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A Brusselator with a continuous Turing-unstable torus mode

Statement

Consider the Brusselator field

F (u, v) = (1 - \frac{5}{2} u + u^{2} v, \frac{3}{2} u - u^{2} v)

with diagonal diffusivity $D = diag (1 / 10, 2)$ . Its homogeneous equilibrium is $(u_{*}, v_{*}) = (1, 3 / 2)$ . On the one-dimensional torus of length $L = 4 π / 3$ , the continuous reaction-diffusion linearization has an unstable Fourier mode with wave number $k = 1$ .

Facts & Assumptions

Given: The reaction field and diffusivities in the Statement, and the dispersion criterion of Two-species continuous and discrete dispersion criteria.

[F1]

Two-species continuous and discrete dispersion criteria: for a mode with Laplacian eigenvalue $λ$ , the linearized matrix is $J - λ D$ ; with negative trace, it has one positive eigenvalue when its determinant $q (λ)$ is negative.

[A1]

The example uses $A = 1$ , $B = 3 / 2$ in the standard Brusselator $F (u, v) = (A - (B + 1) u + u^{2} v, B u - u^{2} v)$ .

Proof

1.1F1givenalgebra

Direct substitution gives $F (1, 3 / 2) = (0, 0)$ . Differentiating the two components gives $J = D F (1, 3 / 2) = (\begin{array}{cc} 1 / 2 & 1 \\ - 3 / 2 & - 1 \end{array}), tr J = - 1 / 2, \det J = 1.$

2.1step 1.1givenalgebra

The coefficient in the dispersion polynomial is $β_{T} = d_{2} J_{11} + d_{1} J_{22} = 9 / 10$ and $d_{1} d_{2} = 1 / 5$ . Therefore $q (λ) = \frac{1}{5} λ^{2} - \frac{9}{10} λ + 1 = \frac{1}{5} (λ - 2) (λ - \frac{5}{2}) .$

3.1F1step 2.1algebra∎

For the continuous torus mode $k = 1$ , $λ = (2 π / L)^{2} = 9 / 4$ . This lies strictly between $2$ and $5 / 2$ , so step 2.1 gives $q (9 / 4) < 0$ . By [F1], the mode matrix has one positive eigenvalue and one negative eigenvalue. This example proves a linear mode instability only; it does not assert nonlinear pattern existence or stability.

Depends on

Used by

A three-cell periodic grid suppresses the Brusselator's unstable continuous mode Example

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Sources

Luca Doria, Numerical Algorithms, lecture notes, Johannes Gutenberg-Universität Mainz, 2022 (standard reference, not scraped)