Nonlinear Systems and Chaos

After serious doubts in the beginning of chaos research with respect to its "scientificality" until the 1990s this research area now, undoubtlessly, substantially contributes to a deeper understanding of the mechanisms of complex phenomena, in particular of living systems. Chaos research left its tarnished reputation to only produce mathematical beauty behind it. Of course, one can still say that chaos research and related areas are at the edge of mysticism (cf. Christoph Holzhey). So be it! If this is a disadvantage at all, in my opinion it does not matter as long as substantial insight into complex systems is generated. Particle physics with their numerous allusions to Buddhistic conceptions appears to me much closer to mysticism but nobody doubts the important contribution to knowledge.

Rapid-prototyping sculptur of the Rössler attractor by Florian Grond

An icon of chaos research is the Rössler attractor here depicted as rapid-prototyping form produced by Florian Grond. This chaotic attractor is named after my doctoral supervisor Otto E. Rössler who lives and works in Tübingen. An excellent introduction into chaos can be found at scholarpedia.org.

Lyapunov exponents

The so-called Lyapunov exponents are essential characteristics to describe the stability of dissipative systems. These entities are measures for the exponential divergence or convergence of neighbored trajectories of nonlinear systems. The importance of the Lyapunov exponents results from the fact that an exact measurement of the actual state is virtually impossible. In chaotic systems this uncertainty usually increases exponentially with time which is why predictability is reduced (a well-known example is the weather dynamics). This is why Lyapunov exponents are a particular useful measure for predictability.

We work on the improvement of the numerics for the estimation of these important entities since it has turned out that commonly used methods are insufficient. The upper figure to the right shows the so called positive manifold of the Rössler attractor. Within the attractor the local Lyapunov exponents are almost everywhere positive (red band) which is why a small area of states grows exponentially in time. Orthogonal to the attractor, in contrast, the exponents are almost everywhere negative (blue band in the lower figure) which is why the attractor attracts points from the outside. I am grateful to Florian Grond for his collaboration in this project in form of a final year project.

Florian Grond, Hans H. Diebner, Sven Sahle, Adolf Mathias, Sebastian Fischer and Otto E. Rossler: A robust, locally interpretable algorithm for Lyapunov exponents. Chaos, Solitons & Fractals 16, 841-852 (2003). (pdf)

Florian Grond and Hans H. Diebner: Local Lyapunov Exponents for Dissipative Continuous Systems. Chaos, Solitons & Fractals 23, 1809-1817 (2005). (pdf)

Chaotic Itinerancy

On the occasion of a short term research exchange at the Hokkaido University of Sapporo from January 21 - March 29, 2005, I dealt with the "Chaotic Itinerancy" approach to brain dynamics by Ichiro Tsuda. This led to the following common publication:

Hans H. Diebner and Ichiro Tsuda: Fundamental Interfaciology: Indistinguishability and Time's Arrow. In: Michel Petitjean (Ed.): Proceedings of FIS2005 - The Third Conference on the Foundations of Information Science. MDPI online (2005). (pdf)

Environmental Systems Sciences

On the occasion of a visiting professorships each summer semester at the University of Graz since the summer semester 2004, nonlinearity in ecological and population modelling moved into the focus of my studies. See the list of useful links to population dynamics sites (in German).

Chaos Control and Synchronisation

One of the most important applications of chaos research is chaos control and synchronisation. Control processes play important roles with respect to both technological as well as bio-ecological applications. In a sense, nature performs a kind of communication with the aid of synchronisation mechanisms of nonlinear systems. A technological application is possible, for example, in the field of reducing fluctuations in combustion engines. In fact, those fluctuations have a deterministic-chaotic part, which can be significantly reduced by means of chaos control. In the context of chaos control I also would like to refer to our research into cognitive systems modelling. Please also confer:

Kazuhiro Matsumoto, Hans H. Diebner, Ichiro Tsuda, and Yukiharu Hosoi: Application of Chaos Theory to Engine Systems. SAE International 2008.