Comments on the Entropy of Nonequilibrium Steady States

We discuss the entropy of nonequilibrium steady states. We analyze the so-called spontaneous production of entropy in certain reversible deterministic nonequilibrium system, and its link with the collapse of such systems towards an attractor that is of lower dimension than the dimension of phase space. This means that in the steady state limit, the Gibbs entropy diverges to negative infinity. We argue that if the Gibbs entropy is expanded in a series involving 1, 2,... body terms, the divergence of the Gibbs entropy is manifest only in terms involving integrals whose dimension is higher than, approximately, the Kaplan–Yorke dimension of the steady state attractor. All the low order terms are finite and sum in the weak field limit to the local equilibrium entropy of linear irreversible thermodynamics.     

Decay of Correlations,Quantitative Recurrence and Logarithm Law for Contracting Lorenz Attractors

In this paper we prove that a class of skew products maps with non uniformly hyperbolic base has exponential decay of correlations. We apply this to obtain a logarithm law for the hitting time associated to a contracting Lorenz attractor at all the points having a well defined local dimension, and a quantitative recurrence estimation.     

A new local linear prediction model for chaotic time series

A new local linear prediction model is proposed to predict chaotic time series in this Letter. We propose that the parameters—the embedding dimension and the time delay of the local linear prediction model—can take values which are different to those of the state space reconstruction in the procedure of finding the nearest neighbor points. We propose a criterion based on prediction power to determine the optimal parameters of the new local linear prediction model. Simulation results show that the new local linear prediction model can effectively predict chaotic time series and the prediction performance of the new local linear prediction model is superior to that of the local linear prediction.     

Discontinuous and nondifferentiable functions and dimension increase induced by filtering chaotic data

We show that one can use recently introduced statistics for continuity and differentiability to show the effect of filters of infinite extent in time on a chaotic time series. The statistics point to a discontinuous or nondifferentiable function between the unfiltered attractor and the filtered attractor as the origin of attractor dimension increase when the filtering is severe. The density of discontinuities as a function of resolution follows a scaling relation. We present direct visualization of this effect in the filtered Henon attractor where the origin of dimension increase becomes obvious.     

N-dimensional dynamical systems exploiting instabilities in full

We report experimental and numerical results showing how certain N-dimensional dynamical systems are able to exhibit complex time evolutions based on the nonlinear combination of N-1 oscillation modes. The experiments have been done with a family of thermo-optical systems of effective dynamical dimension varying from 1 to 6. The corresponding mathematical model is an N-dimensional vector field based on a scalar-valued nonlinear function of a single variable that is a linear combination of all the dynamic variables. We show how the complex evolutions appear associated with the occurrence of successive Hopf bifurcations in a saddle-node pair of fixed points up to exhaust their instability capabilities in N dimensions. For this reason the observed phenomenon is denoted as the full instability behavior of the dynamical system. The process through which the attractor responsible for the observed time evolution is formed may be rather complex and difficult to characterize. Nevertheless, the well-organized structure of the time signals suggests some generic mechanism of nonlinear mode mixing that we associate with the cluster of invariant sets emerging from the pair of fixed points and with the influence of the neighboring saddle sets on the flow nearby the attractor. The generation of invariant tori is likely during the full instability development and the global process may be considered as a generalized Landau scenario for the emergence of irregular and complex behavior through the nonlinear superposition of oscillatory motions. (c) 2000 American Institute of Physics.     

A simpler and elegant algorithm for computing fractal dimension in higher dimensional state space

Chaotic systems are now frequently encountered in almost all branches of sciences. Dimension of such systems provides an important measure for easy characterization of dynamics of the systems. Conventional algorithms for computing dimension of such systems in higher dimensional state space face an unavoidable problem of enormous storage requirement. Here we present an algorithm, which uses a simple but very powerful technique and faces no problem in computing dimension in higher dimensional state space. The unique indexing technique of hypercubes, used in this algorithm, provides a clever means to drastically reduce the requirement of storage. It is shown that theoretically this algorithm faces no problem in computing capacity dimension in any dimension of the embedding state space as far as the actual dimension of the attractor is finite. Unlike the existing algorithms, memory requirement offered by this algorithm depends only on the actual dimension of the attractor and has no explicit dependence on the number of data points considered.     

Pointwise dimensions of the lorenz attractor

We discuss a connection between two complementary views of the Lorenz attractor: the first is the accepted view where the attractor has a smooth measure on a fractal support. This complex system is alternatively manifest as a self-similar curve for the pointwise dimension alpha. We describe why the latter approach is accessible for the analysis of an experimental signal.     

Detection of determinism and randomness in time series: A method based on phase space overlap of attractors

The space overlap of an attractor reconstructed from a time series with a similarly reconstructed attractor from a random series is shown to be a sensitive measure of determinism. Results for the time series for Henon, Lorenz and Rössler systems as well as a linear stochastic signal and an experimental ECG signal are reported. The overlap increases with increasing levels of added noise, as shown in the case of Henon attractor. Further, the overlap is shown to decrease as noise is reduced in the case of the ECG signal when subjected to singular value decomposition. The scaling behaviour of the overlap with bin size affords a reliable estimate of the fractal dimension of the attractor even with limited data.     

On the transition to hyperchaos and the structure of hyperchaotic attractors

Using a recently proposed algorithmic scheme for correlation dimension analysis of hyperchaotic attractors, we study two well-known hyperchaotic flows and two standard time delayed hyperchaotic systems in detail numerically. We show that at the transition to hyperchaos, the nature of the scaling region changes suddenly and the attractor displays two scaling regions for embedding dimension M ≥ 4. We argue that it is an indication of a strong clustering tendency of the underlying attractor in the hyperchaotic phase. Because of this sudden qualitative change in the scaling region, the transition to hyperchaos can be easily identified using the discontinuous changes in the dimension (D ₂) at the transition point. We show this explicitely for the two time delayed systems. Further support for our results is provided by computing the spectrum of Lyapunov Exponents (LE) of the hyperchaotic attractor in all cases. Our numerical results imply that the structure of a hyperchaotic attractor is topologically different from that of a chaotic attractor with inherent dual scales, at least for the two general classes of hyperchaotic systems we have analysed here.     

An optical technique for measuring fractal dimensions of planar poincaré maps

An optical technique for measuring the fractal dimension of two-dimensional Poincaré maps has been developed. The parallel processing feature of this method provides a rapid way to calculate a correlation measure between distributed points in a plane. The method has been used to determine the correlation dimension of the Duffing-Holmes strange attractor for the two-well potential. The optically measured values of fractal dimension agree well with computer calculation of dimension for both numerical and experimental data.     

Nonlinear dynamical analysis of turbulence in a stable cloud layer

An eight mode truncated spectral model based on Burgers' approximation to the one-dimensional Navier-Stokes equations is used to compute the Lyapunov dimension of the dynamical attractor for turbulence in a stable cloud layer. The model results are compared with the correlation dimension obtained earlier from a time series of radar Doppler and reflectivity signals from a turbulent layer in a marine stratus cloud. The analysis supports a weak coupling explanation for the lower correlation dimension found for the reflectivity time series compared with that for the Doppler time series. Turbulent Prandtl number emerges from the analysis as a flow parameter which can enlarge the dimension of the model's dynamical attractor, but the attractor dimension computed for the model remains lower than the radar Doppler correlation dimension. Linear stability analysis of the model's equilibrium states suggests that a nontruncated version of the model will possess an attractor which is also of lower dimension than the radar Doppler correlation dimension. (c) 1995 American Institute of Physics.     

Quantifying information loss on chaotic attractors through recurrence networks

We propose an entropy measure for the analysis of chaotic attractors through recurrence networks which are un-weighted and un-directed complex networks constructed from time series of dynamical systems using specific criteria. We show that the proposed measure converges to a constant value with increase in the number of data points on the attractor (or the number of nodes on the network) and the embedding dimension used for the construction of the network, and clearly distinguishes between the recurrence network from chaotic time series and white noise. Since the measure is characteristic to the network topology, it can be used to quantify the information loss associated with the structural change of a chaotic attractor in terms of the difference in the link density of the corresponding recurrence networks. We also indicate some practical applications of the proposed measure in the recurrence analysis of chaotic attractors as well as the relevance of the proposed measure in the context of the general theory of complex networks.     

Invariant polygons in systems with grazing-sliding

The paper investigates generic three-dimensional nonsmooth systems with a periodic orbit near grazing-sliding. We assume that the periodic orbit is unstable with complex multipliers so that two dominant frequencies are present in the system. Because grazing-sliding induces a dimension loss and the instability drives every trajectory into sliding, the system has an attractor that consists of forward sliding orbits. We analyze this attractor in a suitably chosen Poincare section using a three-parameter generalized map that can be viewed as a normal form. We show that in this normal form the attractor must be contained in a finite number of lines that intersect in the vertices of a polygon. However the attractor is typically larger than the associated polygon. We classify the number of lines involved in forming the attractor as a function of the parameters. Furthermore, for fixed values of parameters we investigate the one-dimensional dynamics on the attractor.     

三维单向拉伸马蹄的寻找及其在Compass行走模型中的应用

对三维映射,因维数较高,混沌吸引子结构复杂.根据混沌吸引子维数相对较低的特点,在选定不稳定周期轨的附近,对吸引子的局部进行曲面拟合,提出了三维空间中单向拉伸拓扑马蹄的“降维-升维”寻找方法,并实现为MATLAB工具箱.以Compass行走模型为例,详细介绍拓扑马蹄的寻找过程,判定了混沌步态的存在性,刻画了混沌不变集的部分结构.     

A universal strange attractor underlying the quasiperiodic transition to chaos

We use a Monte Carlo approach to study the universal properties associated with the breakdown of two-torus attractors for arbitrary winding numbers. We demonstrate that the renormalization equations have a universal strange attractor, compute its critical exponents, and discuss its structure. The fractal dimension of this attractor is 1.8±0.1.     

Lyapunov numbers and the local structure of attractors

We examine an M-dimensional mapping defined by a system of broken linear equations, whose Lyapunov numbers may be prespecified, and whose directions of stretching and compression are the coordinate directions. With K positive and M-K negative Lyapunov exponents, the attractor is locally the product of a K-dimensional continuum and an (M-K)-dimensional Cantor set; the latter is found to be a pseudo-product of Cantor sets or continua or Cantor sets and continua. When seen with finite resolution a pseudo-product may look like a true product, but its fractional dimension is less than the sum of the dimensions of its projections on the coordinate axes. Transitions in the number of Cantor sets and continua involved in the pseudo-product need not correspond to transitions in the integral part of the fractional dimension of the attractor. We speculate as to whether the attractors of continuous mappings and flows have similar structures.     

非线性时间序列的高阶奇异谱分析

基于反映线性相关结构的协方差矩阵的奇异谱分析，本质上是一种线性的方法.奇异谱分析用于吸引子重构的可靠性问题引发了一些争议.本文基于具有盲高斯噪声及体现非线性相关等性质的高阶累积量，提出了一种高阶的奇异谱分析方法.通过对Hénon映射、Logistic映射和Lorenz模型的分析说明了该方法的有效性，并在不同的延时、嵌入维数、抽样时间及有噪声的情况下表现出较好的鲁棒性. 关键词：     

Crossing bifurcations and unstable dimension variability

A crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied. In this Letter, we describe a global bifurcation in three dimensions which can result in a crisis. This bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than 3. We present evidence of unstable dimension variability as a result of the crisis. We then derive a new scaling law describing the density of the new portion of the attractor formed in the crisis. We illustrate this new type of bifurcation with a specific example of a three-dimensional chaotic attractor undergoing a crisis.     

Fractal dimension of Siegel disc boundaries

Using renormalization techniques, we provide rigorous computer-assisted bounds on the Hausdorff dimension of the boundary of Siegel discs. Specifically, for Siegel discs with golden mean rotation number and quadratic critical points we show that the Hausdorff dimension is less than 1.08523. This is done by exploiting a previously found renormalization fixed point and expressing the Siegel disc boundary as the attractor of an associated Iterated Function System. Received: 26 January 1998 / Received in final form: 5 June 1998 / Accepted: 11 June 1998     

A novel hyperchaotic system and its circuit implementation

A four-dimensional hyperchaotic system with five parameters is proposed. Its dynamical properties such as dissipativity, equilibrium points, Lyapunov exponent, Lyapunov dimension, bifurcation diagrams and Poincare maps are analyzed theoretically and numerically. Theoretical analyses and simulation tests indicate that the new system's dynamics behavior can be periodic attractor, chaotic attractor and hyperchaotic attractor as the parameter varies. Finally, the circuit of this new hyperchaotic system is designed and realized by Multisim software. The simulation results confirm that the chaotic system is different from the existing chaotic systems and is a novel hyperchaotic system. The system is recommendable for many engineering applications such as information processing, cryptology, secure communications, etc.