Current research activities and interests of Rainer
Hegger
Nonlinear time series analysis is concerned with the description of
(typically nonlinear) systems by knowing information from measurements,
only. Usually, univariate time series are given. Nevertheless, it is
possible to reconstruct the original phase space from the 'scalar'
information of the time series. This is achieved by means of the
Takens embedding theorem. It turns out that nonlinear time series
analysis is quite a powerful tool if some conditions are
fulfilled. One of the most severe restrictions is a strong 'correlation'
of the allowed dimensionality of the system and the number of data at
hand. From this 'correlation' follows that even well controlled
laboratory systems have to have a dimension smaller than about 5. This
requirement strongly restricts the applicability of nonlinear time
series analysis.
My main interest is to work out ideas and methods which allow to
extend the applicability of nonlinear time series analysis. In this
realm I work on:
Analysing spatially extended,
chaotic systems by means of local measurements
There are quite a lot of spatially extended systems. Maybe the most
prominent one is the weather, which is a system spanning the whole
earth. A local measurement would be the temperature in Frankfurt/Main
(Germany). One could ask whether it is possible to forecast the
weather in New York by just knowing the temperature in Frankfurt. This
sounds absurd if one knows that the meteorologists run the biggest
computers to do the forecasts. However, if the weather is a stricly
deterministic system, it could be possible, in principle, if one had
enough data. Unfortunately, the amount of data would be so huge that
the computers one needed to analyse the data had to be even bigger
(maybe exponentially bigger) than the ones used by the
meteorologists. But what happened if we knew the temperature in
Frankurt and in Berlin and in Paris and in ... The temperature of how
many cities do we need to know to do something meaningful?
Analysing open systems
A more general question is how to analyse open systems by means of
data analysis. An open system is a system which is not only determined
by its internal (autonomous) dynamics, but it is coupled to an
environment (or a heat bath). In principle, this definition holds for all real
systems. But for some systems the influence of the environment is
relevant for others it is not. If you study the short time dynamics of
a LASER, you can neglect the environment, since the time scales of both
parts are well separated. This can be wrong, already, if you look at
the long term behaviour of the same LASER.
Depending on whether one can measure the coupling to the environment
or not, one can speak of input-output or of noisy systems,
respectively.
The Tisean project
Beside the above mentioned projects we try to improve the Tisean software
package in such a way that it will contain more and more useful
programs with less and less bugs.
The work listed on this page is done in collaboration with