Centre manifold theorem

Near a nonhyperbolic equilibrium, a local invariant centre manifold is tangent to the generalized eigenspace with zero-real-part eigenvalues, and its reduced dynamics determine the local bifurcation behaviour.

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.

Appending $\dot{\mu }=0$ turns a system parameter into a centre variable, allowing one invariant graph to describe nearby parameter values.

If a centre manifold is the graph $v=h(u)$ for $\dot{u}=f(u,v)$ and $\dot{v}=g(u,v)$, its coefficients satisfy $Dh(u)f(u,h(u))=g(u,h(u))$.