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

is bounded by a constant we have an O(|a|) estimate on the first term of (2.29).
Next, we recall that the equation for y ′
2 in the Fenichel Normal form is
y ′
2 =ǫg2(z,ǫ)=ǫ(H2(a, b, y,ǫ)ab)
This means that v4 also contains a multiplicative factor of O(|a|) when we esti-
mate it. If we could show that|a| is exponentially small at q then the result will
follow easily. Indeed, Lemma 2.3.1 says that the a-coordinate will expand at
least by a positive rateΛ0>0 inside B. Now we use the hypothesis that the
trajectory through q stays an O(1/ǫ) time in B and therefore|a| must have been
exponentially small at q
|a| 0.
Hence|Z2| is exponentially small which immediately yields that| ˆZ2| is exponen-
tially small.
Step 4: Our next goal is to estimate the size of ˆZi and ˆXi evaluated on the
tangent space to M which evolves under the flow. Our reference solution is the
trajectory starting at q and ending up at ¯q. We know from Lemma 2.5.2 that
|Z1|> ¯Kǫ which means that in a neighbourhood of q the projectivized forms are
well-defined (i.e. the denominator is nonzero). The next lemma provides this
well-definedness inside B.
Lemma 2.5.3. There is a constant C> 0 such that
|Z1| ′ ≥ (Λ− C|a|(1+ˆX+ǫ ˆZ))|Z1| (2.30)
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