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**Karsten Ahnert**

Institut für Physik und Astronomie

Universität Potsdam

Karl-Liebknecht-Strasse 24/25

14476 Potsdam-Golm

Germany

Tel. +49 331 9775986

# Time series analysis

## Nonparametric regression and reconstruction of dynamical systems

Nonparametric regression is tool for the determination of the functional dependency of multivariate data. In combination with embedding methods it can be used to reconstruct nonlinear dynamical systems from time series. As a result of this reconstruction on obtaines a low dimensional ordinary differential equation (ODE) and one can use this ODE for further analysis, e.g. Lyapunov exponents, entropies, spectras... We successfully applied these methods to standard nonlinear system, like the Lorenz and the Chua system as well as real world data sets. The plot on the right shows the Chua system and its reconstruction.

## Numerical differentiation of data

The estimation of derivatives from numerical data is a classical problem of data analysis with a wide range of application. In principal, all methods for numerical differentiation can be divided into local and global ones, which are usually related to regression and filtering problems. We introduced a measure, the smoothness, penelazing curvature of the obtained derivatived. With this measure it is possible to show, that global methods are superior the local ones, if one requires that the derivatives are smooth. The graphic shows four phase space plots of an organ pipe signal, obtained by numerical differentiation. The method is a spectral smoother with a butterworth filter and the plots are taken for different cutoff frequencies of the filter.

## Modeling of dynamical systems, application to chemical oscillators, accoustical systems

Modeling of dynamical systems is an general task in natural sciences. Basically, there are two complementary approaches to accomplish this task: theoretically, by convenient considerations, and empirically, by data analysis. Here we use an mixture of both approaches:we use regression methods to fit the data to a desired model and include prior knowledge to this model. For example, we considered the aero accoustical signal of an organ pipe. It is supposed, that it behaves as a self sustained nonlinear oscillator. So, we assumed an oscillator model and found the nonlinearity from nonparametric regression. The graphic on the right shows the power spectra of an organ pipe signal (red) and its the spectra of ist model (blue). The position and the amplitude of the harmonics coincide very well.

# Nonlinear lattices

## Nonlinear lattices, Hamiltonian lattices

We investigate lattices of nonlinear oscillators and hamiltonian lattices with local coupling. Such lattices can be found in numerous areas of natural sciences, e.g. in physical system like nonlinear wave guides, arrays of josephson junctions or granular medias, biological systems like neuron models oder ecological models. These systems exhibit many interesting phenomena depending on the system under consideration. In some lattice solitons and periodic waves are typical solutions, if the lattice consists of oscillatory units one can observe synchronization effects. Other lattices show localized states and quite often pattern forming processes and space time chaos can be found.

## Traveling waves in nonlinear lattices, i.e. Solitons and Compactons

Here, we consider phase oscillators and Hamiltonian lattice and observe various wave phenomenas, like solitons, periodic waves and kink. A special issue are compactons, a special type of solitary waves with compact superexponential decaying tails. We develop a method for the direct computation of the shape of the waves. Ohter methods for the investigation of traveling waves are simple fixed point analysis and the quasi continuum of the lattice. The plot on the left shows the evolution of an initial pulse into a train of compactons in a phase oscillator lattice with dispersive coupling.