I have this system to study $$ \left\{ \begin{aligned} \frac{dx}{dt} &= y-x - x^2 \\[5pt] \frac{dy}{dt} &= \mu x - y - y^2 \end{aligned} \right. $$
I have derived the Jacobian around fixed point $$ \phi = (0, 0). $$
I have found eigenvalues $$ \lambda_{\pm} = \pm \sqrt{\mu} - 1 $$ and eigenvectors $$ f_{\pm} = (\pm 1,1) $$ in the case of $$ \mu = \mu_c = 1. $$
I compute the system in order to write it in term of $c_+$, $c_-$ which are in the base of $$ f_{\pm} = (\pm 1, 1). $$ I got $$ \left\{ \begin{aligned} \frac{dc_-}{dt} &= -2 c_- - (c_+ + c_-)^2 \\[5pt] \frac{dc_+}{dt} &= -c_+^2 - c_-^2 \end{aligned} \right. $$
Now, my problems begin: In the neighborhood of $\mu_c$, at the dominant order of $\epsilon = \mu -\mu_c$, I must reduce the system made of $c_+$ and $c_-$ to one dimension. I don't know how to start and where to go, my lectures on center manifold don't help me. If someone has an idea.
Regards
Xav