paper

Normal Hyperbolicity and Unbounded Critical Manifolds

arXiv:1204.0947 · doi:10.1088/0951-7715/27/6/1351

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.

15 pages, 3 figures; improved notation and formulation, results unchanged

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