Nonlinear Dynamics and Pattern Formation

Understanding and controlling the spontaneous formation, dynamics, and stability of spatio-temporal patterns in non-equilibrium systems remains a major challenge in nonlinear physics. We study the self-organized emergence of macroscopic spatio-temporal order in physical, chemical and biological systems far from equilibrium. Systems under consideration consist of spatially interacting nonlinear units with bistable, excitable or oscillatory kinetics. As a whole, these spatially extended systems are capable of showing a variety of pattern formation and unexpected behavior impossible under equilibrium conditions.

In more detail our research is focused on the following issues:

1. Propagation, interaction, and control of traveling nonlinear wave-like excitations as interfaces, solitary pulses, periodic pulse trains, spiral waves, scroll waves, and localized spots. In particular, we are interested in the control of rotating excitation waves by external forcing and feedback-mediated parameter modulation. Recently, we started to study the propagation of geometrically confined scroll waves. This research is motivated by supporting novel approaches for the therapy of abnormal electrical wave activity in the heart (tachycardia, fibrillation) or certain brain areas (Parkinson disease, epilepsy). Also we are interested in instabilities in media with long-range (global and non-local) spatial coupling arising quite naturally in neural networks, catalytic surface reactions, electrochemical systems, and others.

2. Contraction patterns and mechano-chemical waves in active matter. In the framework of two-phase models we study poroelastic materials consisting of interpenetrating visco-elastic (with active stresses) and fluid phases as, for example, microplasmodia droplets of the slime mode Physarum polycephalum. In the ladder case, we are interested in a physical minimal model for the onset of directed motion of the droplet.

3. Networks of coupled oscillators. We study the collective dynamics in oscillatory networks, in particular the synchronization transition and emerging phase patterns for strongly coupled relaxation oscillators.

4. We perform experiments with the photosensitive Belousov-Zhabotinsky (PBZ) reaction maintaining stationary out-of-equilibrium conditions in an open reactor. Examples include the control of spiral and scroll waves in PBZ solutions and pattern formation in large populations of photochemically coupled micrometer sized catalytic PBZ beads.