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22
Determining Lyapunov Exponents from a Time Series
 Physica
, 1985
"... We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of n ..."
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We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the longterm growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the BelousovZhabotinskii reaction and CouetteTaylor flow. Contents 1.
Estimating fractal dimension
 Journal of the Optical Society of America A
, 1990
"... Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools ..."
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Cited by 123 (4 self)
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Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor. Confusion is a word we have invented for an order which is not understood.Henry Miller, "Interlude," Tropic of Capricorn Numerical coincidence is a common path to intellectual perdition in our quest for meaning. We delight in catalogs of disparate items united by the same number, and often feel in our gut that some unity must underlie it all.
On the Estimation of Topological Entropy
 Journal of Statistical Physics
, 1993
"... We study here a method for estimating the topological entropy of a smooth dynamical system. Our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system. We present two algorithms for this purpose and we analyze each according to its strengths and pitfalls. ..."
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We study here a method for estimating the topological entropy of a smooth dynamical system. Our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system. We present two algorithms for this purpose and we analyze each according to its strengths and pitfalls. We also contrast these with a method based on the definition of topological entropy, using (n; ffl)spanning sets. 1 Introduction and Preliminaries The topological entropy of a system is a quantitative measure of its orbit complexity. In a certain sense, it is the maximum amount of information lost per unit time by the system using measurements with finite precision. As such, the entropy is an important invariant to know. Since the definition of the entropy requires an exponentially growing number of objects, it is impractical to expect that the definition can effectively be used to estimate it. Fortunately, there are several recent theorems which aid in its estimation. Block and Keesling ...
Fractal basins of attraction associated with a damped Newton's method
 SIAM Rev
, 1998
"... Abstract. An intriguing and unexpected result for students learning numerical analysis is that Newton’s method, applied to the simple polynomial z 3 − 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As an example of an interesting open question that may ..."
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Abstract. An intriguing and unexpected result for students learning numerical analysis is that Newton’s method, applied to the simple polynomial z 3 − 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As an example of an interesting open question that may help to stimulate student interest in numerical analysis, we investigate the question of whether a damping method, which is designed to increase the likelihood of convergence for Newton’s method, modifies the fractal structure of the basin boundaries. The overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting. Key words. Newton’s method, damping, fractal basins of attraction AMS subject classifications. 34C35, 6501, 65H10, 65Y99 1. Introduction. Numerical
Estimating Effective Degrees of Freedom in Motor Systems
, 2006
"... Abstract—Studies of the degrees of freedom and “synergies ” in musculoskeletal systems rely critically on algorithms to estimate the “dimension ” of kinematic or neural data. Linear algorithms such as principal component analysis (PCA) are the most popular. However, many biological data (or realisti ..."
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Abstract—Studies of the degrees of freedom and “synergies ” in musculoskeletal systems rely critically on algorithms to estimate the “dimension ” of kinematic or neural data. Linear algorithms such as principal component analysis (PCA) are the most popular. However, many biological data (or realistic experimental data) may be better represented by nonlinear sets than linear subspaces. We evaluate the performance of PCA and compare it to two nonlinear algorithms [Isomap and our novel pointwise dimension estimation (PDE)] using synthetic and motion capture data from a robotic arm with known kinematic dimensions, as well as motion capture data from human hands. We find that PCA can lead to more accurate dimension estimates when considering additional properties of the PCA residuals, instead of the dominant method of using a threshold of variance captured. In contrast to the single integer dimension estimates of PCA and Isomap, PDE provides a distribution and range of estimates of fractal dimension that identify the heterogeneous geometric structure in the experimental data. A strength of the PDE method is that it associates a distribution of dimensions to the data. Since there is no a priori reason to assume that the sets of interest have a single dimension, these distributions incorporate more information than a single summary statistic. Our preliminary findings suggest that fewer than ten DOFs are involved in some hand motion tasks. Contrary to common opinion regarding fractal dimension methods, PDE yielded reasonable results with reasonable amounts of data. Given the complex nature of experimental and biological data, we conclude that it is necessary and feasible to complement PCA with methods that take into consideration the nonlinear properties of biological systems for a more robust estimation of their DOFs. Index Terms—Data analysis, degrees of freedom (DOFs), dimension estimation, fractal dimension, musculoskeletal synergies. I.
Chaotic time series Part I: Estimation of some invariant properties in state space
 Modeling, Identification and Control, 15(4):205  224
, 1995
"... Certain deterministic nonlinear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows constructing more realistic and better models and thus impro ..."
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Cited by 10 (5 self)
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Certain deterministic nonlinear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows constructing more realistic and better models and thus improved predictive capabilities. This paper provides a review of two main key features of chaotic systems, the dimensions of their strange attractors and the Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations known to display chaotic behaviour are used for illustration. A compilation of applications to real data from widely different fields is given. If chaos is found to be present, one may proceed to build nonlinear models, which is the topic of the second paper in this series.
Scalar observations from a class of highdimensional chaotic systems: Limitations of the time delay embedding
"... The time delay embedding for the reconstruction of a state space from scalar data introduces strong folding of the smooth manifold in which a chaotic attractor is embedded, which is absent in some more natural state space. In order to observe the deterministic nature of data, the typical length s ..."
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The time delay embedding for the reconstruction of a state space from scalar data introduces strong folding of the smooth manifold in which a chaotic attractor is embedded, which is absent in some more natural state space. In order to observe the deterministic nature of data, the typical length scale related to this folding has to be resolved. Above this length scale the data appear to be random. For a particular model class we prove these statements and we derive analytically the dependence of this length scale on the complexity of the system. We show that the number of scalar observations required to observe determinism increases exponentially in the product of the system's entropy and dimension. 1 Outline of the problem Nonlinear time series analysis, i.e. the interpretation and characterization of observed data with irregular, aperiodic time dependence in terms of lowdimensional chaotic motion, has proven to be very successful during the last years [1, 2, 5]. Many physic...
Early Detection of Breast Cancer using Self Similar Fractal Method
"... Breast cancer is one of the major causes of death among women. Small clusters of micro calcifications appearing as collection of white spots on mammograms show an early warning of breast cancer. Early detection performed on Xray mammography is the key to improve breast cancer diagnosis. Image segme ..."
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Breast cancer is one of the major causes of death among women. Small clusters of micro calcifications appearing as collection of white spots on mammograms show an early warning of breast cancer. Early detection performed on Xray mammography is the key to improve breast cancer diagnosis. Image segmentation consists in finding the characteristic entities of an image, either by their contours (edges) or by the region they lie in. Our aim in this paper is to present a method for medical image enhancement based on the well established concept of fractal derivatives and selecting image processing techniques like segmentation of an image with self similar properties. The concept of a fractal is most often associated with geometrical objects satisfying two criteria: selfsimilarity and fractional dimensionality. The method was tested over several images of image databases taken from BSR APPOLO for cancer research and diagnosis, India.
Estimating degrees of freedom in motor systems
 in arXiv:qbio.QM/ 0610058 Oct. 2007 [Online]. Available: http://arxiv.org
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