DFG Collaborative Research Center 910 on
Control of self-organizing nonlinear systems

Special Measures Project

Hysteresis and nonlocal interactions

Research director PD Dr. Pavel Gurevich
Members   Mark Curran     Daria Neverova     Eyal Ron

Summary

The project deals with systems of differential equations exhibiting self-organization 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 reaction-diffusion models with hysteresis.
  
• Travelling waves in neural models and functional differential equations. Dynamics of activator-inhibitor neuron type models are described by the well-studied FitzHugh-Nagumo systems. These are coupled partial differential equations with hysteresis-type slow-fast behaviour. Such equations generate patterns of travelling wave solutions. In order to control these patterns, one can add an augmented transmission capability to the FitzHugh-Nagumo 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 reaction-diffusion systems such as the FitzHugh-Nagumo model with nonlocal coupling.
  
• Reaction-diffusion equations with spatially distributed hysteresis. We study reaction-diffusion 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 spatio-temporal patterns. We are especially interested in the case where the spatial domain is multidimensional.