multiple time scale dynamics with two fast variables and one slow ...

and set Iδ := [s ∗ −δ, s ∗ +δ]. We want to follow W cu close to the slow manifold
S r,ǫ associated to ¯
Cr× Iδ using the Exchange Lemma. Having all definitions and
preliminaries in place we can outline the proof of Theorem 2.6.1.
Proof. (of Theorem 2.6.1, Sketch, see [69]) We now work in the four-dimensional
space (x1, x2, y, s) and with equation (2.45). Forǫ > 0 sufficiently small we
can assume that the center-unstable manifold W cu is close to the singular ob-
ject W cu (0, 0, s). The stability of transverse intersection under perturbation and
Lemma 2.6.2 imply that W cu intersects the stable manifold W s (S r,ǫ) of the slow
manifold S r,ǫ transversely. Now we can apply the Exchange Lemma to follow
W cu close to S r,ǫ and conclude that it can be followed forǫ sufficiently small up
to y≈y ∗ and that it leaves the vicinity of S r,ǫ C 1 -close to W u (S r,ǫ). Note that the
C 1 conclusion is crucial here for the following step.
Since W cu is now C 1 -close to W u (S r,ǫ) it is also C 1 -close to the singular object
Wcu (x∗ 1,r , 0, y). Hence we can use Lemma 2.6.3 and the stability of transversal
intersection (transversality is defined by a C 1 condition!) to conclude that W cu
intersects the stable manifold W s (S l,ǫ) of the compact part of the slow manifold
S l,ǫ associated to Cl×Iδ transversely. Now we can follow W cu close to S l,ǫ. Since
S l,ǫ is very close to Cl×Iδ forǫ > 0 sufficiently small by Fenichel Theory we
have that W cu - after we have followed it around - is close to the center-stable
manifold of the origin W cs . Hence can conclude the transversal intersection of
two-dimensional manifold W cu and the three-dimensional manifold W cs .
Remark: Note that this procedure not only defined a one-dimensional inter-
section curve fixing s0 close to s ∗ but also fixed a value y0 close to y ∗ determining
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