  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.
