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

implies using the Fundamental Theorem of Calculus that
|yi(t)−yi(0)|≤ǫ
t
0
di|a(σ)|dσ (2.20)
Using Lemma 2.3.1, (R3) we can conclude that the right-hand side of (2.20) is
O(ǫ).
Often Theorem 2.3.2 is called a C 0 -Exchange Lemma and we might ask why
there is any need to consider a refined version of it as we just tracked the mani-
fold M near the slow manifold Sǫ. The problem is that every trajectory exits near
¯q almost tangent to the unstable manifold W u (Sǫ). Hence we have no informa-
tion about the part of the tangent spaces of M in the center directions. In this case
we cannot rely on any results about transversality obtained in the singular limit
ǫ= 0 for an intersection of W u (S 0) with some other manifold, say N, to conclude
that M is transversal to N forǫ > 0. We have information about the location
of the manifold M itself (“C 0 -information”) but not sufficient knowledge about
its tangent spaces (“C 1 -information”). As the tangent spaces determine whether
an intersection is transversal the C 0 -Exchange Lemma is insufficient. Note care-
fully that the situation just described occurs for the FitzHugh-Nagumo equation
(2.15) if we try to follow the unstable manifold of the unique equilibrium point
during its second jump; see Figure 2.7.
2.4 The Exchange Lemma II
The C 1 -closeness result we want is the following (cf. Figure 2.6):
Theorem 2.4.1 (Exchange Lemma). Let M be a (k+1)-dimensional invariant mani-
fold. Assume M∩{|b|=δ} intersects{a=0} transversely. Let ¯q∈ M∩{|a|=δ} be the
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