Oscillatory translational instabilities of spot patterns in the Schnakenberg system on general 2D domains

Abstract

Abstract For a bounded 2D planar domain Ω, we investigate the impact of domain geometry on oscillatory translational instabilities of N-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with activator-inhibitor diffusivity ratio. An N-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in Ω except in N well-separated localized regions of extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in -frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a 2N-vector that characterizes the possible modes (directions) of oscillation. The 2 N × 2 N matrix contains terms associated with a certain Green’s function on Ω, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates r = 1 + σ f ( θ ) with 0 < ε ≪ σ ≪ 1 , θ ∈ [ 0 , 2 π ) , and f ( θ ) 2π-periodic, we show that only the mode-2 coefficients of the Fourier series of f impact the bifurcation threshold at leading order in σ. We further show that when f ( θ ) = cos 2 θ , the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains Ω and N-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.