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

withδ>0 sufficiently small. The same holds forǫ> 0 sufficiently small so that
(2.18) is still valid and we can findλǫ> 0 andγǫ< 0 which are the weak unstable
and weak stable eigenvalues near Sǫ. In equations (2.18) we can rectify the slow
flow. Without loss of generality we assume that the slow flow is pointing in
the direction y1 so that we can reduce the problem of analyzing the flow near a
normally hyperbolic slow manifold Sǫ to:
a ′ = Λ(a, b, y,ǫ)a
b ′ = Γ(a, b, y,ǫ)b (2.19)
y ′ = ǫ(U+ H(a, b, y,ǫ)(a, b))
where U = (1, 0,...,0) T . Observe that the stable and unstable manifolds W s
and W u of Sǫ are given by{a=0} and{b=0} respectively. Let M be a (k+1)-
dimensional invariant manifold. M is the manifold we want to follow in phase
space. We remark that the dimensional requirement on M can be generalized
but for simplicity we shall only consider the case of (k+ 1) dimensions. Suppose
M intersects the boundary of the region/box B in{b=δ} at some point q. If q is
close enough to the stable manifold W s (Sǫ)={a=0} then a trajectory starting at
q stays near Sǫ for a long time (e.g. a time that is O(1/ǫ) on the fast time scale t).
We want to find estimates on the fast coordinates (a, b) to quantify the situation
more precisely.
Lemma 2.3.1. There exists constants ca, cb, K> 0 such that for s≤t the following
three results hold
(R1)|b(t)|≤cb|b(s)|e γ0(t−s)
(R2)|a(t)|≥ca|a(s)|e λ0(t−s)
(R3)| s
a(σ)dσ|≤K (independent ofǫ, t, s)
t
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