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

the components of tangent planes of M at points of trajectories in B will always
have non-vanishing components in the ai and the y1 directions. In fact, the hy-
perplane H spanned by the ai and the y1 coordinates can be defined by requiring
that
Pσ1···σk+1 (v)=0 for all v∈Hand (σ1,...,σk+1)(a1,...,ak, y1)
Therefore we can easily restate the Exchange Lemma in an equivalent form us-
ing differential forms.
Theorem 2.4.2 (Exchange Lemma - differential form conclusion). Under the same
assumptions as in Theorem 2.4.1 we conclude that at ¯q:
for all (σ1,...,σk+1)(a1,...,ak, y1).
ˆPσ1···σk+1
= O(ǫ)
In the next section we give a proof of the Exchange Lemma for the simplest
non-trivial case in phase space dimension four for a system with two fast and
two slow variables.
2.5 A Proof in R 4
Here we prove Theorem 2.4.2 for a (2, 2)-fast-slow system in Fenichel Normal
Form near the slow manifold Sǫ with z := (a, b, y1, y2) and a, b, y1, y2∈ R
a ′ = Λ(z,ǫ)a
b ′ = Γ(z,ǫ)b (2.21)
y ′
1 = ǫg1(z,ǫ)=ǫ(1+ H1(a, b, y,ǫ)ab)
y ′
2 = ǫg2(z,ǫ)=ǫ(H2(a, b, y,ǫ)ab)
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