## The amplitude equation of Brusselator

Brusselor reaction can be represented as kinetic equations:

dX/dt = A-(B+1)X + X^2Y
dY/dt = BX - X^2Y

in order to investigate the behavior of the system beyond tje instability(B is used as bifurcation parameter, the critical value is B_H= 1+A^2), I use multiple-scale analysis:

\bftext{X}(t) = (X(t), Y(t))

\bftext{X}(t) = \bftext{X_0} + +\rho\bftext{X_1} + \rho^2\bftext{X_2}

\rho is a very very value.

B = B_H + \rho B_1 + \rho^2 B_2 + ...

d_t = d_{\tau_0} + \rho d_{\tau_1} + \rho^2 d_{\tau_2} +...
in which \tau_i scales as \tau_i = \rho^i t

the solution \bftext{X_1} is a linear combination of the right eigenvectors of the Jacobian martrix L:

L=[B-1 A^2 ; -B -A^2]

\bftext{X_1} = (1; (i-A)/A)Z(t) exp(iAt) + (1; -(i+A)/A)Z^*(t) exp(-iAt)

in order to determine the amplitude Z(t) and Z^*(t), the higher order contribution in \rho should be considered.

Now I know at the order \rho^0, one recovers the reference state (X_0, Y_0)

at the order \rho^1, the linear stability analysis is recovered

So, what will happen at the order \rho^2 and \rho^3, how to identify the amplitude coefficient?? Thank you