Invariant manifolds around equilibria of Newtonian equations: Some pathological examples

AbstractLet the equation x¨=f(t,x) be periodic in time, and let the equilibrium x∗≡0 be a periodic minimizer. If it is hyperbolic, then the set of asymptotic solutions is a smooth curve in the plane (x,x˙); this is stated by the Stable Manifold Theorem. The result can be extended to nonhyperbolic minimizers provided only that they are isolated and the equation is analytic (Ureña, 2007 [6]). In this paper we provide an example showing that one cannot say the same for C2 equations. Our example is pathological both in a global sense (the global stable manifold is not arcwise connected), and in a local sense (the local stable manifolds are not locally connected and have points which are not accessible from the exterior)