Research/PhD Projects
Below is a summary of possible projects for PhD studies.
Extreme Value Thoery and Risk Analysis
For a sequence of random variables X_1,..X_n, extreme
value thoery (EVT) is a study of the limiting distristibution of
max(X_1,..X_n) (or some intermediate order) up to some
linear scaling. A possible research project is to study EVT in the context
of deterministic dynamical systems, where the sequence X_1,..,X_n can
be viewed as an iteration generated from a deterministic map.
Applications of EVT include climate systems, stochastic systems
and finance.
Statistical Properties of Strange Non-chaotic Attractors
Chaotic systems driven by quasiperiodic maps can give rise
to strange non-chaotic attractors: attractors with complex fractal
structure but zero Lyapunov exponents. Such attractors have
been observed numerically in many systems, but many rigorous results
are yet to be obtained. Dynamical questions concern those of
statistical/ergodic properties, fractal structure, and robustness
under perturbation.
Noisy Intermittency Maps
Chaotic maps withe neutral (non-hyperbolic) fixed points display
bursty intermittent phenomenon: periods of chaos interspersed with
long stretches of regular dynamics. Dynamical questions are raised
for multidimensional intermittency maps and the qualitative behaviour
of such maps with the addition of noise. These maps are known to be
useful for generating time series data with long-memory and
applications include modelling queue and network behaviour.