M. Siefert

Extracting model equations from experimental data

This letter wants to present a general data-driven method for formulating suitable model equations for nonlinear complex systems. The method is validated in a quantitative way by its application to experimentally found data of a chaotic electric circuit. Furthermore, the results of an analysis of tremor data from patients suffering from Parkinsons disease, from essential tremor, and from normal subjects with physiological tremor are presented, discussed and compared. They allow a distinction between the different forms of tremor.

On a quantitative method to analyze dynamical and measurement noise

This letter reports on a new method of analysing experimentally gained time series with respect to different types of noise involved, namely, we show that it is possible to differentiate between dynamical and measurement noise. This method does not depend on previous knowledge of model equations. For the complicated case of a chaotic dynamics spoiled at the same time by dynamical and measurement noise, we even show how to extract from data the magnitude of both types of noise. As a further result, we present a new criterion to verify the correct embedding for chaotic dynamics with dynamical noise.

Comment on “Indispensable Finite Time Corrections for Fokker-Planck Equations from Time Series Data”

A Comment on the Letter by Mario Ragwitz and Holger Kantz, Phys. Rev. Lett. 87, 254501 (2001). The authors of the Letter offer a Reply.

RECONSTRUCTION OF THE DETERMINISTIC DYNAMICS OF STOCHASTIC SYSTEMS

We show that based on the mathematics of Markov processes and particularly based on the definition of Kramers–Moyal coefficients, it is possible to estimate the deterministic part of the dynamics for a broad class of nonlinear noisy systems. In particular, we show that for different kinds of noise perturbations, including non-Langevin force with finite correlation time and independent measurement noise, the deterministic part can be reconstructed.

Fat Tail Statistics and Beyond

Based on data from three different systems, namely, turbulence, financial market and surface roughness we discuss methods to analyze their complexities. Scaling analysis and fat tail statistics in the context of Levy distributions are compared with a stochastic method, for which a Fokker-Planck equation can be estimated from data. We show that the last method provides a more detailed characterization of complexity.

A Simple Relation Between Longitudinal and Transverse Increments

The understanding of the complex statistics of fully developed turbulence in detail is still an open problem. One of the central points is to understand intermittency, i.e. to find exceptionally strong fluctuations on small scales. In the last years, the intermittency in different directions has attracted considerable interest. It has been controversial whether there are significant differences in intermittency between the different directions. More specifically one looks at the statistics of increments [u(x + r) − u(x)] e, i.e. at the projection of the differences between two velocities separated by the vector r in a certain direction e. Here we denotes longitudinal increments with u, for which r and e are parallel and transverse increments with v for which r is perpendicular to e.