= Cubic graph expansion of a saddle invariant manifold
{title2=$h'(s)\dot s=\dot h$}
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.
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