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A statistical analysis of the two-point vorticity statistics in the inverse energy cascade of two-dimensional turbulence is presented in terms of probability density functions (PDFs). Evolution equations for the PDFs are derived in the framework of the Lundgren–Monin–Novikov hierarchy, and the unclosed terms are studied with the help of direct numerical simulations (DNS). Furthermore, the unclosed terms are evaluated in a Gaussian approximation and compared to the DNS results. It turns out that the statistical equations can be interpreted in terms of the dynamics of screened vortices. The two-point statistics is related to the dynamics of two point vortices with screened velocity field, where an effective relative motion of the two point vortices originating from the turbulent surroundings is identified to be a major characteristics of the dynamics underlying the inverse cascade.

The formation of regular stripe patterns during the transfer of surfactant monolayers from water surfaces onto moving solid substrates can be understood as a phase decomposition process under the influence of the effective molecular interaction between the substrate and the monolayer, also called substrate-mediated condensation (SMC). To describe this phenomenon, we propose a reduced model based on an amended Cahn-Hilliard equation. A combination of numerical simulations and continuation methods is employed to investigate stationary and time-periodic solutions of the model and to determine the resulting bifurcation diagram. The onset of spatiotemporal pattern formation is found to result from a homoclinic and a Hopf bifurcation at small and large substrate speeds, respectively. The critical velocity corresponding to the Hopf bifurcation can be calculated by means of the marginal stability criterion for pattern formation behind propagating fronts. In the regime of low transfer velocities, the stationary solutions exhibit snaking behavior.

A new method is proposed that allows a reconstruction of time series based on higher order multi-scale statistics given by a hierarchical process. This method is able to model financial time series not only on a specific scale but for a range of scales. The method itself is based on the general n-scale joint probability density, which can be extracted directly from given data. It is shown how based on this n-scale statistics, general n-point probabilities can be estimated from which predictions can be achieved. Exemplary results are shown for the German DAX index. The ability to model correctly the behaviour of the original process for different scales simultaneously and in time is demonstrated. As a main result it is shown that this method is able to reproduce the known volatility cluster, although the model contains no explicit time dependence. Thus a new mechanism is shown how, in a stationary multi-scale process, volatility clustering can emerge.

The clinical diagnosis of epileptic seizures is usually based on an inspection of the electroencephalogram (EEG). The present paper is devoted to a description of a more refined analysis of EEG patterns of petit-mal epilepsy. The treatment is based on the synergetic approach to macroscopic patterns in complex systems which has been inaugurated by H. Haken. This approach aims at an understanding of both spatial as well as dynamical aspects of macroscopic EEG patterns.

We present a stochastic analysis of a data set consisiting of 10^6 quotes of the US Doller - German Mark exchange rate. Evidence is given that the price changes x(tau) upon different delay times tau can be described as a Markov process evolving in tau. Thus, the tau-dependence of the probability density function (pdf) p(x) on the delay time tau can be described by a Fokker-Planck equation, a gerneralized diffusion equation for p(x,tau). This equation is completely determined by two coefficients D_{1}(x,tau) and D_{2}(x,tau) (drift- and diffusion coefficient, respectively). We demonstrate how these coefficients can be estimated directly from the data without using any assumptions or models for the underlying stochastic process. Furthermore, it is shown that the solutions of the resulting Fokker-Planck equation describe the empirical pdfs correctly, including the pronounced tails., Comment: 29 pages, 19 eps figures, misprints corrected, under consideration for publication in Physica A

We present measurements of conditional probability density functions (PDFs) that allow one to systematically bridge from Eulerian to Lagrangian statistics in fully developed 3D turbulence. The transition is investigated for hydro- as well as magnetohydrodynamic flows and comparisons are drawn. Significant differences in the transition PDFs are observed for these flows and traced back to the differing coherent structures. In particular, we address the problem of an increasing degree of intermittency going from Eulerian to Lagrangian coordinates by means of the conditional PDFs involved in this transformation. First simple models of these PDFs are investigated in order to distinguish different contributions to the degree of Lagrangian intermittency.

We investigate the relationship between Eulerian and Lagrangian probability density functions obtained from numerical simulations of two-dimensional as well as three-dimensional turbulence. We show that in contrast to the structure functions of the Lagrangian velocity increment δτv(y)=u(x(y, τ), τ)- u(y, 0), where u(x, t) denotes the Eulerian velocity and x(y, t) the particle path initially starting at x(y, 0)=y, the structure functions of the velocity increment δτw(y)=u(x(y, τ), τ)- u(y, τ) exhibit a wide range of scaling behavior. Similar scaling indices are detected for the structure functions for particles diffusing in frozen turbulent fields. Furthermore, we discuss a connection to the scaling of Eulerian transversal structure functions.

Systems far from equilibrium exhibit the spontaneous emergence of spatial, temporal, and spatio-temporal patterns by the mechanism of self-organization [1], [2], [3]. Recently, considerable interest has been addressed to the study of systems which form coherent spatial structures undergoing a complex and chaotic temporal evolution. Examples may be found e.g, in the fields of hydrodynamic instabilities or chemical reactions. It has been recognized that similar phenomena can also be observed in biological systems. The present paper is devoted to the examination of spatio-temporal patterns connected with the electrical activity of the brain measuredin the form of electroencephalograms (EEG).

We present an analysis of the spatio-temporal patterns connected with the electrical activity of the brain which is measurable at the human scalp in the form of electroencephalograms (EEGs). In the case of the so-called Alpha activity the spatio-temporal EEG patterns show large scale spatial coherence which, however, undergoes a complex temporal evolution. We discuss the change of the spatio-temporal patterns during an epileptic seizure. Based on the synergetic theory of pattern formation in nonequilibrium systems we present a model of the Alpha EEG rhythm which reproduces the observed main spatial as well as temporal features.

We investigate the interaction of self-organized solitary current filaments (dissipative solitons) under the influence of noise in a planar dc gas-discharge system with high-ohmic barrier. These localized structures exhibit interesting properties such as propagation and scattering as well as the transient formation of molecules and clusters. To quantitatively examine these interesting properties, methods of stochastic data analysis are developed to determine interaction laws from experimentally recorded data series. It turns out that in most cases, the sign of the resulting interaction functions alternates with the distance of the dissipative solitons. This phenomenon can be connected to experimentally detectable oscillatorily decaying tails of the filaments, which is in agreement with theoretical predictions.

We report on the experimental detection of a drift bifurcation for dissipative solitons, which we observe in the form of current filaments in a planar semiconductor–gas-discharge system. By introducing a new stochastic data analysis technique we find that due to a change of system parameters the dissipative solitons undergo a transition from purely noise-driven objects with Brownian motion to particles with a dynamically stabilized finite velocity.

We present a stochastic analysis of a data set consisting of 1.25 × 107 samples of the local velocity measured in the turbulent region of a round free jet. We find evidence that the statistics of the longitudinal velocity increment v(r) can be described as a Markov process. This new approach to characterize small-scale turbulence leads to a Fokker–Planck equation for the r-evolution of the probability density function (p.d.f.) of v(r). This equation for p(v, r) is completely determined by two coefficients D1(v, r) and D2(v, r) (drift and diffusion coefficient, respectively). It is shown how these coefficients can be estimated directly from the experimental data without using any assumptions or models for the underlying stochastic process. The solutions of the resulting Fokker–Planck equation are compared with experimentally determined probability density functions. It is shown that the Fokker–Planck equation describes the measured p.d.f.(s) correctly, including intermittency effects. Furthermore, knowledge of the Fokker–Planck equation also allows the joint probability density of N increments on N different scales p(v1, r1, …, vN, rN) to be determined.

We report on spiral type convection patterns obtained from a numerical treatment of a generalized Swift-Hohenberg equation.

A modification of mean-field type models for dynamo action in helical turbulent fluids by the inclusion of local fluctuations of the growth rate of the unstable large scale magnetic field modes exhibits an intermittent temporal behaviour of the magnetic field. The fluctuating dynamo models offer a possible explanation of the various periods of minimal activity of the sun's magnetic field as a result of the very nature of its turbulent generation mechanism.

The order parameter equation for the oscillatory instability of a spatially homogeneous state is used to examine traveling wave patterns in large aspect ratio systems with rectangular as well as circular geometries.

For the first time we give an explanation of the mechanism underlying the generation of modulated traveling wave states in nonequilibrium systems, frequently denoted as ‘blinking state’. The blinking state is generated by two nonlinearly interacting oscillatory modes with slightly different eigenfrequencies. Frequency locking between these modes generate spatial patterns experimentally known as ‘confined states’.

The onset of convection in a fluid contained between a hot inner and a cold outer spherical boundary in a radially symmetric gravity field is examined. Due to the spherical symmetry there are usually (2l+1) unstable modes belonging to the lth irreducible representation of the O(3) group. The actual value of l is determined by a properly defined aspect ratio. There are special values of this aspect ratio where two groups of modes simultaneously become unstable. The instability involving the two groups of modes with l=1 and l=2 is considered by investigating the corresponding generalized Ginzburg-Landau equations. By varying the Rayleigh number and the aspect ratio a variety of interesting spatio-temporal behavior is observed. The occurring static, time periodic, and quasiperiodic fluid motions as well as the various types of chaotic fluid flow are examined in detail.

We derive the generalized Ginzburg-Landau equation for the case of an oscillatory instability of a spatially homogeneous state in systems whose geometry is characterized by two entirely different length scales. This evolution equation is applied to describe the spatio-temporal behaviour of the onset of convection in binary fluid mixtures in large aspect ratio systems. We obtain time periodic traveling wave motions, quasiperiodic fluid motions with two and more frequencies modulating the intensities of the traveling waves as well as chaotic temporal behaviour.

Traveling wave patterns formed at the oscillatory onset of convection in binary mixtures with free slip and permeable horizontal boundary conditions are theoretically derived by means of an order parameter equation for the case of large aspect ratio systems with circular and rectangular geometry. We obtain in addition to ‘Zipper states’ the so called ‘confined states’, which so far has been observed only experimentally. Our present study of traveling wave patterns in circular systems, which are the first theoretical investigations in this respect, exhibit a characteristic change of behaviour. Close to onset the waves travel in radial direction towards the sidewalls. For higher Rayleigh-numbers the waves are confined to the circular boundary traveling in azimuthal direction. The occurrence of this transition should be confirmed experimentally.

We derive the exact stationary probability distribution for the coupled system of Langevin equationsdtu=ɛu−u s,dts=−γs+d2+F(t).

It is by now well known that chaotic temporal behaviour may be generated by completely deterministic evolution laws. The main theoretical interest in chaotic phenomena has been devoted to the examination of the stochastic properties of low dimensional dynamical systems, although experimental confirmations of certain theoretical predictions have been mainly pursued in spatially extended systems, e.g. in lasers, hydrodynamic flows, and chemical reactions. In such systems low dimensional temporal chaotic behaviour evidently has to be strongly interrelated with coherent spatial patterns. The success of synergetics in the explanation of spontaneous pattern formation in selforganizing systems allows a straightforward treatment of time-dependent and chaotic behaviour in spatially extended systems of finite size which is caused by the nonlinear interaction of several order parameters, each connected with a coherent spatial pattern. In these cases a complete examination of the interrelations between temporal disorder and spatial coherence becomes possible. Spatial patterns are naturally connected to the symmetries of a system. Therefore, it can be expected that quite different systems, which, however, possess the same spatial symmetries, may show comparable behaviour. To demonstrate this we shall consider instabilities in systems with 0(3)-symmetry. For the special case of mode interaction between two groups of modes belonging to two different irreducible representations of the 0(3)-group with l=1 and l=2 we shall derive the order parameter equations.

The present article gives a review on the basic concepts of synergetics and their applications to nonequilibrium phase transitions in physical, chemical and biological systems and discusses the synergetic approach to the problem of pattern recognition.

Synergetics is the study and analysis of complex systems which are composed of many subsystems. These systems show the ability to selforganize spontaneously on macroscopic scales, i.e. we may observe highly ordered spatial, temporal, or spatio-temporal structures on scales which are much larger than the corresponding characteristic scales of the subsystems. As a result, we conclude that there must be very many subsystems coherently involved in order to produce these highly regular states on characteristic scales of the composed system. Synergetics offers a unified viewpoint to the emergence of such patterns and the underlying processes. Furthermore, powerful mathematical methods have been developed in this field to understand and predict the spontaneous occurrence of order, qualitatively as well as quantitatively.

Under well-defined conditions systems with many degrees of freedom like lasers, plasmas and fluids show irregular behaviour in their time evolution. These chaotic motions are usually produced by few degrees of freedom only and the basic question arises how to determine these relevant variables and to describe their dynamics. It turns out that the occurrence of low-dimensional chaos in complex systems can qualitatively be explained by using the concepts of synergetics [1], [2]. The degrees of freedom are drastically reduced due to the slaving principle, i. e. the temporal behaviour of the system is governed by the dynamics of only few order parameters. We applied these ideas to a specific hydrodynamic system. This system especially has the advantage that all steps and conclusions of the above principle can be accomplished explicitly

In this paper we wish to analyze a multichannel alpha EEG recording from a normal person. In these EEG measurements, time series (Fig. 1) were derived from various locations at the skull as schematically shown in Fig. 2.

von Rudolf Magenau, Magenau: Philosophiestudent, Beilage: Gedicht "Vor dem Genuß des h. Abendmals" (Nr. 189a)

Brief aus Bern, ETH-Bibliothek

Fetscherin, Stämpfli, Verfasser auf Seite 16 genannt: "Bern, den 21. Dezember 1836. Der Präsident der Außer-Krankenhausdirektion: Fetscherin. Der Sekretär: Stämpfli."

Fetscherin, Verfasser auf Seite 16 genannt: "Bern, den 23. November 1836. Der Präsident der Außer-Krankenhausdirektion: Fetscherin. Der Sekretär: Stämpfli."

von J.R.F. Ith

Fetscherin, Stämpfli, Verfasser auf Seite 7 genannt: "Bern, den 28. Mai 1838. Der Präsident der Inseldirektion: Fetscherin. Der Sekretär: Stämpfli."

http://tartu.ester.ee/record=b2448003~S1*est