MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract
Bär, Markus
Two examples of multiscale pattern formation in reaction-diffusion systems will be described and analyzed. The first part concerns the impact of small scale obstacles upon wave properties in excitable media with application to cardiac propagation. For small densities of such heterogeneities, the properties of excitation waves are well described by an effective equation obtained from homogenization. If the density of the obstacles becomes large enough, excitation waves breakup eventually developing into spiral wave fragments (“reentry”) and irregular patterns (“fibrillation”). It is shown that the critical density that leads to a qualitative change of wave behavior is closely related to the percolation threshold of the obstacles [1]. The second part deals then with competing pattern-forming instabilities with different length scales. Inspired by phenomena found in models for biomembranes and chemical reaction-diffusion systems, we introduce two simple models based on the Swift-Hohenberg and Cahn-Hilliard equations respectively and analyse their behavior by linear stability analysis and numerical simulations [2]. The coupled Swift-Hohenberg equations reveal a codimension-2 Turing-wave point and an extended region of hysteresis between wave and Turing patterns in its vicinity. For the coupled Cahn-Hilliard equations, we find, depending on parameters, two different instabilities that are characteristic for static and oscillatory phase separation and arrested coarsening . beginitemize item[[1]] S. Alonso and M. Bär (2013). emphReentry near the percolation threshold in a heterogeneous discrete model for cardiac tissue. Phys. Rev. Lett. 110, 158103-1 - 158103-5. item[[2]] D. Schüler, S. Alonso, A. Torcini, and M. Bär (2014). emphSpatio-temporal dynamics from codimension-two bifurcations in asymmetrically coupled nonlinear evolution equations. In preparation. enditemize