For the Brusselator model with parameters

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

A

B

and diffusion coefficients

D_u

D_v

(dimensionless units), the Turing threshold is

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

, the Hopf (oscillatory) threshold is

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

, and the critical wavenumber obeys

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

. Take

A=1.5

D_u=1

D_v=10

. ((a)) Find

B_{\rm T}

and

B_{\rm H}

, and confirm that the Turing instability sets in before the Hopf instability as

B

is raised from below. ((b)) Find the critical wavenumber

q_{\rm c}

and the pattern wavelength

\Lambda_{\rm c}

. ((c)) Assume that, as for the Swift-Hohenberg equation, the Brusselator's stationary bifurcation here is supercritical, so that past threshold the pattern amplitude grows as the square root of the distance above it,

A^*(B)\propto\sqrt{B-B_{\rm T}}

(confirming this needs a weakly nonlinear calculation not done here). Using this scaling, find the ratio

A^*(3.0)/A^*(2.5)

of the pattern amplitudes at

B=3.0

and

B=2.5

(both values lie between

B_{\rm T}

and

B_{\rm H}

, so the system is in the stationary pattern regime at each).

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