DFG
Collaborative Research Center 910 on
Control of selforganizing nonlinear systems
Special Measures Project
Hysteresis and nonlocal interactions
Summary
The project deals with systems of differential equations exhibiting selforganization and pattern formation due to the presence of hysteresis and nonlocal effects. These effects can be both in time (delay and hysteresis) and in space (deviations of spatial arguments and spatially distributed hysteresis). The following topics are covered:
 Differential equations with hysteresis and delay. We develop a framework for the study of equations with hysteresis and delay. In particular, we focus on stability of a periodic solution. As an application of the theory, we apply Pyragas control for reactiondiffusion models with hysteresis.
 Travelling waves in neural models and functional differential equations. Dynamics of activatorinhibitor neuron type models are described by the wellstudied FitzHughNagumo systems. These are coupled partial differential equations with hysteresistype slowfast behaviour. Such equations generate patterns of travelling wave solutions. In order to control these patterns, one can add an augmented transmission capability to the FitzHughNagumo model. This results in an additional feedback loop, which can be nonlocal in time and in space.
We are interested both in developing the general theory of nonlocal differential equations and in its application to reactiondiffusion systems such as the FitzHughNagumo model with nonlocal coupling.
 Reactiondiffusion equations with spatially distributed hysteresis. We study reactiondiffusion systems with hysteresis, which is an operator "with memory" in time. When defined at every spatial point, it also becomes spatially distributed. Such systems are known to produce different spatiotemporal patterns. We are especially interested in the case where the spatial domain is multidimensional.
