An Invariant Manifold Theory for ODEs and Its Applications

TL;DR

This paper develops a new invariant manifold theory for ODEs that allows for the analysis of invariant manifolds under broader conditions, including cases with vanishing normal hyperbolicity, and demonstrates its application to complex systems.

Contribution

The paper introduces a novel invariant manifold theory applicable to a wider class of ODE systems, including those with non-hyperbolic behavior, expanding the scope of classical methods.

Findings

Established conditions for invariant manifolds as $C^r$ graphs.

Applied theory to three examples with novel results.

Extended invariant manifold analysis to systems with vanishing hyperbolicity.

Abstract

For a system of ODEs defined on an open, convex domain $U$ containing a positively invariant set $\Gamma$, we prove that under appropriate hypotheses, $\Gamma$ 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.