- Two Weeks, Period 1
- Arno Berger (University of Canterbury, New Zealand)
- Thursdays 12–13h Carslaw Room 526.
- Two assignments, one each week.
- A good working knowledge in introductory real analysis and linear algebra
is essential. Some prior exposure to (ordinary) differential equations as well as to fundamental
concepts (such as e.g. convergence, continuity, compactness, completeness, connectedness) in the
context of metric spaces would be beneficial.
- This course provides a concise introduction to dynamics, with an emphasis on
geometric and topological aspects. Guided by illustrative examples throughout, we will study
how apparently simple systems can exhibit complex and unpredictable (“chaotic”) long-time
behaviour, and we will develop the mathematical terminology and tools necessary to describe
and quantify this complexity in various ways. Topics discussed (with varying level of detail)
- stability, instability, bifurcations;
- signatures of chaos: horseshoes, symbolic dynamics, entropy;
- equivalence of systems, classification, robustness;
- topological dynamics: transience, recurrence, expansivity.
Depending on time and interest, we may also have a first look at some more advanced topic, e.g. Conley
index, multiple recurrence, and Furstenberg’s diophantine theorem.
- Highly readable introductions to dynamics are Devaney: An Introduction to Chaotic Dynamical
Systems and Hasselblatt & Katok: A First Course in Dynamics; the course will considerably
overlap with these textbooks. Most of the material covered (and much more) can be found in
the more advanced texts Brown: Ergodic Theory and Topological Dynamics, Irwin: Smooth
Dynamical Systems, and Katok & Hasselblatt: Introduction to the Modern Theory of Dynamical