The Brusselator model has parameters and

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Pattern formation: instabilities that choose a wavelength

Problem

The Brusselator model has parameters

A

and

B

, with diffusion coefficients

D_u

and

D_v

for the activator and inhibitor. Its Hopf and Turing thresholds and critical wavenumber are

B_{\rm H} = 1+A^2

B_{\rm T} = \left(1+A\sqrt{D_u/D_v}\right)^2

and

q_{\rm c}^2 = A/\sqrt{D_uD_v}

(all quantities dimensionless, in the model's own units). Take

A=3

D_u=1

D_v=6

. ((a)) Compute

B_{\rm T}

and

B_{\rm H}

, and state which instability occurs first as

B

is raised from zero. ((b)) Compute

q_{\rm c}

and the critical wavelength

\Lambda_{\rm c} = 2\pi/q_{\rm c}

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