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

Proof. First, we work near q, then|Z1|0 and so we calculate:
|Z1| ′ = d
Z
dt
2
1
2Z′
=
2
= ΛZ2 1
|Z1| +η11Z1
|Z1|
Using Lemma 2.5.1 we then find that
|Z1| ′ ≥ Λ|Z1|−(|F1|+|G1|)
1 Z1
Z 2
1
=Λ|Z1|+ η11Z1
|Z1|
≥ (Λ− ˜C|a|)|Z1|− ˜C|a|X−ǫ ˜K|a|(Z+X)
Now choose C such that C> ˜C+ǫ ˜K and C> ˜K and we obtain (2.30) near q. By
making the box B sufficiently small we can make|a| small enough so that, since
Λ≥Λ0> 0, we always have
(Λ− C|a|(1+ˆX+ǫ ˆZ))>0
Hence we find near q that|Z1| ′ > 0 and so|Z1| is increasing; if we now leave a
neighbourhood of q then we still have|Z1|> ¯Kǫ. Repeating the argument above
in a compact set away from q and covering B by finitely many compact sets
yields|Z1|0 inside B and the estimate (2.30).
Step 5: The next lemma is fundamental to control ˆZi inside B.
Lemma 2.5.4. There are constants C, K> 0, where C is as in Lemma 2.5.3, so that the
following inequalities hold
| ˆZi| ′ ≤ (α(t)+2C|a|)| ˆZi|+α(t) (2.31)
t
ˆXi ≤ ¯M ˆX0e 0 β1(s)ds
t t
+ e s β1(r)dr
β2(s)ds
(2.32)
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