227-1030-00L  Complex Systems: Computable Chaos in Dynamical Systems

SemesterSpring Semester 2015
LecturersR. Stoop
Periodicityyearly recurring course
Language of instructionGerman


AbstractIntroduction to the theory of both discrete and continuous dynamical systems: Detailed description of the theoretical concepts, simulations in Mathematica, applications from electronics to celestial mechanics.
ObjectiveChaos in dynamical systems is due to a nonlinearity contained in the system. This severly limits the applicability of the more traditional linear analysis tools to predict the behavior of the system. In the course, we introduce the mathematical tools that allow, the prediction of the system behavior, despite its chaotic nature.
With the help of the concepts of Lyapunov exponents, fractal dimensions, invariant density, and the Frobenius-Perron approach, we will achieve predictions on the horizon of predictability, the distribution of states, the possibility of reliably simulating such systems on the computer, and the changes such systems undergo when systems parameters change.
From the technical aspects, the lectures equally focus on analytical as well as on on numerical approaches. All essential aspects of the lectures are exemplified by means of distributed programs written in the simulation environment Mathematica, for which we provide a short introduction.
The lectures aim at providing a basic set of systems for which the origins of the complex behavior are well understood, from the theoretical as well as from the practical viewpoints and will enable the appropriate analysis of new systems, which is critical to today's science and technology.
ContentThe lectures provide a basic introduction into chaotic systems, where no compromise in the mathematical exactness of the treatment is made.
The lectures comprise an in-depth treatment of the classical foci on dynamical systems and include all basic examples from the literature. Additional foci relate to questions like the computability of such systems as well as the reliability of computers.

The fundamental phenomena are exemplified by short, complete, computer programs, written in the programming environment Mathematica, which allow for an easy understanding and experimentation.
Bibliographies of key scientific protagonists are also included.
Lecture notesA detailed script is provided.
LiteratureAdditional and supplementary literature:

R. Stoop und W.H. Steeb, Berechenbares Chaos in Dynamischen Systemen, Birkhäuser 2006.
A. Lasota and M.C. Mackey, Chaos, fractals, and noise : stochastic aspects of dynamics, Springer 1995