Calculate one-dim stable manifold

Question

$\dot x = -x+y+y^2$
$\dot y=3x+y+3xy$

I am asked to show that the system has a one-dimensional stable manifold of the form $y=\xi(x)$ with $\xi(0)=0$, and also calculate $\xi(0)'$ and $\xi(0)''$.

Where should I start to find the $\xi(x)$?

Answer 1

Linearization shows that $(0, 0)$ is a saddle point. There are two separatrices going through the saddle point, one of which is the stable manifold. Take a series approximation $y = a_1 x + a_2 x^2$ for a separatrix. The coefficient $a_1$ can be found by considering that $(1, a_1)$ is an eigenvector corresponding to the negative eigenvalue of the linearized system. Then substitute $y = a_1 x + a_2 x^2$ into the system. Eliminating $\dot x$ and equating the coefficient at $x^2$ to zero gives $$a_1^3 - 3 a_1 - 3 a_2 + 3 a_1 a_2 = 0.$$