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Complex Systems: Computable Chaos in Dynamical Systems

Lecturer Prof. Dr. Ruedi Stoop
Time

Spring semester

Mittwoch 10:45-12:30 Vorlesung
Mittwoch 12:45-13:30 Übungen

Place ETH Hönggerberg, HIT

Beschreibung:

Die Vorlesung bietet eine grundlegende Einführung in chaotische Systeme, welche keinerlei Abstriche an mathematischer Exaktheit macht. Sie umfasst einerseits in ansprechender Tiefe die klassischen theoretischen Gesichtspunkte der dynamischen Systeme, wobei alle wesentlichen Beispiele der Literatur ausführlich behandelt werden. Daneben werden modernere Fragestellungen behandelt, etwa nach der Natur der Berechenbarkeit oder der Verlässlichkeit des Computers. Zu den Kernphänomenen werden kurze, aber vollständige Programme in der Programmiersprache Mathematica abgegeben, welche leicht zu verstehen sind und für das individuelle Experimentieren übernehmbar sind. Biographien von historischen Schlüsselpersönlichkeiten bereichern die Vorlesung.

Description:

Chaos 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.