Given:
$ \dot{x}=x+y^2 = f(x,y),$
$ \dot{y}=-y+4x^3+xy = g(x,y)$
Find a 4th order polynomial approximation for the stable/unstable manifolds at the fixed point $(0,0)$
My attempt: I calculated the linear system with the associated eigen-value/vectors to find the stable and unstable subspace. For the stable subspace I found $\lambda = -1$ and $\vec{v}=(0,1)^T$.
I then attempted to solve $\triangledown H \cdot F = 0 $ where $F=(f(x,y), g(x,y))$ and $H = x - p(y)$ and is a level curve along the flow.
I use a polynomial approximation $p(y)= a+by+cy^2+dy^3$ and determine $a=b=0$ by using that $p(0)=0, p'(0)=0.$
And then I get lost.