scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Measuring the Strangeness of Strange Attractors

TL;DR: In this paper, the correlation exponent v is introduced as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise, and algorithms for extracting v from the time series of a single variable are proposed.
About: This article is published in Physica D: Nonlinear Phenomena.The article was published on 1983-10-01. It has received 5239 citations till now. The article focuses on the topics: Correlation dimension & Lyapunov exponent.
Citations
More filters
Journal ArticleDOI
TL;DR: A review of the main mathematical ideas and their concrete implementation in analyzing experiments can be found in this paper, where the main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions).
Abstract: Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

4,619 citations


Cites background or methods from "Measuring the Strangeness of Strang..."

  • ...It is given by if these limits exist (see Grassberger and Procaccia, 1983a)....

    [...]

  • ...We describe here another approach due to Grassberger and Procaccia (1983a); see also Cohen and Procaccia (1984)....

    [...]

  • ...We concentrate on the determination of the information dimension, using the method advocated by Young (1982), and Grassberger and Procaccia (1983b)....

    [...]

  • ...This algorithm has been successfully used in several experiments, e.g., Malraison et al. (19831, Abraham et al. (1984), Grassberger and Procaccia (1983b); it has indicatedfinite dimensions in hydrodynamic systems, even though the phase space is infinite dimensional and the system therefore could…...

    [...]

  • ...Starting from different ideas, Grassberger and Procaccia (1983a,1983b) have arrived at a very similar way of computing the information dimension dimHp of the measure p....

    [...]

Journal ArticleDOI
TL;DR: The issue of determining an acceptable minimum embedding dimension is examined by looking at the behavior of near neighbors under changes in the embedding dimensions from d\ensuremath{\rightarrow}d+1 by examining the manner in which noise changes the determination of ${\mathit{d}}_{\math it{E}}$.
Abstract: We examine the issue of determining an acceptable minimum embedding dimension by looking at the behavior of near neighbors under changes in the embedding dimension from d\ensuremath{\rightarrow}d+1. When the number of nearest neighbors arising through projection is zero in dimension ${\mathit{d}}_{\mathit{E}}$, the attractor has been unfolded in this dimension. The precise determination of ${\mathit{d}}_{\mathit{E}}$ is clouded by ``noise,'' and we examine the manner in which noise changes the determination of ${\mathit{d}}_{\mathit{E}}$. Our criterion also indicates the error one makes by choosing an embedding dimension smaller than ${\mathit{d}}_{\mathit{E}}$. This knowledge may be useful in the practical analysis of observed time series.

3,375 citations

Journal ArticleDOI
TL;DR: The aim of this work is to provide the readers with the know how for the application of recurrence plot based methods in their own field of research, and detail the analysis of data and indicate possible difficulties and pitfalls.

2,993 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present a test of independence that can be applied to the estimated residuals of any time series model, which can be transformed into a model driven by independent and identically distributed errors.
Abstract: This paper presents a test of independence that can be applied to the estimated residuals of any time series model that can be transformed into a model driven by independent and identically distributed errors. The first order asymptotic distribution of the test statistic is independent of estimation error provided that the parameters of the model under test can be estimated -consistently. Because of this, our method can be used as a model selection tool and as a specification test. Widely used software1 written by Dechert and LeBaron can be used to implement the test. Also, this software is fast enough that the null distribution of our test statistic can be estimated with bootstrap methods. Our method can be viewed as a nonlinear analog of the Box-Pierce Q statistic used in ARIMA analysis.

2,723 citations


Cites background or methods from "Measuring the Strangeness of Strang..."

  • ...absolutely regular with k = 0 for k > m, and by Theorem 1 of Denker and Keller (1983) for generalized U{statistics we get...

    [...]

  • ...Let G j i be {fui;ui+1;:::;ujg for 1 i Denker and Keller (1983) ) if...

    [...]

  • ...In Grassberger and Procaccia (1983) the correlation integral was introduced as a method of measuring the fractal dimension of deterministic data....

    [...]

  • ...Cm;n in equation (2.2) is a generalized U-statistic (c.f., Sering (1980, Chapter 5), and Denker and Keller (1983) ) with symmetric kernel (kx yk)....

    [...]

Journal ArticleDOI
TL;DR: The various applications of HRV and different linear, frequency domain, wavelet domain, nonlinear techniques used for the analysis of the HRV are discussed.
Abstract: Heart rate variability (HRV) is a reliable reflection of the many physiological factors modulating the normal rhythm of the heart. In fact, they provide a powerful means of observing the interplay between the sympathetic and parasympathetic nervous systems. It shows that the structure generating the signal is not only simply linear, but also involves nonlinear contributions. Heart rate (HR) is a nonstationary signal; its variation may contain indicators of current disease, or warnings about impending cardiac diseases. The indicators may be present at all times or may occur at random-during certain intervals of the day. It is strenuous and time consuming to study and pinpoint abnormalities in voluminous data collected over several hours. Hence, HR variation analysis (instantaneous HR against time axis) has become a popular noninvasive tool for assessing the activities of the autonomic nervous system. Computer based analytical tools for in-depth study of data over daylong intervals can be very useful in diagnostics. Therefore, the HRV signal parameters, extracted and analyzed using computers, are highly useful in diagnostics. In this paper, we have discussed the various applications of HRV and different linear, frequency domain, wavelet domain, nonlinear techniques used for the analysis of the HRV.

2,344 citations

References
More filters
Journal ArticleDOI
TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into consider­ably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

16,554 citations

Journal ArticleDOI
10 Jun 1976-Nature
TL;DR: This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.
Abstract: First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.

6,118 citations