Abstract
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 theta -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.
Highlights
Many spatially extended physical, chemical and biological systems form so-called excitable media, in which a supercritical perturbation from a stable equilibrium triggers an excitation that is transferred to its neighbours, followed by refractory return to the rest state.Perhaps the simplest dynamical system that realises a caricature excitable medium is the 1D Greenberg-Hastings cellular automata (GHCA) [25,26]
From a dynamical systems viewpoint the non-wandering set is of prime relevance, and it turns out that for the GHCA it can be decomposed into invariant sets with different wave dynamics [31]
Motivated by studies of the Greenberg-Hastings cellular automaton (GHCA) as a caricature of excitable systems, in this paper we have considered the θ -equations describing oscillatory phase dynamics, as the perhaps simplest partial differential equations (PDE) model of excitable media
Summary
Chemical and biological systems form so-called excitable media, in which a supercritical perturbation from a stable equilibrium triggers an excitation that is transferred to its neighbours, followed by refractory return to the rest state. In this paper we consider suitable scalar parabolic PDE for periodic phase dynamics as models for excitable media, and study similarities and differences between this and the local pulse dynamics of the GHCA These so-called θ -equations for oscillator phase dynamics [8,41] are given by θt = θxx + f (θ ), θ (t, x) ∈ S1 = R/2π Z, x ∈ R , t > 0. The model (1) is suited for a comparison with GHCA as it allows to study the combination of strong interactions, when a kink and an antikink collide, and weak interactions when considering consecutive kinks This observation already suggests a multiscale nature, which turns out to imply a fundamental difference for the long time dynamics of excitation pulse positions. In the appendices we collect some technical proofs as well as notes on the numerical implementations
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