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
References:
- 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
References:
- 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)
3. Fractal measures in the dissipative baker map
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
References:
- 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
References:
- 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
References:
- 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