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

since|a| is exponentially small,|b|≤δ and ˆZ≤1 (see Lemma 2.5.4). Then we
can use (2.32) and get
ˆXi ≤ ¯M ˆX0e
≤ ¯M
1
≤ ¯M
1
¯Kǫ e−µ 2 t + 4K
µ (ǫ+δ)
t
e s β1(r)dr
β2(s)ds
t
0 β1(s)ds
t
+
0
¯Kǫ e−µ 2 t t
+ e
0
−µ
2 (t−s) 2K(ǫ+δ)ds
(2.38)
Looking at (2.38) we see that the first term is exponentially small for T1= O(1/ǫ)
and so we haveˆX≤O(ǫ+δ). Since|a| is exponentially small we also have
α(t)+2C|a|≤K2e −c2/ǫ =:κ (2.39)
for some K2, c2> 0. Using (2.31) and the usual Gronwall Lemma we compute
| ˆZi(t)| ≤ | ˆZi|e κt +
t
0
e κ(t−s) κds
= | ˆZi(0)|e κt + e κt − 1 (2.40)
But sinceκis exponentially small and t≤O(1/ǫ) we see thatκt must be expo-
nentially small. But e κt − 1 ≈ κt which means that the second term of (2.40)
is exponentially small. In Lemma 2.5.2 we showed that| ˆZi(0)| is exponenen-
tially small and this implies that| ˆZi| is exponentially small. This establishes the
two estimates we want for Step 1 as long as we can show that ˆZ ≤ 1 and
ˆX≤1/( ¯Kǫ). Indeed, we know that| ˆZi(0)| is exponenentially small and so there
is a time 0