Abstract
Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift–Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction–diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg–Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.
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