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H Kantz

Publications and source records attributed to H Kantz.

14 recordsLinked to original sources
PubMed2001

Invariant densities of delayed maps in the limit of large time delay.

The marginal invariant density of chaotic attractors of scalar systems with time delayed feedback has an asymptotic form in the limit of large delay. It is well known that the dimension and the entropy of such attractors obey interesting scaling laws in this limit, but very little has been said about properties of the invariant density. We present general considerations, detailed analytical results in low order perturbation theory for a particular model, and numerics for understanding the asymptotic behavior of the projections of the invariant density. Our approach clarifies how the analytical properties of the model determine the behavior of the marginal invariant densities for large delay times.

Journal Article↗
PubMed2001

Indispensable finite time corrections for Fokker-Planck equations from time series data.

The reconstruction of Fokker-Planck equations from observed time series data suffers strongly from finite sampling rates. We show that previously published results are degraded considerably by such effects. We present correction terms which yield a robust estimation of the diffusion terms, together with a novel method for one-dimensional problems. We apply these methods to time series data of local surface wind velocities, where the dependence of the diffusion constant on the state variable shows a different behavior than previously suggested.

Journal Article↗
PubMed2001

Reconstruction of the parameter spaces of dynamical systems.

Parameter variations in the equations of motion of dynamical systems are identified by time series analysis. The information contained in time series data is transformed and compressed to feature vectors. The space of feature vectors is an embedding for the unobserved parameters of the system. We show that the smooth variation of d system parameters can lead to paths of feature vectors on smooth d-dimensional manifolds in feature space, provided the latter is high-dimensional enough. The number of varying parameters and the nature of their variation can thus be identified. The method is illustrated using numerically generated data and experimental data from electromotors. Complications arising from bifurcations in deterministic dynamical systems are shown to disappear for slightly noisy systems.

Journal Article↗
PubMed2001

Periodic orbits and topological entropy of delayed maps.

The periodic orbits of a nonlinear dynamical system provide valuable insight into the topological and metric properties of its chaotic attractors. In this paper we describe general properties of periodic orbits of dynamical systems with feedback delay. In the case of delayed maps, these properties enable us to provide general arguments about the boundedness of the topological entropy in the high delay limit. As a consequence, all the metric entropies can be shown to be bounded in this limit. The general considerations are illustrated in the cases of Bernoulli-like and Hénon-like delayed maps.

Journal Article↗
PubMed2000

Coping with nonstationarity by overembedding

We discuss how nonstationarity in observed time series data due to pronounced fluctuations of system parameters can be resolved by making use of embedding techniques for scalar data. If a D-dimensional deterministic system is driven by P slowly time dependent parameters, a (D+P)-dimensional manifold has to be reconstructed from the scalar time series, which is done by an m>2(D+P)-dimensional time delay embedding. We show that in this space essential aspects of determinism are restored. We demonstrate the validity of the idea heuristically, for numerical examples and for human speech data.

Journal Article↗
PubMed2000

Denoising human speech signals using chaoslike features.

A local projective noise reduction scheme, originally developed for low-dimensional stationary deterministic chaotic signals, is successfully applied to human speech. This is possible by exploiting properties of the speech signal which resemble structure exhibited by deterministic dynamical systems. In high-dimensional embedding spaces, the strong inherent nonstationarity is resolved as a sequence of many different dynamical regimes of moderate complexity.

Humans↗
PubMed2000

Local estimates for entropy densities in coupled map lattices

We present a method to derive an upper bound for the entropy density of coupled map lattices with local interactions from local observations. To do this, we use an embedding technique that is a combination of time delay and spatial embedding. This embedding allows us to identify the local character of the equations of motion. Based on this method we present an approximate estimate of the entropy density by the correlation integral.

Journal Article↗
PubMed2000

Analysis of vocal disorders in a feature space.

This paper provides a way to classify vocal disorders for clinical applications. This goal is achieved by means of geometric signal separation in a feature space. Typical quantities from chaos theory (like entropy, correlation dimension and first lyapunov exponent) and some conventional ones (like autocorrelation and spectral factor) are analysed and evaluated, in order to provide entries for the feature vectors. A way of quantifying the amount of disorder is proposed by means of a healthy index that measures the distance of a voice sample from the centre of mass of both healthy and sick clusters in the feature space. A successful application of the geometrical signal separation is reported, concerning distinction between normal and disordered phonation.

Algorithms↗
PubMed2000

Relation between coupled map lattices and kinetic ising models

A spatially one-dimensional coupled map lattice possessing the same symmetries as the Miller-Huse model is introduced. Our model is studied analytically by means of a formal perturbation expansion which uses weak coupling and the vicinity to a symmetry breaking bifurcation point. In parameter space four phases with different ergodic behavior are observed. Although the coupling in the map lattice is diffusive, antiferromagnetic ordering is predominant. Via coarse graining the deterministic model is mapped to a master equation which establishes an equivalence between our system and a kinetic Ising model. Such an approach sheds some light on the dependence of the transient behavior on the system size and the nature of the phase transitions.

Journal Article↗
PubMed2000

Chaos or noise: difficulties of a distinction

In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set, it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension. These restrictions limit our ability to distinguish between signals generated by different systems, such as regular, chaotic, or stochastic ones, when analyzed from a time series point of view. We propose to classify the signal behavior, without referring to any specific model, as stochastic or deterministic on a certain scale of the resolution epsilon, according to the dependence of the (epsilon,tau) entropy, h(epsilon, tau), and the finite size Lyapunov exponent lambda(epsilon) on epsilon.

Journal Article↗
PubMed1999

Improved false nearest neighbor method to detect determinism in time series data.

The false nearest neighbor method introduced by Kennel et al. [Phys. Rev. A 45, 3403 (1992)] is revisited and modified in order to ensure a correct distinction between low-dimensional chaotic data and noise. Still, correlated noise processes can yield vanishing percentages of false nearest neighbors for rather low embedding dimensions and can be mistaken for deterministic signals. Therefore, the false nearest neighbors method has always to be combined with a surrogate data test.

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