Author

# H. G. Schuster

Bio: H. G. Schuster is an academic researcher from University of Kiel. The author has contributed to research in topics: Series (mathematics) & Attractor. The author has an hindex of 7, co-authored 7 publications receiving 758 citations.

##### Papers

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TL;DR: In this article, the first minimum of the logarithm of the generalized correlation integral C1(τ) provides an easily evaluable criterion for the proper choice of the time delay τ that is needed to reconstruct the trajectory in phase space from chaotic scalar time series data.

Abstract: It is shown that the first minimum of the logarithm of the generalized correlation integral C1(τ) provides an easily evaluable criterion for the proper choice of the time delay τ that is needed to reconstruct the trajectory in phase space from chaotic scalar time series data.

310 citations

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TL;DR: A new method is introduced to obtain optimal delay coordinates for data from chaotic dynamic systems by determining simultaneously the minimal necessary embedding dimension as well as the proper delay time to achieve optimal reconstructions of attractors.

Abstract: Guided by topological considerations, a new method is introduced to obtain optimal delay coordinates for data from chaotic dynamic systems. By determining simultaneously the minimal necessary embedding dimension as well as the proper delay time we achieve optimal reconstructions of attractors. This can be demonstrated, e.g., by reliable dimension estimations from limited data series.

195 citations

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TL;DR: This article generalized the Grassberger and Procaccia correlation integral method to yield the whole spectrum of dimensions and entropies from a measured time series with a numerical effort which is only insignificantly larger than that needed to determine the original correlation integral.

Abstract: The correlation-integral method of Grassberger and Procaccia is generalized to yield the whole spectrum of dimensions ${D}_{q}$ and entropies ${K}_{q}$ from a measured time series with a numerical effort which is only insignificantly larger than that needed to determine the original correlation integral. It is shown that our method yields reliable numerical results for the tent map and for the Mackey-Glass equation.

182 citations

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TL;DR: It is demonstrated that the method works and provides reliable values of the fractal dimensions for systems that are described by maps or differential equations and for real experimental data.

Abstract: We present a method that allows one to decide whether an apparently chaotic time series has been filtered or not. For the case of a filtered time series we show that the parameters of the unknown filter can be extracted from the time series, and thereby we are able to reconstruct the original time series. It is demonstrated that our method works and provides reliable values of the fractal dimensions for systems that are described by maps or differential equations and for real experimental data.

41 citations

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TL;DR: In this paper, the most dominant unstable periodic orbits are extracted from a measurement of a time-continuous flow and used to fit models of the flow and to predict the orbit.

Abstract: A hierarchical approximation of a generic chaotic attractor can be formulated in terms of unstable periodic orbits. We demonstrate the possibility of extracting the most dominant unstable periodic orbits from a measurement of a time-continuous flow. Since unstable periodic orbits not only represent the static properties of the system but also dominate the dynamics, they can be used to fit models of the flow and to predict the orbit. We show that predictions for the Roessler system using unstable periodic orbits extracted from a time series of moderate length are significantly better than those from other approaches.

37 citations

##### Cited by

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08 Feb 1999TL;DR: Support vector machines for dynamic reconstruction of a chaotic system, Klaus-Robert Muller et al pairwise classification and support vector machines, Ulrich Kressel.

Abstract: Introduction to support vector learning roadmap. Part 1 Theory: three remarks on the support vector method of function estimation, Vladimir Vapnik generalization performance of support vector machines and other pattern classifiers, Peter Bartlett and John Shawe-Taylor Bayesian voting schemes and large margin classifiers, Nello Cristianini and John Shawe-Taylor support vector machines, reproducing kernel Hilbert spaces, and randomized GACV, Grace Wahba geometry and invariance in kernel based methods, Christopher J.C. Burges on the annealed VC entropy for margin classifiers - a statistical mechanics study, Manfred Opper entropy numbers, operators and support vector kernels, Robert C. Williamson et al. Part 2 Implementations: solving the quadratic programming problem arising in support vector classification, Linda Kaufman making large-scale support vector machine learning practical, Thorsten Joachims fast training of support vector machines using sequential minimal optimization, John C. Platt. Part 3 Applications: support vector machines for dynamic reconstruction of a chaotic system, Davide Mattera and Simon Haykin using support vector machines for time series prediction, Klaus-Robert Muller et al pairwise classification and support vector machines, Ulrich Kressel. Part 4 Extensions of the algorithm: reducing the run-time complexity in support vector machines, Edgar E. Osuna and Federico Girosi support vector regression with ANOVA decomposition kernels, Mark O. Stitson et al support vector density estimation, Jason Weston et al combining support vector and mathematical programming methods for classification, Bernhard Scholkopf et al.

5,506 citations

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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.

Abstract: Recurrence is a fundamental property of dynamical systems, which can be exploited to characterise the system's behaviour in phase space. A powerful tool for their visualisation and analysis called recurrence plot was introduced in the late 1980's. This report is a comprehensive overview covering recurrence based methods and their applications with an emphasis on recent developments. After a brief outline of the theory of recurrences, the basic idea of the recurrence plot with its variations is presented. This includes the quantification of recurrence plots, like the recurrence quantification analysis, which is highly effective to detect, e. g., transitions in the dynamics of systems from time series. A main point is how to link recurrences to dynamical invariants and unstable periodic orbits. This and further evidence suggest that recurrences contain all relevant information about a system's behaviour. As the respective phase spaces of two systems change due to coupling, recurrence plots allow studying and quantifying their interaction. This fact also provides us with a sensitive tool for the study of synchronisation of complex systems. In the last part of the report several applications of recurrence plots in economy, physiology, neuroscience, earth sciences, astrophysics and engineering are shown. 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. We therefore detail the analysis of data and indicate possible difficulties and pitfalls.

2,993 citations

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TL;DR: A new method for calculating the largest Lyapunov exponent from an experimental time series is presented that is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level.

Abstract: Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.

2,942 citations

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TL;DR: Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field as mentioned in this paper, and many tools have been developed for the analysis of such data.

Abstract: Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements. They discuss methods for (1) separating the signal of physical interest from contamination ("noise reduction"), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the effects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. While much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. It is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.

1,691 citations

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TL;DR: A practical method to determine the minimum embedding dimension from a scalar time series that has the following advantages: does not contain any subjective parameters except for the time-delay for the embedding.

Abstract: A practical method is proposed to determine the minimum embedding dimension from a scalar time series. It has the following advantages: (1) does not contain any subjective parameters except for the time-delay for the embedding; (2) does not strongly depend on how many data points are available; (3) can clearly distinguish deterministic signals from stochastic signals; (4) works well for time series from high-dimensional attractors; (5) is computationally efficient. Several time series are tested to show the above advantages of the method.

1,485 citations