The Unstable Manifold of an Overflowing Invariant Manifold

Abstract

It is reasonable to consider the existence of the unstable manifold (but not the stable manifold) of an overflowing invariant manifold. In this chapter, under appropriate hypotheses, we will construct the unstable manifold of the overflowing invariant manifold . Intuitively, we think of the unstable manifold of an invariant set as the set of points in phase space that approach the invariant set as t → – ∞. This notion requires careful interpretation since has a boundary, and trajectories starting on . in forward time by crossing ∂M (this is the reason that it does not make sense to consider a stable manifold). We also want to emphasize that initially we will not be dealing with a perturbation problem; we will be concerned with constructing the unstable manifold of . Afterward, we will show that this unstable manifold also satisfies the hypotheses of the persistence theorem for overflowing invariant manifolds. Hence, will also have an unstable manifold under appropriate hypotheses. We begin developing the setting in much the same way as earlier.KeywordsVector FieldInvariant ManifoldUnstable ManifoldStable ManifoldLocal Unstable ManifoldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.