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I've encountered a problem like this: \begin{eqnarray} \left\{ \begin{array}{l} \dot{y_1}=-y_1+y_2^2,\\ \dot{y_2}=y_2,\\ \dot{y_3}=y_1y_2 \end{array}\right.\notag \end{eqnarray} The linerized matrix has a zero,so clearly, it should have a center manifold at $(0,0,0)$.

It's easy to solve that \begin{eqnarray} \left\{ \begin{array}{l} y_1=\left(y_1(0)-\frac{y_2^2(0)}{3}\right)e^{-t}+\frac{y_2^2(0)}{3}e^{2t},\\ y_2=y_2(0)e^{t},\\ y_3=y_2(0)\left(y_1(0)-\frac{y_2^2(0)}{3}\right)t+\frac{y_2^3(0)}{3}e^{3t}+y_3(0)-\frac{y_2^3(0)}{9} \end{array}\right.\notag \end{eqnarray} But I have no idea how to get the parametric representation of the center manifold.

Thank you for your attention.

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It's easy in this case: $(0,0,y_3)$, which is a line of fixed points.

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    But how to prove that? I should know $y_3(0,0)=0$ as well as $D y_3(0,0)=0$,which drives me mad..2012-12-14