Brusselator

The Brusselator model was proposed by Prigogine and co-workers in Brussels in 1971.
  • P. Glandsdorff and I. Prigogine, Thermodynamic theory of structure, stability and fluctuations, Wiley, New York (1971).
  • G. Nicolis, Advan. Chem. Phys. 19, 209 (1971).
  • Prigogine, I.; Lefever, R. 1968, "Symmetry Breaking Instabilities in Dissipative Systems". II,J. Chem. Phys. 48, 1695-1700.
  • Tyson, J. 1973., "Come Further Studies of Nonlinear Oscillators in Chemical Systems", J. Chem. Phys. 58, 3919-3930.

Scheme:
A --> X ; k1
B + X --> Y + D ; k2
2X + Y --> 3X ; k3
X --> E ; k4
ODEs:
dX/dt = k1A - k2BX + k3X2Y - k4X
dY/dt = k2BX - k3X2Y
Translation: X -> u, Y -> v, a = A k1/k3, b = B k2/k3, k4 = k3, t -> t k3; DX/k3 -> Du, DY/k3 -> Dv.

This model always has a supercritical Hopf bifurcation because Re(g)>0. In the reduced form can be expressed by

The normal form for Hopf before rescaling

The normal form for 1-D Turing

The normal form for hexagonal Turing


This model can produce oscillatory square pattern:

We had used a coupled Brusselator model to study interacting Turing modes and reproduce the “black-eye” patterns:

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We had used such a simple model to produce segmented spirals.


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