Masters Thesis Projects

PLEASE NOTE: If you are interested in doing your MSc project with me in Summer 2017 please contact me. However, I might be on research leave starting from some time next year, so we need to set up a plan B in case this happens. My availability should become clear by Spring 2017.

- Please see here for general requirements if you are interested to work with me on a project.
- The projects below are sorted according to the estimated degree of difficulty (easier -> more demanding)
- These are only some suggestions! What I usually do is to first discuss with you in order to find about what your interests are, and what your pre-knowledge is; and then to decide together with you about a project that hopefully fits your needs. This project will probably be in the spirit of the ones listed below, but the topic might in detail be quite different.
- If
you are willing to do a more computer-based project on chaotic
dynamical systems you would be particularly
welcome! This does not mean that these projects are purely numerically:
Typically, they will involve an interplay between simple computer
simulations of a chaotic dynamical system, and matching your numerical
results to analytical approximations that you have to derive. I have a
number of very interesting projects along these lines; please see below
for details.

- If you wish to get a taste of it, please see here for a full list of previous masters thesis projects that I have supervised, with the best theses being available as pdf-files; see also here for (some) previous collaborators of mine.

1. Deterministic chaos in the Bernoulli shift

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 project 4 below.

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 Ljapunov 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.

Prerequisites:

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 following courses:

- MTH6107 Chaos and Fractals
- MTH744U Dynamical Systems
- MTH743U Complex Systems

- 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 University Press, Cambridge, 1999)
- E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993)
- A. Lasota, M.C. Mackey, Chaos, Fractals, and Noise (Springer,
Berlin, 1994)

2. Deterministic chaos in the baker map

Style and difficulty:

Textbook-based, but requires to learn about some concepts that are more demanding than for project 1. Could again be purely analytical or in combination with computer work. Again, will be difficult, though not impossible, to score an A with it. If done well should be safe for an (upper) B.

Contents:

The baker map is perhaps the simplest two-dimensional map exhibiting chaotic behavior. Based on different tetxbooks, summarize important dynamical systems properties of this paradigmatic model such as its dynamical instability, ergodicity, mixing behavior, being a K-system, and being Bernoulli. This requires to calculate analytically Lyapunov exponents, dynamical entropies and to use symbolic dynamics. Finally, construct analytically the SRB measure for a dissipative baker's map. If you like, also discuss Arnold's cat map. You may support your analytical results by computer simulations,

Prerequisites:

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 following courses:

- MTH6107 Chaos and Fractals
- MTH744U Dynamical Systems
- MTH743U Complex Systems

- J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, Cambridge, 1999)
- V.I. Arnold, A. Avez, Ergodic problems of classical mechanics (W.A. Benjamin, New York, 1968)

- M.Toda, R.Kubo, N.Saito, Statistical Physics 1 (Springer, Berlin, 1992)
- A. Lasota, M.C. Mackey, Chaos, Fractals, and Noise (Springer, Berlin, 1994)
- T. Tel, M. Gruiz, Chaotic Dynamics (Cambridge University Press, Cambridge, 2006)

Style and difficulty:

If you are not sure what you are capable of doing research-wise, you may start this project like project 2 above. Then either continue with project 2, 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.

Prerequisites:

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 following courses:

- MTH6107 Chaos and Fractals
- MTH744U Dynamical Systems
- MTH743U Complex Systems

- J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, Cambridge, 1999)
- V.I. Arnold, A. Avez, Ergodic problems of classical mechanics (W.A. Benjamin, New York, 1968)

- A. Lasota, M.C. Mackey, Chaos, Fractals, and Noise (Springer, Berlin, 1994)
- S.Tasaki, T.Gilbert, 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)

4. Chaotic diffusion in deterministic Langevin dynamics

Style and difficulty:

If you are not sure what you are capable of doing research-wise, you may start this project like project 1 above. Then either continue with project 1, or dive into more demanding aspects according to this project as explained below.

Accordingly, the core of this project is based on simple and advanced textbooks, then moves on to research papers. Degree of difficulty can be tuned from simple to very demanding. Mostly analytical but may involve some easy computer work later on. For an excellent student, this should lead to some new research.

Contents:

If you want to be on the safe side, you may start like project 1 above by discussing basic chaos properties of the Bernoulli shift (Devaney's definition, Ljapunov exponents, ergodicity). Then explore the concept of Brownian motion by explaining what a Langevin equation is. Put this dynamics into context of stochastic theory by explaining what a Wiener process and an Ornstein-Uhlenbeck process is. This defines the textbook part of the project.

On this basis, study research articles by my colleague Prof.Christian Beck and summarize the idea of a deterministic Langevin equation. This dynamical system can be used to generate chaotic diffusion: describe what this concept means. If you are very clever, you can now start to do research on this type of system: Calculate the model's diffusion coefficient analytically by a so-called Takagi function technique. If you can, compare your analytical findings to results from own computer simulations.

Prerequisites:

This project consists of a mix between dynamical systems theory (70%) and basic concepts of stochastic theory (10%) and statistical mechanics (20%). Hence you should have interest and some working knowledge of this theory as provided by at least one of the following courses:

- MTH6107 Chaos and Fractals
- MTH744U Dynamical Systems
- MTH743U Complex Systems

- J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics, Chapter 14 (Cambridge University Press, Cambridge, 1999)
- R.Klages, Deterministic diffusion in one-dimensional chaotic dynamical systems (Wissenschaft und Technik Verlag, Berlin, 1996); Chapter 5 (Ph.D. thesis available on my homepage)
- C.Beck, Dynamical systems of Langevin type, Physica A 233, 419 (1996)
- see also a previous precursor MSc thesis; the research part of this thesis should be continued in more depth

5. Anomalous stochastic processes and fluctuation relations

Style and difficulty:

This is a new, more advanced project that is more closely related to research.

Contents:

This project requires some familiarity with stochastic processes, though not on a deep mathematical level, and the motivation to generally learn more about stochastic dynamics. You should start by explaining the concepts of a Langevin equation, anomalous diffusion, and so-called fluctuation relations. The first topic is about 100 years old, the second and third ones became very active fields of research over the past two decades. All of this material can be extracted from (advanced) textbooks and reviews, see Refs.[1,2] below. Then explain the idea of generalized Langevin dynamics by possibly touching upon fractional derivatives.

After this introductory part, try to roughly understand what has been done in the recent research paper of mine Ref.[3] below (see also [4], to some extent). Recalculate as much as possible what has been stated in Section 4 therein.

Hopefully there is some time left to now do some research in terms of new calculations. They may be performed by considering a slightly different version of the generalized Langevin equation studied in Ref.[3] below, checking for an anomalous fluctuation relation by using the very same kind of methods you have familiarized yourself with in the task before.

This is a nice project in this area of research that possibly could even be continued as a PhD project.

Prerequisites:

This project can be done purely analytically by using concepts of stochastic theory. Some familiarity with Fourier-Laplace transform techniques would be helpful but is not absolutely necessary. You should have interest and some working knowledge in stochastic processes, which relates to material covered by some of our UG and PG modules.

References:

- M.Toda, R.Kubo, N.Saito, Statistical Physics 1 (Springer, Berlin, 1992)
- R.Klages, A.V.Chechkin, P.Dieterich,
*Anomalous fluctuation relations*, book chapter in: R.Klages, W.Just, C.Jarzynski (Editors),*Nonequilibrium Statistical Physics of Small Systems*(Wiley-VCH, Weinheim, February 2013), p.259-282; ISBN 978-3-527-41094-1 [preprint as pdf-file] - A.V.Chechkin, F.Lenz, R.Klages,
*Normal and anomalous fluctuation relations for Gaussian stochastic dynamics*, J.Stat.Mech. L11001/1-13 (2012) (Letter) [link to journal|article as pdf-file] - A.V.Chechkin, R.Klages,
*Fluctuation relations for anomalous dynamics*, J.Stat.Mech. L03002/1-11 (2009) (Letter) [link to journal|article as pdf-file]

6. Anomalous diffusion in weakly chaotic dynamical systems

Style and difficulty:

This is a challenging project that goes mostly beyond textbooks. It is more based on scientific reviews and research papers. It can be performed purely analytically or in combination with computer work.

Although the main outcomes of this project are known, it poses a very challenging task that connects directly with active research. Hence, this is an ideal project for someone thinking of doing a PhD later on. If successful, it could immediately be expanded into a PhD project.

Contents:

Anomalous diffusion defines a very active field of current research. What does it mean to say that a system exhibits anomalous diffusion? Outline the basic idea. Show that anomalous diffusion can be generated by weakly chaotic dynamical systems: Introduce the so-called Pomeau-Manneville map and qualitatively discuss its intermittent dynamics. Explain the concept of weak chaos and argue that this map is weakly chaotic. By using this map, construct a deterministic model that exhibits subdiffusion. Calculate the anomalous diffusion coefficient of this model by continuous time random walk theory by explaining what this theory is about. This assumes familiarity with Fourier-Laplace transformations and the like. Explain the idea of fractional derivatives and derive a fractional diffusion equation for this model.

Prerequisites:

This project consists of a mix between dynamical systems theory (20%) and basic concepts of stochastic theory (70%) and statistical mechanics (10%). Hence you should have interest and some working knowledge of this theory as provided by at least one of the following courses:

- MTH6107 Chaos and Fractals
- MTH744U Dynamical Systems
- MTH743U Complex Systems

- R.Metzler, J.Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000)
- R.Klages,
*Weak chaos, infinite ergodic theory, and anomalous dynamics*, book chapter in: X.Leoncini and M.Leonetti (Eds.),*From Hamiltonian Chaos to Complex Systems*(Springer, Berlin, July 2013), p.3-42. - R.Klages,
*Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics*, monograph, Advanced Series in Nonlinear Dynamics Vol.24 (World Scientific, Singapore, 2007) - N. Korabel, R. Klages, A.V. Chechkin, I.M. Sokolov,
V.Yu. Gonchar, Fractal properties of anomalous diffusion in
intermittent maps, Phys. Rev. E
**75,**036213 (2007)

Further topics

Style and difficulty:

The following projects can be tuned from very basic to very demanding. I am happy to discuss details with you upon request.

- some very nice computer-based projects like

- computer simulation of dynamical instability in the Sinai billiard
- computer simulation of chaos in the bouncing ball problem

- computer simulation of chaos in the rotating disk model
- simulate probability densities in one-dimensional maps (very basic!)
- chaotic ratchets (combination of numerics and analytics)
- review fractional derivatives (stochastic theory)

last update: Nov. 2016