Holger Kantz

Practical implementation of nonlinear time series methods: The TISEAN package

We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature. (c) 1999 American Institute of Physics.

A robust method to estimate the maximal Lyapunov exponent of a time series

Abstract A very simple method to compute the maximal Lyapunov exponent of a time series is introduced. The algorithm makes use of the statistical properties of the local divergence rates of nearby trajectories. It does not depend explicitly on the knowledge of the correct embedding dimension or on other parameters.

On noise reduction methods for chaotic data

Recently proposed noise reduction methods for nonlinear chaotic time sequences with additive noise are analyzed and generalized. All these methods have in common that they work iteratively, and that in each step of the iteration the noise is suppressed by requiring locally linear relations among the delay coordinates, i.e., by moving the delay vectors towards some smooth manifold. The different methods can be compared unambiguously in the case of strictly hyperbolic systems corrupted by measurement noise of infinitesimally low level. It was found that all proposed methods converge in this ideal case, but not equally fast. Different problems arise if the system is not hyperbolic, and at higher noise levels. A new scheme which seems to avoid most of these problems is proposed and tested, and seems to give the best noise reduction so far. Moreover, large improvements are possible within the new scheme and the previous schemes if their parameters are not kept fixed during the iteration, and if corrections are included which take into account the curvature of the attracting manifold. Finally, the fact that comparison with simple low-pass filters tends to overestimate the relative achievements of these nonlinear noise reduction schemes is stressed, and it is suggested that they should be compared to Wiener-type filters.

Nonlinear Analysis of Physiological Data

This book surveys recent developments in the analysis of physiological time series. The authors, physicists and mathematicans, physiologists and medical researchers, have succeeded in presenting a review of the new field of nonlinear data analysis as needed for more refined computer-aided diagnostics. Together with the techniques, they actually propose a new approach to the problems. The practitioners may find the many applications to the cardio-respiratory system, EEG analysis, motor control and voice signals very useful.

Repellers, semi-attractors, and long-lived chaotic transients

Abstract We study the chaotic transients observed in many deterministic systems. In general, they are related to strange repellers (or “semi-attractors”, if they are repelling in some and attracting in other directions). We propose formulas relating the average life time of the transient to dimensions of the repeller, and to Lyapunov exponents of the flow on it. The formulas are tested numerically in a number of cases.

Extreme events in nature and society

Extreme Events: Magic, Mysteries, and Challenges.- Extreme Events: Magic, Mysteries, and Challenges.- General Considerations.- Anticipating Extreme Events.- Mathematical Methods and Concepts for the Analysis of Extreme Events.- Dynamical Interpretation of Extreme Events: Predictability and Predictions.- Endogenous versus Exogenous Origins of Crises.- Scenarios.- Epilepsy: Extreme Events in the Human Brain.- Extreme Events in the Geological Past.- Wind and Precipitation Extremes in the Earths Atmosphere.- Freak Ocean Waves and Refraction of Gaussian Seas.- Predicting the Lifetime of Steel.- Computer Simulations of Opinions and their Reactions to Extreme Events.- Networks of the Extreme: A Search for the Exceptional.- Prevention, Precaution, and Avoidance.- Risk Management and Physical Modelling for Mountainous Natural Hazards.- Prevention of Surprise.- Disasters as Extreme Events and the Importance of Network Interactions for Disaster Response Management.

Recurrence time analysis, long-term correlations, and extreme events.

The recurrence times between extreme events have been the central point of statistical analyses in many different areas of science. Simultaneously, the Poincaré recurrence time has been extensively used to characterize nonlinear dynamical systems. We compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series. In particular, we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function, on the interval whereto the recurrences are observed, and on the temporal correlations of time series. In the case of long-term correlations, we verify the validity of the stretched exponential distribution, which is uniquely defined by the exponent gamma, at the same time showing that it is restricted to the class of linear long-term correlated processes. Simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different gamma, which shows a lack of invariance under the change of observables.

Dimension estimates and physiological data

Dimension estimates for data from physiological systems are notoriously difficult since the data are far from ideal in the sense of deterministic dynamical systems. Possible pitfalls and necessary precautions are pointed out and a recipe is given which is viable for those researchers who want to use the Grassberger-Procaccia algorithm but who are not familiar with the vast existing literature on dimension estimates. The relevance of dimension estimates for the characterization of physiological data is discussed, where both the cases of finding and not finding a low dimension are considered. (c) 1995 American Institute of Physics.

Modified method of simplest equation and its application to nonlinear PDEs

Abstract We search for traveling-wave solutions of the class of PDEs ∑ p = 1 N 1 A p ( Q ) ∂ p Q ∂ t p + ∑ r = 2 N 2 B r ( Q ) ∂ Q ∂ t r + ∑ s = 1 N 3 C s ( Q ) ∂ s Q ∂ x s + ∑ u = 2 N 4 D u ( Q ) ∂ Q ∂ x u + F ( Q ) = 0 where A p ( Q ) , B r ( Q ) , C s ( Q ) , D u ( Q ) and F ( Q ) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto–Sivashinsky equation, reaction–diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.

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.