Module
- Mathematische Methoden der analytischen Mechanik
| Di. |
07:30-09:00 (Vorlesung) |
H3/125 |
online |
| Do. |
15:00-16:30 (Vorlesung/Übung) |
H3/326 |
online |
- Tutorium (Analysis 2)
| Mi. |
13:00-15:00 (Tutorium) |
AE24/SR1 |
Office Hours (during teaching semesters)
Currently on Tuesdays 09:00-10:45 in U69-H3-426.
If you prefer online or another time please contact me by email for an
appointment.
Projects (Bachelor, Master, PhD)
I think, topics for projects cannot be taken
off the shelf. Quite the contrary, a project should be tailored according
to the needs and interests of the particular candidate.
For further details please contact me:
Wolfram Just, Room U69-H3-426,
wolfram.just@uni-rostock.de
If of interest here a list of potential topics with brief descriptions:
PhD topics
- Transfer operator methods in dynamical systems
For a class of one-dimensional expanding maps we relate the two fundamental
notions in the theory of dynamical systems: sensitivity to initial conditions (quantified
by Lyapunov exponent or entropy) and mixing (measured via decay of correlation
functions). More precisely, for piecewise linear expanding Markov maps on the interval
observed via piecewise analytic functions, we show that the Lyapunov exponent
provides a barrier to the exponential rate of mixing, by establishing a lower bound on
the subleading eigenvalue of the transfer operator.
Motivated by the question whether a similar bound in terms of the Lyapunov
exponent can be obtained in the nonlinear setting, we construct a family of expanding
maps for which the entire spectrum of the associated transfer operator is explicitly
known. Considered on the interval, these maps provide counterexamples to an old conjecture
on the reality of spectra.
These examples belong to a special class of circle maps arising from finite Blaschke
products. Their analytic features allow us to determine the entire spectrum of the
associated transfer operators (on spaces of holomorphic functions) in terms of multipliers
of attracting fixed points. This is achieved by deriving a natural representation of
the respective adjoint operators in terms of certain composition operators. Using this
explicit spectral information we then obtain examples of nonlinear expanding interval
maps with arbitrarily fast exponential mixing but bounded Lyapunov exponent.
- Piecewise linear stochastic systems
Piecewise-smooth stochastic systems are widely used in engineering science. However,
the theory of these systems is only in its infancy. We take as a motivation the model
of Brownian motion with dry friction to illustrate some dynamical
properties of these systems. We show
that the weak-noise approximation of the path integral correctly reproduces the known
propagator of the SDE at lowest order in the noise power, as well as the main features of
the exact propagator with higher-order corrections, provided that the singularity of the
path integral is treated with some heuristics. We also consider a smooth regularisation
of this piecewise-constant SDE and study to what extent this regularisation can rectify
some of the problems encountered in the non-smooth case. In addition, we provide analytic
solutions to the FPT problem of the model of Brownian motion with dry friction, using
two different but closely related approaches which are based on eigenfunction
decompositions on the one hand and on the backward Kolmogorov equation on the other. For the
pure dry friction case, a phase transition phenomenon in the spectrum is found which
relates to the position of the exit point and affects the tail of the FPT distribution.
For the model with dry and viscous friction the impact of the corresponding stick-slip
transition and of the transition to ballistic exit is evaluated quantitatively.
- Dynamics of triangular billiards
Polygonal billiards represent one of the simplest examples of systems with anomalous
dynamics. So far, they have resisted numerous attempts to fully describe their dynamical
behaviour. There is still a lack of complete understanding of even some basic features,
such as ergodicity or the decay of correlations. We study the dynamical
properties of triangular billiards using numerical means. We highlight the importance of
the billiard table geometry, more specifically symmetry, for the resulting dynamics. We
show that while typical triangular billiards appear to show correlation decay as expected
by the community, symmetric billiards may not even be ergodic with respect to the uniform
distribution in phase space. We provide compelling evidence that symmetry plays
a decisive role in the dynamics.
- Extended dynamical mode decomposition
Extended dynamic mode decomposition (EDMD) provides a class of
algorithms to identify patterns and effective degrees of freedom in
complex dynamical systems. We show that the modes identified by EDMD
correspond to those of compact Perron-Frobenius and Koopman operators
defined on suitable Hardy-Hilbert spaces when the method is applied to
classes of analytic maps. Our findings elucidate the interpretation of the
spectra obtained by EDMD for complex dynamical systems. We illustrate
our results by numerical simulations for analytic maps.
- Time delay dynamics
The finite propagation speed of signals causes time delays
in transmitting signals and forces. These effects become relevant
in fast dynamical systems and render the traditional description in terms
of differential equations invalid. In contrast the motion is captured by
equations with time delay, or in general by functional differential equations.
On the one hand such equations pose a considerable theoretical challenge since the
phase space becomes infinite dimensional, while on the other hand
the analysis of these dynamical systems huge potential for applications.
In particular theoretical and numerical bifurcation analysis gives insight
into the impact of time delay on control and synchronisation with applications
e.g. for establishing secure communication channels.
-
A. Amann, E. Schöll, and W. Just;
Some basic remarks on eigenmode expansions of time-delay dynamics,
Physica A 373 (2007) 191-202
-
W. Just, B. Fiedler, M. Georgi, V. Flunkert, P. Hövel, and
E. Schöll;
Beyond the odd number limitation: a bifurcation analysis
of time-delayed feedback control, Phys. Rev. E 76 (2007) 026210.
-
H. Erzgräber and W. Just;
Global view on a generic nonlinear oscillator
subject to time-delayed feedback control,
Physica D 238 (2009) 1680-1687.
-
C. von Loewenich, H. Benner, and W. Just;
Experimental verification of Pyragas-Schöll-Fiedler control,
Phys. Rev. E 82 (2010) 036204.
General topics (predominantly Master and BSc projects)
- Bifurcation analysis of oscillator systems
The simplest example of an oscillator is a pendulum which can be described by a linear second order differential equation. Such linear systems are fairly well understood and can be studied using ideas from linear algebra. If one considers oscillators where nonlinear behaviour becomes important, and which are described by nonlinear systems of differential equations the motion becomes more complex. Depending on the parameters of the system the motion may change considerably and instabilities may occur. The project aims at investigating such changes in dynamical behaviour, a subject which is known in mathematics under the notion of bifurcation theory. The project can be given either theoretical and rigorous flavour by summarising results on various types of bifurcations, or a more applied flavour by studying systems of differential equations by numerical means.
- Chaos
The term chaos refers to dynamical behaviour which looks like a random process, while the underlying system does not contain any random component. A main feature of chaos is the so called sensitivity on initial conditions ("butterfly effect"), i.e., that small changes may trigger dramatic changes. Chaos can be found in systems which are as basic as one-dimensional maps, where one can on the one hand uncover universal behaviours, and on the other hand may even apply analytical tools such as symbolic dynamics to study chaotic behaviour from a more rigorous perspective. The project aims at studying the chaotic dynamics of simple models either by analytical or by computational means.
- Financial Time Series Analysis
Predicting and forecasting market evolution is one of the holy grails in economics and financial mathematics. Within this project we look at time series data of the electricity system price of the Nord Pool spot market. A particular feature of electricity prices is an their inherent periodic component which reflects the variation of electricity consumption during the day. To cope with such trends we employ an idea developed in the context of signal processing where one uses concepts from complex variables and Fourier transforms to discount for such periodicities by computing a time dependent amplitude and a corresponding phase. The project then will look at a statistical analysis of these two quantities (amplitude and phase) to uncover hidden structures of electricity price fluctuations. Of particular interest is to understand whether extreme price fluctuations are related with particular time instances, such as rush hours or busy days within the week, and to uncover correlations in the time series as for instance those induced by electricity consumption as well as to uncover the relevance of non-stationary features. To predict and explain extreme events, i.e., massive price fluctuations we look at the time intervals between such extreme events which display an intermittent pattern. Dynamical systems theory predict a certain distribution of such time intervals (so called on-off intermittency) and the projects aims at testing such an hypothesis by computing the histogram of time intervals between extreme events.
- Football data analysis
Mathematical tools, such as those introduced in Statistics, Graph Theory, and Network Analysis can be used to understand the performance of football teams and to uncover the strength and weaknesses of players and team managers' strategies. Nowadays data sets are publicly available which can serve as an input for the mathematical analysis of football matches. The project aims at applying various mathematical data analysis tools to study the performance of teams and players for different leagues and competitions.
- Fractals and iterated function systems
Which kind of mathematics can be used to describe e.g. the shape of a tree or a leave, and how is that related to computer generated images used e.g. in contemporary video games. The answer can be found to some extent in modules such as Chaos and Fractals where a mathematical concept, so called iterated function systems, are introduced to generate fractal sets which share some properties with natural shapes. The theoretical aspect of the project aims at a deeper understanding and classification of fractal sets and iterated function systems, while the applied aspect aims at using iterated function systems to obtain computer generated images which resemble natural shapes.
- Surveys and Field Studies
Data collection is one of the key issue in any Applied Statistics task. The project aims at addressing a question of your choice by collecting relevant data, e.g., within a survey, and to perform a statistical data analysis to test various research hypothesis. Examples are for instance career opportunities of graduates from different higher education institutions, or the impact of Covid-19 on travel behaviour.
- Swarm Formation
The motion of flock of birds, a school of fish, or a herd of horses are examples of swarm behaviour, a peculiar example of collective motion. Surprisingly there are general mathematical mechanisms which support the formation of swarms. Their study is the main interest of this topic, and the results can then be used as well in a technological setting, for instance, by optimising the motion pattern of robots.
The most fundamental model of swarm formation is the so called Vicsek model which is a simple illustration of agent based modeling. Each agent obeys simple realistic movement rules which finally result by self-organisation in swarm formation. The project studies the theoretical background of this model and provides the opportunity to illustrate swarm formation with numerical simulations.
- Synchronisation and the Kuramoto model
Synchronisation is a ubiquitous phenomenon in science and engineering. Without synchronisation your mobile phone would not work, and in fact, you would be dead as your heartbeat needs to synchronised with your breathing, the so called, cardio-respiratory synchronisation. In a simple form synchronisation can be nicely illustrated by the coherent flashing of fireflies. Such counter-intuitive behaviour can be explained and modeled with the help of some basic and fundamental mathematical concepts. Within the project synchronisation phenomena are analytically studied using a simple model of phase coupled oscillators developed by Kuramoto. Conditions for the existence and the stability of the synchronised state will be obtained by methods developed, e.g., in dynamical systems theory.
- The German Tank Problem
Estimating the size of a population can be a matter of life or death. The tanks produced by Germany in WWII had equipment with a consecutive serial number, i.e., tanks were essentially labeled by integers. The Allies knew about this practise and they were interested to get a good estimate about the number of tanks produced. They had serial numbers from captured or destroyed tanks, i.e., they had a random sample of serial numbers. Given such an information what would be a suitable estimate for the number of tanks produced?
Problems of this type, which appear frequently and which are summarised under the notion of population size estimates, are at the heart of Statistics, when the setup is cast into a proper mathematical language. Within this project we are going to illustrate and apply various basic and advanced Statistics concepts, such as unbiased estimation, hypothesis testing, maximum likelihood estimation, to determine the size of a population from a random sample. In addition, within this setup one can illustrate as well the fundamentals of Statistics, i.e., the distinction between Bayesians and frequentists. The project can be given a theoretical flavour which does not involve any data analysis or computing, or it can be cast into an applied data analysis project using, e.g., the statistics software R.
- Time Delay Dynamics
A large class of dynamical systems can be mathematically modeled by differential equations, for instance the equations of motion in mechanics. Such an approach fails when the propagation time of signals becomes important, e.g., in contemporary communication technology, where the speed of light becomes a relevant quantity in fast signal processing networks. Such propagation effects result in equations of motions where the right hand side contains a time delay. Historically the first example of such delay equations occurred in the context of the balancing problem, i.e., when studying the impact of the physiological delay on the ability of humans to balance a stick (or even their own body). Within this project we study the simplest installment of delay equations, i.e., linear equations where the analysis can be done to a good deal by analytical methods. Among others this analysis helps to understand how time delay impacts on the stability of motion.
- Traffic Jams and the Nagel Schreckenberg model
Traffic jams are not just annoying, they do cost the economy an enormous amount of money. jams occur of course, e.g., during rush hour but they are not limited to car traffic. jams may occur as well, e.g., on the internet, in production lines, or even on your DNA (affecting your life at a very fundamental level). Hence understanding and modeling traffic jams from a mathematical perspective has far reaching consequences. We will analyse a simple car traffic model, the so called Nagel-Schreckenberg model, to obtain some insight into the formation of traffic jams. The project uses tools from Analysis, Combinatorics, and Probability. Depending on your interests it is possible to base the project also on numerical simulations and programming.