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

Recall from Lemma 2.5.1 that we have the estimate
t
exp (B−ΛId)dr ≤ ¯Me
s
−µ(t−s)
Therefore we can now estimateˆX as follows:
t
ˆX ≤ J(t) −1 ˆX0+
≤ ¯M e −µt ˆX0+
J(t) −1
J(s)
η2
−
Z1
η11
ˆX
Z1
ds
e −µ(t−s)
(β3(s) ˆX+β2(s))ds
0
t
0
(2.35)
The last equation is in a form to apply a generalized Gronwall inequality which
states that if u, v, c≥0, c is differentiable and v(t)≤c(t)+ t
u(s)v(s)ds then
t t
v(t)≤c(0) exp u(s)ds+ c
0
0
′ t
(s) exp u(r)dr ds
s
For a proof see [20]. If we multiply (2.35) by e µt and apply the generalized Gron-
wall inequality we obtain:
e µt
t
ˆX≤ ¯M ˆX0e 0 β3(s)ds
t
+ e
0
µs t
β2(s)e s β3(r)dr
ds
We know from the size of the neighbourhood/box B that|a|