= Centre manifold theorem for a discrete dynamical system
{title2=$|\lambda|=1$}
For a sufficiently <smooth> local map fixing the origin, a local invariant <centre manifold> is tangent to the <generalized eigenspaces> with <eigenvalues> of <modulus> one. Writing the map as $(u,v)\mapsto(F(u,v),G(u,v))$ and the manifold as $v=h(u)$ gives the invariance equation $h(F(u,h(u)))=G(u,h(u))$. Finite <Taylor series> coefficients can be found from this equation. If all transverse <eigenvalues> have <modulus> less than one, local asymptotic stability reduces to that of the map on the <centre manifold>.
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