Abstract
This chapter extends the coordinate-independent GSPT beyond Fenichel theory. Loss of normal hyperbolicity is one essential ingredient for a singularly perturbed system to switch between slow and fast dynamics as observed in many relaxation oscillator models; see Chap. 2. Geometrically, loss of normal hyperbolicity occurs generically along (a union of) codimension-one submanifold(s) of S where a nontrivial eigenvalue of the layer problem crosses the imaginary axis.
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Notes
- 1.
This matrix is also referred to as the adjugate.
- 2.
We have \((\Pi ^S_d)^2=-\det (D\!f N) \Pi ^S_d\), i.e., this is only a ‘projection’ operator up to the time rescaling factor \(-\det (D\!f N)\) which does not cause any troubles from a dynamical systems point of view.
- 3.
A blow-up analysis near contact points justifies this formal approach.
- 4.
For z ∈ S h, the projectors ΠS and \(\Pi ^S_d\) are equivalent and have the same rank which is k ≥ 1.
- 5.
This part follows the same steps as in the proof of Lemma 4.1.
- 6.
A simple translation of the contact point suffices.
- 7.
This is left as an exercise for the reader; see, e.g., [103].
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Wechselberger, M. (2020). Loss of Normal Hyperbolicity. In: Geometric Singular Perturbation Theory Beyond the Standard Form. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-030-36399-4_4
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