My research focusses on the following main themes:
Parametrized KAM Theory.
Kolmogorov Arnold Moser Theory deals with the persistent occurrence of multi-periodicity in dynamical systems. Examples can be found in the solar system, or in coupled oscillators as these occur in many contexts. Multi- or quasi-periodic dynamics usually occurs as an orderly `state' in a scenario in between orderly (say, periodic) and chaotic. Our interest is to describe these scenarios with parametrized dynamical systems.
In the mathematics of KAM Theory one meets `small divisors': certain perturbation series diverge on a dense set of resonances. In the global geometry also singularity theory shows up, where the classical semi-algebraic geometry is `Cantorized' by the resonances. Quasi-periodic dynamics takes place on invariant tori. Of special interest are the quasi-periodic bifurcations, which often involves tori of different dimension.
Resonance is a periodic interaction of oscillatory subsystems with rational frequency ratios. Such phenomena are most visible when these ratios have the format 1:1, 1:2, 2:3, etc., so with small integers. They occur as tidal resonances in the solar system (think of the Moon locked to the Earth always showing the same face), or in synchronized clocks according to Christiaan Huygens, as well as in many biological circumstances. Think of synchronizing fire flies on a bush or of the entrainment of biological clocks by the circadian rhythm.
Resonances are best understood in terms of parameter dependent dynamical systems. In the parameter space each resonance usually corresponds to an open domain (or `gap'), as part of a bifurcation set determined by singularity theory. The overall geometry then often is fractal, due to the Cantorization described above.
Dynamical systems theory is extremely useful for many modelling purposes. Apart from classical areas as mechanics, optics, etc., also meteorology and biology are suitable from mathematical modelling from dynamical systems.
In meteorology we are working on several subjects regarding climate variability. A current subject regards the statistics of extreme events
in cooperation with groups from the University of Exeter and the Royal Netherlands Meteorological Institute
Advanced dynamical systems theory is being used to describe certain phenomena, e.g., concerning the Atlantic Multidecadal Oscillation. In this description chaos and intermittency occur in several forms.
In biology the current interest is formed by applications in chronobiology, see under the item Resonance. Here we cooperate with the Groningen group Chronobiology.
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