle is responsible for chaotic transients that mimic the dynamics of the post-transition attractor.     

Multi-step-prediction of chaotic time series based on co-evolutionary recurrent neural network

This paper proposes a co-evolutionary recurrent neural network （CERNN） for the multi-step-prediction of chaotic time series, it estimates the proper parameters of phase space reconstruction and optimizes the structure of recurrent neural networks by coevolutionary strategy. The searching space was separated into two subspaces and the individuals are trained in a parallel computational procedure. It can dynamically combine the embedding method with the capability of recurrent neural network to incorporate past experience due to internal recurrence. The effectiveness of CERNN is evaluated by using three benchmark chaotic time series data sets： the Lorenz series, Mackey-Glass series and real-world sun spot series. The simulation results show that CERNN improves the performances of multi-step-prediction of chaotic time series.     

Chaotic scattering in the gravitational three-body problem

We summarize some results of an ongoing study of the chaotic scattering interaction between a bound pair of stars (a binary) and an incoming field star. The stars are modeled as point masses and their equations of motion are numerically integrated for a large number of initial conditions. The global features of the resulting initial-value space maps are presented, and their evolution as a function of system parameters is discussed. We find that the maps contain regular regions separated by rivers of chaotic behavior. The probability of escape within the chaotic regions is discussed, and a straightforward explanation of the scaling present in these regions is reviewed. We investigate a statistical quantity of interest, namely the cross section for temporarily bound interactions, as a function of the third star's incoming velocity and mass. Finally, a new way of considering long-lived trajectories is presented, allowing long data sets to be qualitatively analyzed at a glance.     

输油管道压力时间序列混沌特性研究

输油管道压力波动的内在动态特征,采用非线性的分析方法探讨了实测输油管道压力波动中存在混沌动态特性的可能性. 以6个典型的输油管道压力实测数据集为研究对象,重构相空间、求得了分形维数和Lyapunov指数谱、验证了数据的平稳性和非线性. 通过对所得结果的分析,从理论上证明了输油管道压力信号具有严格混沌动态特性. 为基于输油管道压力时间序列的研究提供了混沌理论基础. 关键词： 压力时间序列 Lyapunov指数 分形维数 混沌     

Monitoring changes in time of chaotic nonlinear systems

We extend our method for classifying signals from chaotic nonlinear dynamical systems to the problem of monitoring chaotic nonlinear dynamical systems with the goal of detecting that the state of a system has changed. One potential application would be to systems where the changes are not easily detectable by spectral analysis or other linear techniques. The method is expected to be most useful in comparison to other techniques when there are other signals or noise present, some of which have a broad band frequency spectrum, and the signal of interest is associated with either a low dimensional dynamical system or a low dimensional chaotic attractor. The method is applied to data from a laboratory model of a fluidized bed reactor and to data from a gyroscope as well as to numerically generated signals from mathematical models. For the dynamical systems considered in the paper, the proposed method provides significantly better discrimination than spectral analysis. (c) 1995 American Institute of Physics.     

Strange saddles and the dimensions of their invariant manifolds

Nonattracting chaotic sets play a fundamental role in typical dynamical systems. They occur, for example, in the form of chaotic transient sets and fractal basin boundaries. The subject of this paper is the dimensions of these sets and of their stable and unstable manifolds. Numerical experiments are performed to determine these dimensions. The results are consistent with a conjectured formulae expressing the dimensions in terms of Lyapunov exponents and the transient life-time associated with the strange saddle.