Dynamical systems

Duration

Two Weeks, Period 1

Lecturer

Arno Berger (University of Canterbury, New Zealand)

Consultation Hours

Thursdays 12–13h Carslaw Room 526.

Assessment

Two assignments, one each week.

Assumed Knowledge

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.

Course Outline

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) include:

• 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.

References

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 Systems.