The research activities of the group Dynamical Systems and Mathematical Physics cover a broad and diverse spectrum of subjects in the fields of fundamental and applied dynamical systems theory, classical, statistical and quantum mechanics and their interfaces in the light of dynamics. Applications to life sciences, meteorology, physics and chemistry play a crucial role.
General description
The primary interest of dynamical systems theory is the long term behaviour of systems that evolve in time. This concerns stationary, periodic, multi-periodic and chaotic dynamics, but also transient behaviour. Also bifurcations or transitions between such asymptotic states -- in particular transitions between regular and chaotic motions -- under variation of parameters is of great importance. We develop mathematical tools using methods from analysis, geometry and measure theory to grasp, study and develop the structures involved. Moreover, we develop methods to detect and understand the dynamics in specific models, employing numerical and graphical tools and computer algebra.
Many applications are from the field of mechanics. This concerns the motion of point masses like planets and their satellites in celestial mechanics, and also the motion of atoms and molecules which again can be described as point masses or rigid bodies. Here also relativistic or quantum effects may play a role. This is a wide area with great outreach, also in the direction of life sciences. If the number of constituent particles is huge then such systems are best described by statistical means. Statistical mechanics deals with the question how global observables, like temperature, can be explained from the microscopic behaviour. There is a close relationship with dynamical systems theory in particular with regard to random and chaotic behaviour and the so-called non-equilibrium systems.
Mathematical physics is the encompassing discipline of all the above and still larger areas of theoretical physics.
Specific subjects studied in the group include
- KAM theory
- Normal form theory
- Bifurcation theory
- Computation of invariant manifolds
- Integrable systems
- Times series analysis
- Probability theory
- Thermodynamic formalism
- Renormalisation group theory
- Theory of Metastates
