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

Abstract For a bounded 2D planar domain Ω, we investigate the impact of geometry on oscillatory translational instabilities N -spot equilibrium solutions for singularly perturbed Schnakenberg reaction-diffusion system with activator-inhibitor diffusivity ratio. An is characterized by an activator concentration that exponentially small everywhere in Ω except well-separated localized regions $\mathcal{O}(\varepsilon)$?> extent. We use method matched asymptotic analysis to analyze Hopf bifurcation thresholds above which becomes unstable perturbations, result $\mathcal{O}(\varepsilon^2)$?> -frequency oscillations locations spots. find stability these perturbations governed nonlinear matrix-eigenvalue problem, eigenvector 2 -vector characterizes possible modes (directions) oscillation. The $2N\times 2N$?> 2 N × matrix contains terms associated certain Green’s function encodes geometric effects. special case disk radius polar coordinates $r = 1 + \sigma f(\theta)$?> r = 1 + σ f ( θ stretchy="false">) $0\lt \varepsilon 1$?> 0 < ε ≪ , $\theta \in [0,2\pi)$?> ∈ stretchy="false">[ , π and $f(\theta)$?> π -periodic, show only mode-2 coefficients Fourier series f threshold at leading order σ . further when $f(\theta) \cos2\theta$?> cos dominant mode oscillation direction parallel longer axis disk. Numerical investigations full PDE are performed various domains equilibria confirm results also demonstrate how impacts modes.