Normal Hyperbolicity and Unbounded Critical Manifolds

TL;DR

This paper extends the theory of normal hyperbolicity to unbounded critical manifolds in fast-slow systems, providing a scaling law and technical improvements to the blow-up method, with applications to autocatalytic reaction models.

Contribution

It introduces a general theoretical framework for unbounded manifolds losing hyperbolicity, including a scaling law and an enhanced blow-up technique.

Findings

Derived a scaling law for unbounded manifolds with power-law decay.

Extended the blow-up method with an optimality criterion for exponents.

Provided a rigorous analysis applicable to autocatalytic reaction models.

Abstract

This work is motivated by mathematical questions arising in differential equation models for autocatalytic reactions. In particular, this paper answers an open question posed by Guckenheimer and Scheper [SIAM J. Appl. Dyn. Syst. 10-1 (2011), pp. 92-128] and provides a more general theoretical approach to parts of the work by Gucwa and Szmolyan [Discr. Cont. Dyn. Sys.-S. 2-4 (2009), pp. 783-806]. We extend the local theory of singularities in fast-slow polynomial vector fields to classes of unbounded manifolds which lose normal hyperbolicity due to an alignment of the tangent and normal bundles. A projective transformation is used to localize the unbounded problem. Then the blow-up method is employed to characterize the loss of normal hyperbolicity for the transformed slow manifolds. Our analysis yields a rigorous scaling law for all unbounded manifolds which exhibit a power-law decay for the alignment with a fast subsystem domain. Furthermore, the proof also provides a technical extension of the blow-up method itself by augmenting the analysis with an optimality criterion for the blow-up exponents.