Derivation of the Amplitude Equation for Reaction–Diffusion Systems via Computer-Aided Multiple-Scale Expansion

Abstract

The amplitude equation describes a reduced form of a reaction–diffusion system, yet still retains its essential dynamical features. By approximating the analytic solution, the amplitude equation allows the examination of mode instability when the system is near a bifurcation point. Multiple-scale expansion (MSE) offers a straightforward way to systematically derive the amplitude equations. The method expresses the single independent variable as an asymptotic power series consisting of newly introduced independent variables with differing time and space scales. The amplitude equations are then formulated under the solvability conditions which remove secular terms. To our knowledge, there is little information in the research literature that explains how the exhaustive workflow of MSE is applied to a reaction–diffusion system. In this paper, detailed mathematical operations underpinning the MSE are elucidated, and the practical ways of encoding these operations using MAPLE are discussed. A semi-automated MSE computer algorithm Amp_solving is presented for deriving the amplitude equations in this research. Amp_solving has been applied to the classical Brusselator model for the derivation of amplitude equations when the system is in the vicinity of a Turing codimension-1 and a Turing–Hopf codimension-2 bifurcation points. Full open-source Amp_solving codes for the derivation are comprehensively demonstrated and available to the public domain.