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Instructors: Dr. Ivan Yaroslavtsev
Event type:
Lecture
Displayed in timetable as:
WP-DGLDyn-V
Hours per week:
4
Language of instruction:
German
Min. | Max. participants:
- | -
Comments/contents:
Processes with a time axis can be conceived as dynamical systems and treated with the methods of this broad field. Dynamical systems theory also has applications in pure mathematics, such as number theory, and seamlessly transitions into virtually every natural science, especially physics.
The focus in this lecture is on mathematically rigorous theory, but we also look at examples of applications and software use. Is the dynamics regular, stable, periodic, chaotic? How does the behavior depend on parameters? Basic concepts of the theory are elaborated and the generation of dynamical systems from differential equations is studied.
The dynamical systems treated here are abstractly a mapping together with a set (phase space) and a group operation as a time axis, which can be continuous (real numbers) or discrete (integers).
In the discrete case, important classes are interval mappings and iterated function systems, leading to fractals. For the continuous case, theory of ordinary differential equations is needed and repeated and extended from the perspective of dynamical systems: Existence and uniqueness (theorems of Picard-Lindelöf and Peano), stability (theorems of Floquet and Liouville), planar systems (theorem of Poincaré-Bendixson), local bifurcations (fold, pitchfork, Hopf bifurcation), Hamiltonian systems (nonlinear pendulums). Various examples of model equations from physics, chemistry and biology are considered, and numerical simulations are used for illustration. At the end, the Lorenz system is considered as an example of complicated, ``chaotic'' dynamics in ODE.
Literature:
- M. Denker: Einführung in die Analysis dynamischer Systeme. Springer, 2005.
- Jan Prüss, Mathias Wilke, Gewöhnliche Differentialgleichungen und dynamische Systeme, Birkhäuser 2010
- B. Hasselblatt & A. Katok: A First Course in Dynamics. Cambridge University Press, 2003.
- A. Katok & B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1995.
- V.I. Arnold, Ordinary Differential Equations, Springer, Berlin, 1992.
- J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
- J. Hale, Ordinary Differential Equations, Krieger, Malabar, 1980.
- Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Systems and an introduction to chaos. , Elsevier 2004.
- Clark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, 1995.
- Clark Robinson, Introduction to Dynamical Systems: Discrete and Continuous, Prentice Hall, New York, 2004.
Additional examination information:
Exercises:
- Regular and active participation and attendance required
Module final exam:
- Dates: Written exam Do 25.07.2024 9-11 Uhr; Fr 27.09.2024 13-15
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