Note:
* I will not offer any masters projects in the academic year 2010/2011, because I will be on sabbatical leave.
*
This list needs to be updated! Projects 1, 2 and 5 have been done,
new projects have to be added. This page thus gives you an indication
of what kind of masters projects I have to offer. For further examples
please see the list of the 9 masters projects that have been done with
me over the past few years. An update
will follow in due time.
1. The projects below are sorted
according to the estimated degree of difficulty (easier -> more
demanding)
2. 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 preknowledge is; and then to jointly 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.
1. * booked * Bifurcations in deterministic dynamical
systems (textbook-based,
perhaps easy computer
simulations)
Review the Feigenbaum (period doubling) bifurcation in the
logistic map, perhaps supplementing your presentation by own results
from computer simulations. Then describe further types of bifurcations
and related phenomena (tangent bifurcations, inverse period doubling,
crisis). If you wish, discuss also basic bifurcations in
higher-dimensional dynamical systems such as Hopf and saddle node
bifurcations.
Prerequisite courses:
- Chaos and Fractals and/or Introduction to Dynamical Systems
References (basically any
introductory textbook on dynamical systems):
- Edward Ott, Chaos in Dynamical Systems (Cambridge Univ
Press, 1993)
- Robert L. Devaney, An Introduction to Chaotic Dynamical Systems
(Westview Press, 2003)
- Kathleen T. Alligood, Tim D. Sauer, James A. Yorke, Chaos (Springer, 1996)
2. * booked * Fractals and fractal dimensions (textbook-based, purely analytical)
What is a fractal? Review definitions of
fractal dimensions and related quantities, which apply to fractal
objects, particularly box counting, Hausdorff
dimension, information dimension, correlation dimension, Renyi
dimensions, perhaps Hurst exponents. Illustrate your discussion by
simple
examples of fractal objects.
Prerequisite courses:
- Chaos and Fractals and/or Introduction to Dynamical Systems
References:
- Edward Ott, Chaos in Dynamical Systems (Cambridge Univ
Press, 1993)
- Kathleen T. Alligood, Tim D. Sauer, James A. Yorke, Chaos (Springer, 1996)
- Kenneth Falconer, Fractal Geometry (Wiley, New York, 1990)
- Benoit M. Mandelbrot, The fractal geometry of nature (Freeman,
New York, 1977)
- Peitgen, Juergens, Saupe, Chaos and Fractals (Springer, Berlin,
1992)
3. The Bernoulli shift (textbook-based, but requires to learn
about some advanced concepts; purely analytical)
Many fundamental
concepts of dynamical systems theory can be exemplified by the
Bernoulli shift. Summarize the 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 system. Discuss Pesin's theorem and the
so-called escape rate formula, which both establish relations between
ergodic properties and dynamical instability, and verify these formulas
for the Bernoulli shift.
Prerequisite courses:
- Chaos and Fractals and/or Introduction to Dynamical Systems
References:
- J.R. Dorfman, An Introduction to Chaos in Nonequilibrium
Statistical Mechanics (Cambridge University Press, 1999)
- Edward Ott, Chaos in Dynamical Systems (Cambridge Univ
Press, 1993)
- A. Lasota, M.C. Mackey, Chaos, Fractals, and Noise (Springer,
1994)
4. The baker map (textbook-based,
but requires to learn
about some more complicated concepts)
The baker map is one of
the simplest dynamical systems 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.
Prerequisite courses:
- Chaos and Fractals and/or Introduction to Dynamical Systems
References:
- J.R. Dorfman, An Introduction to Chaos in Nonequilibrium
Statistical Mechanics (Cambridge University Press, 1999)
- V.I. Arnold, A. Avez, Ergodic problems of classical mechanics
- M.Toda, R.Kubo, N.Saito, Statistical Physics 1 (Springer, 1992)
- A. Lasota, M.C. Mackey, Chaos, Fractals, and Noise (Springer,
1994)
- T. Tel, M. Gruiz, Chaotic Dynamics (Cambridge, 2006)
5. * booked * (however, a similar new project is still available) Chaotic diffusion and
fractal
functions (advanced
textbooks, perhaps some research papers, mostly analytical but may
involve some easy computer work)
Briefly
outline the idea of chaotic diffusion and define the concept of a
diffusion coefficient. Then derive the Taylor-Green-Kubo formula for
diffusion in maps. By using this formula, calculate the diffusion
coefficient for a lifted Bernoulli shift. It is obtained in terms of
the famous Takagi function, which is a continuous but nowhere
differentiable fractal that can be calculated by solving a functional
recursion relation. Consider a generalized, parameter-dependent version
of the lifted Bernoulli shift and try to calculate the diffusion
coefficient again by the same method (this is new, I have not done
this). Calculate the fractal dimension of all (generalized) Takagi
functions involved (also new).
Prerequisite courses:
- Chaos and Fractals and/or Introduction to Dynamical Systems
References:
- J.R. Dorfman, An Introduction to Chaos in Nonequilibrium
Statistical Mechanics, Chapter 14 (Cambridge University Press, 1999)
- R.Klages, Deterministic diffusion in one-dimensional chaotic
dynamical systems, Chapter 5 (Ph.D. thesis available on my homepage)
- Kenneth Falconer, Fractal Geometry (Wiley, New York, 1990)
6. What is anomalous diffusion? (goes mostly beyond textbooks: reviews and
research papers, purely analytical work)
Anomalous diffusion defines a
very
active field of current research.
What
does it mean to say that a system is anomalously diffusive? Outline the
basic idea. Introduce a simple subdiffusive map and qualitatively
discuss its dynamics. Then 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. - Although the results of this project are known, it poses
a very challenging task that connects directly with active research.
Prerequisite courses:
- Chaos and Fractals and/or Introduction to Dynamical Systems
References:
7. Further topics (more advanced; details upon request)
- Periodic orbit theory for
deterministic diffusion
- Introduction to thermodynamic formalism
- 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
last update: 25 February 2010