Development of a set-oriented numerical technique which allows to compute low-dimensional invariant sets for infinite dimensional dynamical systems.

Systems of interest:

• Delay Differential Equations (DDEs): type of differential equation in which the time derivative of the unknown function depends not only on the current state but also on previous times.
  Focus of interest: Analysis of long term behavior of DDEs
  Applications: e.g. signal processing, disease transition, population models
• Partial Differential Equations (PDEs): type of differential equation in which the time derivative of the unknown function depends on multivariable functions and their partial derivatives.
  Focus of interest: Analysis of long term behavior of PDEs
  Applications: e. g. fluid dynamics, quantum mechanics, electrodynamics

In engineering applications, one of the major challenges today is to develop reliable and robust control algorithms for complex net- worked systems. Controllability and observability of such systems play a crucial role in the design process. The underlying network structure may contain symmetries – caused, for example, by the cou- pling of identical building blocks – and these symmetries lead to repeated eigenvalues in a generic way. This complicates the design of controllers since repeated eigenvalues decrease the controllability of the system. We analyze the relationship between the controllability and observability of complex networked systems and symmetries using results from group representation theory.

Time-dependent transport maps for the Southern Ocean are useful in many ways: they permit precise mass, heat and freshwater transport calculations for the Antartic Circumpolar Current.

Cooperations: University of New South Wales, Australia; Centre National de la Recherche Scientifique, Brest, France

Development of a set-oriented numerical methodology which allows to perform uncertainty quantification (UQ) for dynamical systems from a global point of view.

• State space approach: Dynamical Systems with uncertain parameters
  Focus of interest: Analysis of the statistical behavior of the system to capture the long-term uncertainty propagation

Efficient solutions of multiobjective optimization problems and systems of nonlinear equations require new theory and algorithms. These problems can be of hierarchical structure or depend on additional parameters.

Cooperations: it's OWL - Intelligent Technical Systems Ostwestfalen Lippe; Hella KGaA Hueck & Co.