Estimation of noise levels for models of chaotic dynamical systems

We investigate how far it is possible to identify and separate dynamical noise from measurement noise in observed nonlinear time series. Using Bayesian methods, we derive estimates for the two noise levels, and find that, given a good model of the dynamics, these can give accurate results even if the dynamical noise level is orders of magnitude smaller than the measurement noise level, whereas a simple calculation of root mean square error badly understates the dynamical noise. We argue that this allows better estimates of the underlying dynamical time series, and so better predictions of its future and of its fundamental dynamical properties.     

A “No-Go” theorem for the existence of a discrete action principle

In this paper, we study the problem of the existence of a least-action principle for invertible, second-order dynamical systems, discrete in time and space. We show that, when the configuration space is finite and arbitrary state transitions are allowed, a least-action principle does not exist for such systems. We dichotomize discrete dynamical systems with infinite configuration spaces into those of finite type for which this theorem continues to hold, and those not of finite type for which it is possible to construct a least-action principle. We also show how to recover an action, by restriction of the phase space of certain second-order discrete dynamical systems. We provide numerous examples to illustrate each of these results.     

Constructing dynamical systems with specified symbolic dynamics

In this paper we demonstrate how to construct signals (time series) of continuous-time dynamical systems that exhibit a given symbolic dynamics. This is achieved without construction of the ordinary differential equations that generate the flow. This construction is of theoretical interest and is useful as a source of dynamical data that can be used to test various data analysis algorithms.     

Stability and the maintenance of balance following a perturbation from quiet stance

We investigate stability and the maintenance of balance with the use of tools from dynamical systems. In particular we investigate the application of such tools to the study of the ground reaction forces resulting from an athlete being perturbed from quiet stance. We develop a nonlinear model consisting of a set of coupled vector fields for the derivative with respect to time of the angles between the resultant ground reaction forces and the vertical in the anteroposterior and mediolateral directions. This model contains a basin of attraction bound by a closed curve which we call the critical curve. It is only inside this curve that perturbations can be corrected, with the orbit spiraling onto an attractor corresponding to quiet stance. We show how the critical curve and also the strength of the attractor found in the basin of attraction can be fit to model the experimental data (time series) for an individual athlete. We also discuss how our model can be used to identify nonsymmetric behavior caused by muscle imbalances and differences in the ranges of motion on either side of the body.     

Poisson Involutions, Spin Calogero-Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models

We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary dynamical Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars-Schneider equations which correspond to the N-soliton solutions of affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid.     

The construction of an Ito model for geoelectrical signals

The Ito stochastic differential equation governs the one-dimensional diffusive Markov process. Geoelectrical signals measured in seismic areas can be considered as the result of competitive and collective interactions among system elements. The Ito equation may constitute a good macroscopic model of such a phenomenon in which microscopic interactions are adequately averaged. The present study shows how to construct an Ito model for a geoelectrical time series measured in a seismic area of southern Italy. Our results reveal that the Ito model describes the whole time series quite well, but it performs better when one considers fragments of the data set with lower variability range (absent or rare large fluctuations). Our findings show that generally detrended geoelectrical time series can be considered as approximations of Markov diffusion processes.     

两相流相空间多元图重心轨迹动力学特征

本文提出了一种新的混沌时间序列高维相空间多元图重心轨迹动力学特征提取方法. 在确定了最佳嵌入维数和延迟时间后, 将相空间中高维矢量点映射到二维平面的雷达图上, 相应地将相空间中高维矢量点变换为对应的几何多边形. 通过提取几何多边形的重心位置得到重心轨迹动力学演化特性, 并利用重心轨迹矩特征量区分不同性质的混沌时间序列. 在此基础上, 处理分析了气液两相流电导传感器动态信号, 发现高维相空间多元图重心轨迹矩特征量不仅可以辨识泡状流、段塞流和混状流, 而且为流型动力学演化机理提供了新的分析途径.     

Recurrence analysis of quasiperiodicity in experimental fluid data

We propose a procedure to analyze experimental data which exhibit different dynamical regimes, namely, periodicity, chaoticity and quasiperiodicity by means of Recurrence Plots (RPs). We show that, based on the recurrence properties captured by RPs, we are able to characterize successfully the type of dynamics. This approach is particularly useful for detecting the existence of quasiperiodic motion in short time series. We show the capability and validity of this method by analyzing time series from fluid experiments.     

Optimal instruments and models for noisy chaos

Analysis of finite, noisy time series data leads to modern statistical inference methods. Here we adapt Bayesian inference for applied symbolic dynamics. We show that reconciling Kolmogorov's maximum-entropy partition with the methods of Bayesian model selection requires the use of two separate optimizations. First, instrument design produces a maximum-entropy symbolic representation of time series data. Second, Bayesian model comparison with a uniform prior selects a minimum-entropy model, with respect to the considered Markov chain orders, of the symbolic data. We illustrate these steps using a binary partition of time series data from the logistic and Henon maps as well as the R?ssler and Lorenz attractors with dynamical noise. In each case we demonstrate the inference of effectively generating partitions and kth-order Markov chain models.     

Partial transfer entropy on rank vectors

For the evaluation of information flow in bivariate time series, information measures have been employed, such as the transfer entropy (TE), the symbolic transfer entropy (STE), defined similarly to TE but on the ranks of the components of the reconstructed vectors, and the transfer entropy on rank vectors (TERV), similar to STE but forming the ranks for the future samples of the response system with regard to the current reconstructed vector. Here we extend TERV for multivariate time series, and account for the presence of confounding variables, called partial transfer entropy on ranks (PTERV). We investigate the asymptotic properties of PTERV, and also partial STE (PSTE), construct parametric significance tests under approximations with Gaussian and gamma null distributions, and show that the parametric tests cannot achieve the power of the randomization test using time-shifted surrogates. Using simulations on known coupled dynamical systems and applying parametric and randomization significance tests, we show that PTERV performs better than PSTE but worse than the partial transfer entropy (PTE). However, PTERV, unlike PTE, is robust to the presence of drifts in the time series and it is also not affected by the level of detrending.     

Detecting dynamical nonstationarity in time series data

Nonlinear time series analysis is becoming an ever more powerful tool to explore complex phenomena and uncover underlying patterns from irregular data recorded from experiments. However, the existence of dynamical nonstationarity in time series data causes many results of such analysis to be questionable and inconclusive. It is increasingly recognized that detecting dynamical nonstationarity is a crucial precursor to data analysis. In this paper, we present a test procedure to detect dynamical nonstationarity by directly inspecting the dependence of nonlinear statistical distributions on absolute time along a trajectory in phase space. We test this method using a broad range of data, chaotic, stochastic and power-law noise, both computer-generated and observed, and show that it provides a reliable test method in analyzing experimental data. (c) 1999 American Institute of Physics.     

Gauge invariance based on the extrinsic geometry in the rigid string

The string model with the extrinsic curvature is studied which is a gauge invariant field theory with higher order derivatives. We present an equivalent action without any higher order derivatives which keeps the gauge invariance. We point out the difficulty caused by the second class constraints in Dirac's canonical method. Following a new method for dynamical systems with second class constraints, we construct an equivalent model which has no second class constrants but as a new gauge invariance. This gauge invariance guarantees the equivalence between the original model and the new one. We show that the model can be quantized in this formalism. We find the unitarity violation of the model.     

Reservoir Computing with Delayed Input for Fast and Easy Optimisation

Reservoir computing is a machine learning method that solves tasks using the response of a dynamical system to a certain input. As the training scheme only involves optimising the weights of the responses of the dynamical system, this method is particularly suited for hardware implementation. Furthermore, the inherent memory of dynamical systems which are suitable for use as reservoirs mean that this method has the potential to perform well on time series prediction tasks, as well as other tasks with time dependence. However, reservoir computing still requires extensive task-dependent parameter optimisation in order to achieve good performance. We demonstrate that by including a time-delayed version of the input for various time series prediction tasks, good performance can be achieved with an unoptimised reservoir. Furthermore, we show that by including the appropriate time-delayed input, one unaltered reservoir can perform well on six different time series prediction tasks at a very low computational expense. Our approach is of particular relevance to hardware implemented reservoirs, as one does not necessarily have access to pertinent optimisation parameters in physical systems but the inclusion of an additional input is generally possible.     

Formal model of internal measurement: Alternate changing between recursive definition and domain equation

We sketch a paradox generally resulting from recursivity, and propose a novel model to express evolutionary processes that requires identification of an interaction with internal measurement. In this model, a paradox is not resolved and the notion of relativity of any resolution is implicit. In a dynamical system a certain transition rule is used recursively along time. If one takes the foundation (or context) of recursivity into consideration, one obtains a fixed point or one confronts a paradox. In order to resolve this paradox, we adopt Scott's technical way to identify the form of a fixed point with a domain equation and to obtain a reflective domain, however we simultaneously show that any resolution is destined to be relative. In utilizing this notion, we construct a model of dynamical process by embedding a measurement process in one time step. Any time transition involves the process of doubting the foundation of a transition rule leading to a fixed point. Solving it and obtaining a reflexive domain is used as a new transition rule. Also, this process perpetually proceeds along time, and then the system perpetually proceeds while any solution is destined to be relative. We illustrate this type of model by using a dynamically changing contraction mapping as the interface of state and transition rule. Finally, we show that one can formalize emergent properties by using this model and discuss the relationship between endo-physics and internal measurement.     

基于稀疏贝叶斯学习的复杂网络拓扑估计

提出了一种噪声环境下复杂网络拓扑估计方法, 仅利用含噪时间序列估计未知结构混沌系统的动力学方程和参数, 以及由混沌系统组成的复杂网络的拓扑结构、节点动力学方程、所有参数、 节点间耦合方向和耦合强度.通过采用动力学方程的统一形式, 将动力系统方程结构和参数估计看成线性回归问题的系数估计, 该估计问题利用贝叶斯压缩传感的信号重建算法求解, 含噪信号的模型重建使用相关向量机方法,即通过稀疏贝叶斯学习求解稀疏欠定线性方程得到上面提到的可估计对象.以单个Lorenz系统及由200个 Lorenz系统组成的无标度网络为例说明方法的有效性. 仿真结果表明,提出的方法对噪声有很强的鲁棒性,收敛速度快,稳态误差极小, 克服了最小二乘估计方法收敛速度慢、 稳态误差大以及压缩传感估计方法对噪声鲁棒性不强的缺点.     

非线性时间序列的动力结构突变检测的研究

基于非线性时间序列分析方法——动力学相关因子指数，提出一种新的动力结构突变的检测方法——动力学指数分割算法.通过理想时间序列试验，验证了该方法检测动力结构突变的有效性，同时发现相对少量的尖峰噪声对该方法的影响较小，但连续分布的随机白噪声对其具有一定的影响，并与传统的滑动T检验法和Yamamoto法进行比较，进而讨论它们各自的优缺点. 关键词： 动力学相关因子指数 动力学指数分割算法 噪声 滑动T检验 Yamamoto法     

Droplet dynamics on patterned substrates

We present a lattice Boltzmann algorithm which can be used to explore the spreading of droplets on chemically and topologically patterned substrates. As an example we use the method to show that the final configuration of a drop on a substrate comprising hydrophobic and hydrophilic stripes can depend sensitively on the dynamical pathway by which the state is reached. We also consider a substrate covered with micron-scale posts and investigate how this can lead to superhydrophobic behaviour. Finally we model how a Namibian desert beetle collects water from the wind.     

Noise reduction by recycling dynamically coupled time series

We say that several scalar time series are dynamically coupled if they record the values of measurements of the state variables of the same smooth dynamical system. We show that much of the information lost due to measurement noise in a target time series can be recovered with a noise reduction algorithm by crossing the time series with another time series with which it is dynamically coupled. The method is particularly useful for reduction of measurement noise in short length time series with high uncertainties.     

非一致通信时滞动力学网络上的接连滞后同步

在动力学网络中,节点与节点之间的通信通常存在时滞,并且不同节点之间的通信时滞往往是不同的(即非一致通信时滞),研究非一致通信时滞动力学网络上的接连滞后同步,更具现实意义.为此,本文首先构建含有非一致通信时滞的动力学网络模型.其次分别设计线性反馈控制和自适应反馈控制,利用Lyapunov函数方法,重点分析了该网络的接连滞后同步的稳定性,得到了同步稳定的充分条件.最后,选取蔡氏电路作为局部动力学,又分别选取了链式网络和星型网络这两种拓扑结构来验证理论结果的正确性和有效性.     

Networks with time structure from time series

We propose a method of constructing a network, in which its time structure is directly incorporated, based on a deterministic model from a time series. To construct such a network, we transform a linear model containing terms with different time delays into network topology. The terms in the model are translated into temporal nodes of the network. On each link connecting these nodes, we assign a positive real number representing the strength of relationship, or the “distance,” between nodes specified by the parameters of the model. The method is demonstrated by a known system and applied to two actual time series.