Polynomial approximation of a local stable manifold

Question

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.

Answer 1

Since $x=p(y)$, we have $$ \begin{align} x' &=p'(y)y'\\ &=\left(2cy+3dy^2+\cdots\right)\left(-y+4x^3+xy\right)\\ &=\left(2cy+3dy^2+\cdots\right)\left(-y+4\left(cy^2+dy^3+\cdots\right)^3+(cy^2+dy^3+\cdots)y\right) \end{align} $$ On the other hand, $$ x'=x+y^2=cy^2+dy^3+\cdots+y^2. $$ Now you must choose the constants $c$, $d$, $\ldots$ so that the two series are equal. For example, $$ x'=-2cy^2+\cdots=(c+1)y^2+\cdots $$ and so $-2c=c+1$, etc.