Maths Project Repository Database

Dr Rainer Klages

Style and difficulty:

• If you are not sure what you are capable of doing research-wise, you may start this project like the project Deterministic chaos in the Baker map. Then either continue with the project above, or dive into more demanding aspects according to this project as explained below. The core of this project is based on simple and advanced textbooks, then moves on to research papers. The degree of difficulty can be tuned from simple to very demanding. This project starts with analytical basics that need to be reviewed, the research part requires simple computer simulations. For an excellent student, this should lead to some new research.

Contents:

• The baker map is perhaps the simplest two-dimensional map exhibiting chaotic behavior. Start with a very brief summary of basic dynamical systems properties of this model by particularly reviewing the concept of an invariant probability measure. Then consider a slight variant of the original map, which is the dissipative baker map. Discuss basic dynamical systems properties of this model. Construct analytically the fractal SRB measure of this map and verify your results by computer simulations. Finally, compute numerically the measure by projecting onto an arbitrary direction in phase space. Try to verify your numerical findings by working out a simple approximate analytical theory.

• J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics. Cambridge Univ. Press, 1999.
• V.I. Arnold and A. Avez, Ergodic problems of classical mechanics. W.A. Benjamin, New York, 1968.
• A. Lasota and M.C. Mackey, Chaos, Fractals, and Noise. Springer, Berlin, 1994.
• S. Tasaki, T. Gilbert, and J.R. Dorfman, An analytical construction of the SRB measures for Baker-type maps, Chaos 8 424 (1998).

More research articles to be provided in the course of this project.

MTH6107 Chaos and Fractals

MTH744U/MTH744P Dynamical Systems

MTH743U/MTH743P Complex Systems