Motivated by studies of the Greenberg-Hastings cellular automata (GHCA) as a caricature of excitable systems, in this paper we study kink-antikink dynamics in the perhaps simplest PDE model of excitable media given by the scalar reaction diffusion-type -equations for excitable angular phase dynamics. On the one hand, we qualitatively study geometric kink positions using the comparison principle and the theory of terraces. This yields the minimal initial distance as a global lower bound, a well-defined sequence of collision data for kinks- and antikinks, and implies that periodic pure kink sequences are asymptotically equidistant. On the other hand, we study metastable dynamics of finitely many kinks using weak interaction theory for certain analytic kink positions, which admits a rigorous reduction to ODE. By blow-up type singular rescaling we show that distances become ordered in finite time, and eventually diverge. We conclude that diffusion implies a loss of information on kink distances so that the entropic complexity based on positions and collisions in the GHCA does not simply carry over to the PDE model.
@article{PAUTHIER_RADEMACHER_ULBRICH_2023,title={Weak and strong interaction of excitation kinks in scalar parabolic equations},author={Pauthier, A. and Rademacher, J. D. M. and Ulbrich, D.},journal={Journal of Dynamics and Differential Equations},volume={35},doi={https://doi.org/10.1007/s10884-021-10040-2},number={3},pages={2199--2235},year={2023},publisher={Springer}}
In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with e excited and r refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large e,r.
@article{KESSEBÖHMER_RADEMACHER_ULBRICH_2021,title={Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata},volume={41},doi={10.1017/etds.2020.18},number={5},journal={Ergodic Theory and Dynamical Systems},author={Kesseböhmer, M. and Rademacher, J. D. M. and Ulbrich, D.},year={2021},pages={1397–1430}}
Ongoing work
2026
Discrete hypocoercivity for a nonlinear kinetic reaction model without initial close-to-equilibrium assumption
L. Liu, M. Pirner, and D. Ulbrich
2026
In preparation
Theses
2021
Ergodic theory of nonlinear waves in discrete and continuous excitable media
In this thesis, we analyze discrete and continuous models of excitable media with the intention to reveal similarities between both approaches in terms of wave propagation and interaction. While the discrete perspective is represented by the one-dimensional Greenberg-Hastings cellular automata (GHCA), as a continuous model we consider the theta-equations which are basic partial differential equations (PDE) for pure phase dynamics. On the one hand, qualitatively, collision and annihilation of waves can be observed in both models in striking resemblance. However, on the other hand, it turns out that a quantitative comparison of discrete and continuous wave interactions is limited due to weak wave interactions in the PDE. Specifically, complexity considerations show that a direct comparison of discrete and continuous strong wave interactions is problematic.
@misc{Ulbrich_2021,title={Ergodic theory of nonlinear waves in discrete and continuous excitable media},doi={10.26092/elib/1053},author={Ulbrich, D.},year={2021},note={Dissertation}}