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

Theorem 2.3.2. Let ¯q ∈ M∩{|a| = δ} be the exit point of a trajectory starting at
q∈ M∩{|b|=δ} that spends a time t that is O(1/ǫ) in B. Let V be a neighborhood of q
in M. If V is sufficiently small then the image of V under the time t map is close to
{|a|=δ, yi− yi(0)=0, i>1}
in the C 0 -norm where yi(0) denotes the y-coordinates of q.
The situation is illustrated in Figure 2.6.
a=0
Sǫ
V
q
b=0
¯q
φt(V)
Figure 2.6: In this picture we have suppressed all coordinates yi with
i > 1. The image of the neighborhood V near the exit point
¯q is denoted byφt(V); it is very close to the unstable manifold
W u (Sǫ)={|b|=0} near the exit point.
Proof. From Lemma 2.3.1, (R1) we find that b(t) is small. Hence we are left with
the yi coordinates with i>1. Since b(t) is small we clearly have for i>1:
y ′ ⎛ k
⎞
i≤ǫ ⎜⎝ Hiuau⎟⎠
:=ǫ ¯Hi· a
u=1
where ¯Hi is a k-vector of functions. As ¯Hi is smooth and B is compact we can let
di be a bound for| ¯Hi|. Therefore
t
0
yidσ≤ǫ
34
t
0
¯Hi· adσ
b
y
a