Evolving chaos: Identifying new attractors of the generalised Lorenz family

In a recent paper, we presented an intelligent evolutionary search technique through genetic programming (GP) for finding new analytical expressions of nonlinear dynamical systems, similar to the classical Lorenz attractor's which also exhibit chaotic behaviour in the phase space. In this paper, we extend our previous finding to explore yet another gallery of new chaotic attractors which are derived from the original Lorenz system of equations. Compared to the previous exploration with sinusoidal type transcendental nonlinearity, here we focus on only cross-product and higher-power type nonlinearities in the three state equations. We here report over 150 different structures of chaotic attractors along with their one set of parameter values, phase space dynamics and the Largest Lyapunov Exponents (LLE). The expressions of these new Lorenz-like nonlinear dynamical systems have been automatically evolved through multi-gene genetic programming (MGGP). In the past two decades, there have been many claims of designing new chaotic attractors as an incremental extension of the Lorenz family. We provide here a large family of chaotic systems whose structure closely resemble the original Lorenz system but with drastically different phase space dynamics. This advances the state of the art knowledge of discovering new chaotic systems which can find application in many real-world problems. This work may also find its archival value in future in the domain of new chaotic system discovery.     

Safety preserving control synthesis for sampled data systems

In sampled data systems the controller receives periodically sampled state feedback about the evolution of a continuous time plant, and must choose a constant control signal to apply between these updates; however, unlike purely discrete time models the evolution of the plant between updates is important. In this paper we describe an abstract algorithm for approximating the discriminating kernel (also known as the maximal robust control invariant set) for a sampled data system with continuous state space, and then use this operator to construct a switched, set-valued feedback control policy which ensures safety. We show that the approximation is conservative for sampled data systems. We then demonstrate that the key operations–the tensor products of two sets, invariance kernels, and a pair of projections–can be implemented in two formulations: one based on the Hamilton–Jacobi partial differential equation which can handle nonlinear dynamics but which scales poorly with state space dimension, and one based on ellipsoids which scales well with state space dimension but which is restricted to linear dynamics. Each version of the algorithm is demonstrated numerically on a simple example.     

低噪声水平混沌时序的预测技术及其应用研究

研究含有噪声的混沌时序的除噪及其重构技术,基于除噪混沌数据的预测技术及其应用．应用混沌时序的奇异值分解技术对混沌时序的噪声进行了剥离,将混沌时序的相空间分解成为值域空间和虚拟的噪声空间,在值域空间内重构了原混沌时序,并在此基础上,确立了非线性模型的阶,利用所提出的非线性模型对时序进行了预测研究工作,研究结果表明,该非线性模型具有很强的函数逼近能力,所采用的混沌预测方法对相应的实际问题有着一定的指导意义．     

混沌时序相空间重构的分析和应用研究

在国内外学者工作的基出上,应用Legendere坐标法重构动力系统的相空间,研究了时序时隔τ的取值范围,讨论了时序间隔τ对相空间重构工作的影响,并用所提方法重构了系统的吸引子.算例表明所提方法是有效的.     

低维混沌时序非线性动力系统的预测方法及其应用研究

主要研究由低维混沌时序所确定的非线性动力系统的预测方法及其应用。在国外学者研究工作的基础上,应用一种非线性混沌模型在相空间内对时序进行重构工作,先通过改进的最小二乘方法来估计模型的参数,满足一定精度后,再采用最优化方法来估计模型的参数,并用所求得的混沌时序模型在其相空间内对时序的未来值进行预测。给出了非常有代表性的实例对文中模型和算法进行验证。结果发现采用该算法能较准确地求得模型的参数,在相空间中对混沌时序进行预测,将传统方法中的外推变成了相空间中的内插,及选取最佳的模型阶数等工作都能增加预测的准确程度,且混沌时序不可能进行长期的预测。     

Short-Term Prediction of the Sea State Dynamics

Forecasting of the sea level plays a key role to control on- and offshore facilities. First, we start with a determinstic time series method based on the state space embedding to determine the vector field of the nonlinear dynamical system and deduce the solution of its corresponding high-order differential equation. Second, We assume that the sea state is a stochastic process governed by a deterministic part and by noise so that this dynamical system can be modelled by the Langevin equation. We extract the nonlinear dynamical system considering fluctuations directly from a measured time series by estimating the drift vector and the diffusion matrix of the Fokker-Planck equation. In order to determine the prediction accuracy, the numerical solutions of the deterministic model and the Langevin equation are compared to the data values at future time. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transition Manifolds of Complex Metastable Systems

We consider complex dynamical systems showing metastable behavior, but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.     

AN ENCOMPASSING APPROACH TO MODELING FISHERY DYNAMICS: MODELING DYNAMIC NONLINEAR SYSTEMS

Uncertainty about the nature and significance of nonlinearities and the manner in which dynamics affect future realizations makes model specification the most difficult aspect of modeling dynamic systems. By interpreting several popular fishery models as subcases of a nesting dynamic Taylor series approximation, we isolate the specification differences between these models in a way that accounts for commonalities. On the argument that the differences due to alternative nonlinear forms are likely to be small compared to more mundane considerations such as delay difference and general dynamic lag specification, we propose an alternative model that uses the terms from the first order approximation common to all models combined with a data-based determination of the appropriate lags using the methods of state space time series analysis. Finally, the success of the alternative models is judged in an application to Pacific halibut data.     

Optimizing Markovian modeling of chaotic systems with recurrent neural networks

In this paper, we propose a methodology for optimizing the modeling of an one-dimensional chaotic time series with a Markov Chain. The model is extracted from a recurrent neural network trained for the attractor reconstructed from the data set. Each state of the obtained Markov Chain is a region of the reconstructed state space where the dynamics is approximated by a specific piecewise linear map, obtained from the network. The Markov Chain represents the dynamics of the time series in its statistical essence. An application to a time series resulted from Lorenz system is included.     

High-speed image analysis reveals chaotic vibratory behaviors of pathological vocal folds

Laryngeal pathology is usually associated with irregular dynamics of laryngeal activity. High-speed imaging facilitates direct observation and measurement of vocal fold vibrations. However, chaotic dynamic characteristics of aperiodic high-speed image data have not yet been investigated in previous studies. In this paper, we will apply nonlinear dynamic analysis and traditional perturbation methods to quantify high-speed image data from normal subjects and patients with various laryngeal pathologies including vocal fold nodules, polyps, bleeding, and polypoid degeneration. The results reveal the low-dimensional dynamic characteristics of human glottal area data. In comparison to periodic glottal area series from a normal subject, aperiodic glottal area series from pathological subjects show complex reconstructed phase space, fractal dimension, and positive Lyapunov exponents. The estimated positive Lyapunov exponents provide the direct evidence of chaos in pathological human vocal folds from high-speed digital imaging. Furthermore, significant differences between the normal and pathological groups are investigated for nonlinear dynamic and perturbation analyses. Jitter in the pathological group is significantly higher than in the normal group, but shimmer does not show such a difference. This finding suggests that the traditional perturbation analysis should be cautiously applied to high speed image signals. However, the correlation dimension and the maximal Lyapunov exponent reveal a statistically significant difference between normal and pathological groups. Nonlinear dynamic analysis is capable of quantitatively describing the aperiodic vocal fold vibrations and may be helpful for understanding disordered behaviors in pathological laryngeal systems.     

MR image reconstruction from undersampled data by using the iterative refinement procedure

Magnetic resonance imaging (MRI) reconstruction from sparsely sampled data has been a difficult problem in medical imaging field. We approach this problem by formulating a cost functional that includes a constraint term that is imposed by the raw measurement data in k-space and the L¹ norm of a sparse representation of the reconstructed image. The sparse representation is usually realized by total variational regularization and/or wavelet transform. We have applied the Bregman iteration to minimize this functional to recover finer scales in our recent work. Here we propose nonlinear inverse scale space methods in addition to the iterative refinement procedure. Numerical results from the two methods are presented and it shows that the nonlinear inverse scale space method is a more efficient algorithm than the iterated refinement method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time-series analysis of TCP/RED computer networks,an empirical study

Packet-level observations show that the TCP/RED congestion control systems exhibit complex non-periodic oscillations which vary with the network/RED parameter variations. In this paper, it is investigated whether such complex behaviors are due to nonlinear deterministic chaotic dynamics or do they originate from nonlinear stochastic dynamics. To do this, various methods of linear and nonlinear time series analyses have been applied to the packet-level data gathered from a typical network simulated in ns-2. The results of the analysis for a wide range of variations in averaging weight of RED (as the most important bifurcation factor in TCP/RED networks) show that such behaviors are not due to deterministic chaos in the system, but originate from the stochastic nature of the network.     

Delay Embeddings for Forced Systems. I. Deterministic Forcing

Summary. Takens Embedding Theorem forms the basis of virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems. It typically allows us to reconstruct an unknown dynamical system that gives rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal separation, and most recently for control and targeting. Current versions of Takens Theorem assume that the underlying system is autonomous. Unfortunately this is not the case for many real systems; in the laboratory we often force an experimental system in order for it to exhibit interesting behaviour, whilst in the case of naturally occurring systems it is very rare for us to be able to isolate the system to ensure that there are no external influences. In this paper we therefore prove two versions of Takens Theorem relevant to forced systems: one applicable to the case where the forcing is unknown, and the other to the situation where we are able to determine independently the state of the forcing system (usually because we are responsible for the forcing ourselves). In a subsequent paper we shall show how to extend these results to give an analogue of Takens Theorem for randomly forced systems, leading to a new framework for the analysis of time series arising from nonlinear stochastic systems. Received March 13, 1995; final revision received April 3, 1998; accepted April 21, 1998     

Multiscale recurrence analysis of long-term nonlinear and nonstationary time series

Recurrence analysis is an effective tool to characterize and quantify the dynamics of complex systems, e.g., laminar, divergent or nonlinear transition behaviors. However, recurrence computation is highly expensive as the size of time series increases. Few, if any, previous approaches have been capable of quantifying the recurrence properties from a long-term time series, while which is often collected in the real-time monitoring of complex systems. This paper presents a novel multiscale framework to explore recurrence dynamics in complex systems and resolve computational issues for a large-scale dataset. As opposed to the traditional single-scale recurrence analysis, we characterize and quantify recurrence dynamics in multiple wavelet scales, which captures not only nonlinear but also nonstationary behaviors in a long-term time series. The proposed multiscale recurrence approach was utilized to identify heart failure subjects from the 24-h time series of heart rate variability (HRV). It was shown to identify the conditions of congestive heart failure with an average sensitivity of 92.1% and specificity of 94.7%. The proposed multiscale recurrence framework can be potentially extended to other nonlinear dynamic methods that are computationally expensive for large-scale datasets.     

动力系统实测数据的非线性混沌模型重构

动力系统实测非线性混沌数据的模型重构技术是相空间重构的重要内容。在判定了实测数据的非线性混沌特征,计算了实测数据的分维数,Lyapunov指数,并对其进行了本征值分解和噪声去除及确定其模型阶数以后,提出了一个动力系统实测数据的非线性混沌模型,给出了相应的模型参数辨识方法,并用其确立的混沌模型进行了预测工作,计算结果表明:模型参数辨识方法能迅速地将参数估计值带到多峰目标函数的全局最少值附近,然后再采用优化理论能较准确地求出模型的参数,用得到的混沌模型对系统进行预测工作其预测效果良好,且混沌时序不可能作长期预测。     

Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach

State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.     

All-Time Existence of Smooth Solutions to PDEs of Mixed Type and the Invariant Subspace of Uniform States

We prove a general result concerning the all-time existence of smooth solutions of the space-periodic Cauchy problem for a class of PDEs which involve the coupling of a linear with a nonlinear operator. The initial data are assumed to have small deviations from a constant state. Cases of particular interest are hyperbolic-parabolic systems. For their linear part, we develop simple algebraic conditions which guarantee the applicability of our general all-time existence result. Applications include a complex model of magnetogasdynamics, including dispersion due to Hall currents. Results for standard MHD and gasdynamic systems follow as special cases. We also treat multidimensional viscous Boussinesq equations, which are of third order in space. In these applications, the modes corresponding to spatially uniform states will not decay with increasing time, but are associated with a finite dimensional invariant subspace consisting of all constant states. This subspace may consist solely of stationary states or may contain nontrivial dynamics. An example of the latter is provided by the Boussinesq system if Coriolis forces are included. In either case, the technical complications arising from the invariant subspace of constant states makes our result different from corresponding all-time existence results on the whole space or with other boundary conditions.     

Chaos evidence in catecholamine secretion at chromaffin cells

Chromaffin cells secrete catecholamine molecules via exocytosis process. Each exocytotic event is characterized by a current spike, which corresponds to the amount of released catecholamine from secretory vesicles after fusing to plasma membrane. The current spike might be measured by the oxidation of catecholamine molecules and can be experimentally detected through amperometry technique. In this contribution, the secretion of catecholamine, namely adrenaline, of a set of bovine chromaffin cells is measured individually at each single cell. The aim is to study quantitative results of chaotic behavior in catecholamine secretion. For analysis, time series were obtained from amperometric measurements of each single chromaffin cell. Three analysis techniques were exploited: (i) A low-order attractor was generated by means of phase space reconstruction, Average Mutual Information (AMI) and False Nearest Neighbors (FNN) were used to compute embedding lag and embedding dimension, respectively. (ii) The properties of power spectrum density of time series were studied by Fast Fourier Transform (FFT) looking for possible dominant frequencies in power spectrum. (iii) Maximun Lyapunov Exponent (MLE) analysis was done to study the divergence of trajectories of the time series. Nevertheless, in order to dismiss the possibility of positiveness of MLE are due to the inherent noise in experiments, seven surrogate data sets computed using the Amplitude Adjusted Fourier Transform (AAFT) algorithm was computed. The phase space reconstruction showed that, in all cases, the trajectories lie in an embedding subspace suggesting oscillatory nature. The FFT analysis showed high dispersion of the power spectrum without any predominant frequency range. MLE analysis showed that the MLE values are positive for a given orbit time and a defined range of maximum scale values. Moreover, the trajectory of the MLE evolution of all the surrogate data are asymptotic and hold positive along the maximum scale range. These findings are preliminary evidence on detecting chaotic behavior in catecholamine secretion and, in general, their provide a first step towards a deeply understanding of nonlinear behavior of protein releasing dynamics.     

基于相空间重构的降噪技术研究

探讨了基于相空间重构的局部线性映射算法在非线性时间序列降噪技术中的应用,并给出了算法中主要参数的选取方法.实验结果表明,该算法的降噪效果明显优于传统的线性信号滤波技术.并且针对多数实测数据的原始动态模型未知的特点,提出通过计算降噪前后时序信号的关联维数作为评判降噪效果的工具,克服了已有方法中无法计算该类时序信号降噪水平的缺点.     

Mixed hyperbolic-elliptic systems in self-similar flows

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolic-elliptic type, in some regions of space. In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynamics. We show that the mixed systems which arise exhibit complicated nonlinear dependence. In a prototype system, the nonlinear wave system, this behavior is much simplified, and we outline the solution to some typical Riemann problems.Dedicated to Constantine Dafermos on his 60^th birthdayResearch supported by the National Science Foundation, grant DMS-9970310.Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-9973475 (POWRE).Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-0103823.