Caput Dynamical Systems and Chaos

Faculteit Science and Engineering
Jaar 2021/22
Vakcode WMMA004-05
Vaknaam Caput Dynamical Systems and Chaos
Niveau(s) master
Voertaal Engels
Periode semester I a

Uitgebreide vaknaam Caput Dynamical Systems and Chaos
Leerdoelen At the end of the course, the student is able to:

1. reproduce key properties and phenomenology of standard examples of dynamical systems, such as "the doubling map", "the Henon map", "the horseshoe map", "the solenoid", "circle maps", and relate the qualitative features of such examples to more complex systems.

2. apply key concepts, such as "Poincare maps", "suspensions", "symbolic dynamics", "conjugation", "multi-periodic dynamics", "chaotic dynamics", "dispersion exponents", "persistence of dynamical properties", "attractors", and "structural stability" to concrete examples of dynamical systems.

3. Explain the difference between a "regular" and a "singular perturbation", in particular in the context of ordinary differential equations.

4. Apply the concept of "normal hyperbolicity" to concrete examples of dynamical systems.

5. Apply the "geometric desingularization technique" via blow-up to concrete examples of dynamical systems.
Omschrijving Dynamical systems theory concerns the question of how deterministic systems evolve in time. This first of all concerns the longterm behaviour of systems which includes stationary, periodic, multi-periodic, and chaotic dynamics, but also transient behaviour is of interest. Moreover bifurcations or transitions between asymptotic states -- in particular transitions between regular and chaotic motions -- under variation of parameters are of great importance. Special stationary solutions or more generally invariant manifolds can also form the organizing centers for the dynamics in the state space of a dynamical system. Applications of dynamical systems theory range from molecular dynamics to celestial mechanics and extend to the life sciences, climate science, and many other fields.

The first half of the course will consist of lectures based on the text book by Broer and Takens, which will provide a solid background. The second half of the course is devoted to studying a topic in contemporary research on dynamical systems and their applications.

A trending and challenging issue in dynamical systems are singular perturbation problems. Here we will be motivated by singular perturbations that arise when considering dynamics with multiple time scales. Mathematical models involving multiple time scales are relevant in neuroscience, biology, engineering, economy, among many others. In the second half of this course we will provide the basic theory of deterministic dynamical systems with two or more time scales, and learn to distinguish regular from singular problems. We will also discuss a geometric technique that is suitable to analyse such singular problems. This part of the course is based on the books by Kuehn and Wechselberger.
Uren per week
Onderwijsvorm Hoorcollege (LC), Opdracht (ASM)
(The students will give presentations during the tutorials)
Toetsvorm Opdracht (AST)
(The final grade is computed as (A1 + A2) / 2, where A1 and A2, are the grades for the two take-home exam assignments.)
Vaksoort master
Coördinator dr. A.E. Sterk
Docent(en) dr. H. Jardon Kojakhmetov ,dr. A.E. Sterk
Verplichte literatuur
Titel Auteur ISBN Prijs
Dynamical Systems and Chaos.
(Note: a cheaper edition of this book may be obtained from the Dutch publisher Epsilon Uitgaven!) 2011
Henk Broer and Floris Takens 978-1-4614-2712-4 €  68,00
Multiple Time Scale Dynamics

(Note: students may want to check if the book can be consulted online via SpringerLink and the UG library.) 2015
Christian Kuehn 9978-3-319-34417-1 €  50,00
Geometric Singular Perturbation Theory Beyond the Standard Form

(Note: students may want to check if the book can be consulted
online via SpringerLink and the UG library.) 2020
Martin Wechselberger 978-3-030-36398-7 €  50,00
Entreevoorwaarden The course unit assumes prior knowledge acquired from an introductory course to dynamical systems theory like the compulsory course Dynamical Systems in the bachelor curriculum.
Opmerkingen The material will be tailor-made for the projects to be carried out in groups.
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