## BrusselatorThe 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:
dX/dt = k_{1}A - k_{2}BX + k_{3}X^{2}Y - k_{4}XdY/dt = k_{2}BX - k_{3}X^{2}YTranslation: X -> u, Y -> v,
a = A k_{1}/k_{3},
b = B k_{2}/k_{3},
k_{4} = k_{3},
t -> t k_{3};
D_{X}/k_{3} -> D,
_{u}D_{Y}/k_{3} -> D.
_{v}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:
## _We had used such a simple model to produce segmented spirals. |