Maths Project Repository Database

Dr Rainer Klages

Style and difficulty:

• Mostly textbook-based, but requires to learn about some advanced concepts. Could be purely analytical or in combination with computer work. Rather easy and straightforward, so if you're doing a reasonable job, you should be on track for scoring a B. But please note that with a textbook-based project it will be difficult to go for an A! If you wanted to do so, think of the project Chaotic diffusion in deterministic Langevin dynamics.

Contents:

• Many fundamental concepts of dynamical systems theory can be explored by studying the dynamics of simple one-dimensional maps. A famous example is the Bernoulli shift. Start by stating Devaney's definition of chaos, explain what it means, and apply it to the Bernoulli shift. Then summarize dynamical systems properties of this simple model by focusing on Lyapunov exponents, ergodic properties and dynamical entropies. Define these different concepts and apply them to this and related models. Discuss Pesin's theorem and the so-called escape rate formula, which both establish relations between ergodic properties and dynamical instability. Again verify these formulas for the Bernoulli shift and related models.

• R.L. Devaney, An introduction to chaotic dynamical systems. Addison-Wesley, Reading, 1989.
• J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics. Cambridge Univ. Press, 1999.
• E. Ott, Chaos in Dynamical Systems. Cambridge Univ. Press, 1993.
• A. Lasota and M.C. Mackey, Chaos, Fractals, and Noise. Springer, Berlin, 1994.

MTH6107 Chaos and Fractals

MTH744U/MTH744P Dynamical Systems

MTH743U/MTH743P Complex Systems

This is a project about dynamical systems theory, hence you should have interest and some working knowledge of this theory as provided by at least one of the modules above.