An Invariant Manifold Theory for ODEs and Its Applications

For a system of ODEs defined on an open, convex domain $U$ containing a positively invariant set $\mathrm{\Gamma }$, we prove that under appropriate hypotheses, $\mathrm{\Gamma }$ is the graph of a $C^r$ function and thus a $C^r$ manifold. Because the hypotheses can be easily verified by inspecting the vector field of the system, this invariant manifold theory can be used to study the existence of invariant manifolds in systems involving a wide range of parameters and the persistence of invariant manifolds whose normal hyperbolicity vanishes when a small parameter goes to zero. We apply this invariant manifold theory to study three examples and in each case obtain results that are not attainable by classical normally hyperbolic invariant manifold theory.