Abstract
As mentioned in the previous chapter, the local dynamics near regular jump points indicate one possibility for solutions of a general singular perturbation problem (3.10), respectively, (3.15) to switch from slow to fast dynamics (or vice versa) which is key for any global relaxation oscillatory behaviour.
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Notes
- 1.
Although the layer problem is linear, we can only define the point p l implicitly.
- 2.
Otherwise, one has to show that the third derivative does not vanish; see Definition 4.2.
- 3.
Calculate the characteristic polynomial for this singularity: based on the assumptions on the parameters (α 1, α 2, α 3), Descartes Sign Rule shows that there are no real positive roots and the Routh–Hurwitz Test shows that not all roots have negative real parts.
- 4.
Showing this layer problem connections analytically is a nontrivial task.
- 5.
A detailed analysis can be found in [36].
- 6.
The intersection point at the origin can also be analysed with GSPT techniques.
- 7.
If this technique is applied to an isolated nilpotent singularity then it is usually referred to as a spherical blow-up since all variables participate in the blow-up.
- 8.
Blow-up coordinates are denoted in each coordinate chart κ i by a subscript i; see also Remark 3.2.
- 9.
To prove this result rigorously, one needs to perform two additional spherical blow-ups of the two nilpotent equilibria (on the blown-up cylinder): the non-hyperbolic origin and the fold F z; see [36] for details.
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Wechselberger, M. (2020). Relaxation Oscillations in the General Setting. 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_5
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