Center manifold

If \(f\) and \(g\) are identically zero then (6) has the two obvious invariant manifolds \(x=0\) and \(y=0\ .\) The invariant manifold \(x=0\) is called the stable manifold, and on the stable manifold all solutions decay to zero exponentially fast. The invariant manifold \(y=0\) is called the centre manifold. In general, an invariant manifold \(y = h(x)\) for (6) defined for small \(|x|\) with \(h(0)=0\) and \(Dh(0)=0\) is called a centre manifold. In more physical terms, the dynamics of y follows the dynamics of x and one may say that x enslaves the variable y. This interpretation has been called slaving principle.