The analysis courses of the first and second year and some linear algebra. `Analyse 3' (ordinary differential equations) can be seen as an important preparatory course. An equivalent background in calculus-like courses should also be sufficient.
There are various kinds of dynamical systems: discrete maps, smooth, finite dimensional, ordinary differential equations, and infinite dimensional systems such as partial, functional or stochastic differential equations. This introductory course focuses on the second type, dynamical systems generated by ordinary differential equations. However, the ideas developed in this course are central to all types of dynamical systems. First, some fundamental concepts -- asymptotic stability by linearization, topological conjugacy, omega-limit sets, Poincaré maps -- are introduced, building on a basic background in the field of ordinary differential equations. Next, the existence and character of invariant manifolds -- that play an essential role in the theory of dynamical systems -- will be considered. This will give a starting point for the study of bifurcations. Finally, the concept of `chaos' will be discussed, mostly through the definition and basic properties of Lyapunov exponents.
The field of dynamical systems is driven by the interplay between `pure' mathematics and explicit questions and insights from `applications' -- ranging from (classical) physics and astronomy to ecology and neurophysiology. This is also reflected in the way this course will be taught: it will be a combination of developing mathematical theory and working out explicit example systems.
Remarks:
This course can be seen as a basic ingredient of the program chosen by a student who intends to specialize on analysis. However, it also is a relevant subject for students whose main interests lie in geometry, stochastics or numerical mathematics.
More explicitly, this course can be seen as a natural preparation for the courses `Introduction to Asymptotic Analysis', `Bifurcations and Chaos', and several national master courses (such as `Partial Differential Equations').
Week 37
Week 38
Week 39
Week 40 No lectures
Week 41
Assignment Set 2
Deadline: 26 October
Week 42
Week 43
Week 44
Assignment
Set 3
Deadline: 23 November
Week 45 No lectures.
Week 47
Week 48
The explicit construction of a stable manifold (5.3).
Week 49
Week 50 (last week)