Cubic graph expansion of a saddle invariant manifold ($h^{\prime }(s)\dot{s}=\dot{h}$)

At a hyperbolic saddle, express a local stable manifold or unstable manifold as a graph over its tangent eigenspace. Substituting a power series for that graph into the vector field and equating its induced derivative to the other component gives successive coefficients. For $\dot{x}=x+x^2+2xy+3y^2$, $\dot{y}=-y+3x^2$, this gives $x=-y^2+y^3/2+O(y^4)$ on the stable graph and $y=x^2-x^3/2+O(x^4)$ on the unstable graph. These are local expansions, not globally invariant polynomial curves.