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

Then let H(Z, X, t) := X ′ − BX−E(X, t) and use the triangle inequality again. This
shows (iv) and completes the proof.
Step 3: We want to incorporate the assumptions of the Exchange Lemma
into estimates on the initial conditions of Zi and Xi at q. We assumed that the
trajectory spends at least O(1/ǫ) in a box B and we also have a transversality
condition at the entry point q. This will constrain the values of (Z, X) on TqM.
Let ˆXi := Xi/Z1 and ˆ
Z2= Z2/Z1 as suggested in the differential form version of the
Exchange Lemma in Theorem 2.4.2.
Lemma 2.5.2. There exists ¯K> 0 such that at TqM we have|Z1|> ¯Kǫ,| ˆXi|
¯Kǫ. Therefore
| ˆXi(TqM)|= |Xi(TqM)|
|Z1(TqM)| < 1/( ¯Kǫ).
It remains to show that|ˆZ2| is exponentially small and so we must try to estimate
|Z2|. Here we view TqM as spanned by two vectors, one defined by the vector
field (2.21), denoted by v, and a vector orthogonal to it in TqM, say v ⊥ . Up to a
scalar multiple we find that
⎛
|Z2(TqM)|=const·
det
⎜⎝
v1 v4
v ⊥
1
v ⊥
4
⎞
⎟⎠
= const·|v1v ⊥
4
⊥
− v4v1 | (2.29)
since evaluating the differential form da∧dy2= Z2 on two vectors amounts to
picking out the a and y2 components and then applying the determinant. By
the definition of v we look at equation (2.21) and see that|v1| is O(|a|). Since|v ⊥
4 |
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