At a hyperbolic fixed point, stable and unstable manifolds are tangent to the corresponding eigenspaces. Power-series coefficients follow from graph invariance.
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 , , this gives on the stable graph and on the unstable graph. These are local expansions, not globally invariant polynomial curves.
For a sufficiently smooth vector field at a hyperbolic equilibrium, the sums of generalized eigenspaces with negative and positive real parts are tangent to local invariant stable and unstable manifolds of the same dimensions. Their points converge to the equilibrium in forward and backward time respectively. Hyperbolicity excludes eigenvalues with zero real part; center manifolds require a separate theorem.
