Our research focuses on the theory and application of models under time-variable external influences – these are known as nonautonomous dynamical systems. Their analysis requires a combination of different mathematical tools ranging from classic theory of dynamical systems to non-linear analysis, numerics, and simulations. Another focus are equations involving delays.
Dynamical systems are mathematics models of current, temporally variable processes. Starting with the simple movement of a pendulum, examples of dynamical systems include chemical reactions, biological or sociological interactions and complex climate models, and influence all aspects of our daily lives.
They appear in every magnitude from micro to macrocosms and range from elementary linear models to non-linear, chaotic or random systems, which are currently the topic of much discussion.
A range of differential and difference equations are used to describe dynamical systems. As a rule, these are complicated and cannot be solved in their closed form. This applies in particular to problems from applications in which external and internal influences are not known to the finest detail, meaning that geometric and qualitative methods are indispensable for a solid understanding and analysis of these problems. They make it possible to obtain information on how solutions behave without knowing them on an individual basis. The aim is to understand the precise structure of the state space in which solutions with known long-term properties are identified. This in turn provides information on the temporal development of a system and the system’s dependence on starting conditions and external parameters.
