The Exchange Lemma was initially proved by Jones and Kopell [68]. We shall mainly follow their original paper and an expository presentation in [74]. Recall from the problem of finding homoclinic orbits in the FitzHugh-Nagumo with more than one jump discussed in the previous section that we have to track an invariant manifold in phase phase. For the general situation we start with a fast-slow system
ǫ ˙x = f (x, y, ǫ)
˙y = g(x, y, ǫ) (2.17)
with x ∈ Rm and y ∈ Rn. We include in (2.17) possible parameters in the sys-
tem in the vector y as slow variables. In the FitzHugh-Nagumo equation that would mean including the equation for the wave speed ˙s = 0 and re-labeling s to a suitable indexed y-coordinate. Let S0 denote some compact normally hy-
perbolic submanifold of the critical manifold and let Sǫ be the corresponding
slow manifold. Note that we shall assume that S0 is in fact uniformly normally
hyperbolic meaning that Dxf (p)has eigenvalues uniformly bounded away from
zero for each p ∈ S0.
a = 0 b = 0 a b y S0
Figure 2.5: A fast-slow system near a normally hyperbolic critical mani- fold S0 in Fenichel Normal Form. In this picture we have sup-
pressed all coordinates yi with i > 1.
normally hyperbolic critical manifold S0into Fenichel normal form: a′ = Λ(a, b, y, ǫ)a
b′ = Γ(a, b, y, ǫ)b (2.18)
y′ = ǫ(h(y, ǫ) + H(a, b, y, ǫ)(a, b)) with a ∈ Rk
, b ∈ Rl
, y ∈ Rn and k + l = m; Λ and Γ are matrix-valued functions
and H is a bilinear-form valued function which can be written explicitly as
y′i = ǫ hi(y, ǫ) + k X u=1 l X s=1 Hiusaubs
Sometimes also the tensor notation H(a, b, y, ǫ) ⊗ a ⊗ b = H(a, b, y, ǫ)(a, b) is em- ployed. The matrix-valued functions Λ and Γ are defined by separating the fast unstable and fast stable directions. More precisely, in (2.18) the critical manifold
S0 is {a = 0, b = 0} and normal hyperbolicity implies there are λ0, γ0 such that
any eigenvalue λi of Λ(0, 0, y, 0) or any eigenvalue γi of Γ(0, 0, y, 0) satisfies
Re λi > λ0 > 0, Re γi < γ0 < 0
in a region
with δ > 0 sufficiently small. The same holds for ǫ > 0 sufficiently small so that (2.18) is still valid and we can find λǫ > 0and γǫ < 0which 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 Sǫto:
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 Ws
and Wu of S
ǫ 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
Mintersects the boundary of the region/box B in {b = δ} at some point q. If q is
close enough to the stable manifold Ws(S
ǫ) = {a = 0} then a trajectory starting at
qstays 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)|eγ0(t−s)
(R2) |a(t)| ≥ ca|a(s)|eλ0(t−s)
as long as a trajectory remains in B.
Proof. We start by proving (R1). Since the eigenvalues of Λ(a, b, y, ǫ) are close to the ones of Λ(0, 0, y, ǫ) we see that near each point z = (a, b, y) ∈ B there is a neighborhood N of z such that an estimate of the form
|b(t)| ≤ CN|b(s)|eγ0(t−s)
holds if b(σ) ∈ N with σ ∈ (s, t) and ǫ sufficiently small. By compactness of B we can cover B by a finite number of such neighborhoods. In fact, if we con- sider all trajectories segments lying in B we can cover each segment by an open cover. Taking the union of those open covers will cover B and then we extract a finite subcover by compactness. Therefore we get an estimate for an arbitrary trajectory given by
|b(t)| ≤ CN1CN2· · · CNm|b(s)|e
γ0(t−s)
for some finite fixed m ∈ N. Now (R1) follows and (R2) is immediate by using the same method of proof as for (R1). This leaves the estimate (R3). Using (R2) we find
|a(σ)| ≤ 1
ca|a(t)|e
γ0(σ−t)
for σ ≤ t Integrating both sides of the last equation yields:
Z t s |a(σ)|dσ ≤ 1 ca|a(t)| Z t s eλ0(σ−t) dσ ≤ |a(t)| caλ0(1 − e λ0(s−t))
Since |a(t)| ≤ δ and λ0 > 0we can conclude (R3) holds.
Now we can track trajectories inside B. Since the manifold M is invariant a trajectory starting at q ∈ M ∩ {|b| = δ} has to stay inside M for all time. Hence we can follow a neighborhood of q in M under the flow as shown by the next result.
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 C0-norm where y
i(0)denotes the y-coordinates of q.
The situation is illustrated in Figure 2.6.
a = 0 b = 0 a b y Sǫ V φt(V) q ¯q
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
¯qis denoted by φt(V); it is very close to the unstable manifold
Wu(S
ǫ) = {|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′i ≤ ǫ k X u=1 Hiuau := ǫ ¯Hi· a
where ¯Hiis a k-vector of functions. As ¯Hiis smooth and B is compact we can let
dibe a bound for | ¯Hi|. Therefore
Z t 0 yidσ ≤ ǫ Z t 0 ¯ Hi· adσ
implies using the Fundamental Theorem of Calculus that
|yi(t) − yi(0)| ≤ ǫ
Z 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 C0-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
¯qalmost tangent to the unstable manifold Wu(S
ǫ). 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 Wu(S
0)with some other manifold, say N, to conclude
that M is transversal to N for ǫ > 0. We have information about the location of the manifold M itself (“C0-information”) but not sufficient knowledge about
its tangent spaces (“C1-information”). As the tangent spaces determine whether
an intersection is transversal the C0-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.