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

trajectory flows along the completely unstable slow manifold Cm,ǫ. Different
types of MMOs seem to occur very frequently in single- and multi-parameter bi-
furcation problems; see [27] for a recent example. This contrasts with most work
on the analysis of MMOs [76, 98] that focuses on identifying the mechanism
for generating MMOs in an example. The MMOs in the FitzHugh-Nagumo
equation show that a fast-slow system with three or more variables can exhibit
MMOs of different types and that one should not expect a priori that a single
mechanism suffices to explain all the MMO dynamics.
5.7 Additions
We will give some additional details of the computations for the invariant man-
ifolds. The slow manifold Cl,ǫ is of saddle type. Hence a special algorithm is
needed to compute it. We used the boundary value approach (BVP) described
in [55] with boundary conditions well-away from the section
Σ0.09={(x1, x2, y)∈R 3 : y=0.09}
near Cl. The computed piece of Cl,ǫ was then intersected withΣ0.09. At the in-
tersection point the linearization of (5.1) is used to compute the eigenspaces
E s (Cl,ǫ) and E u (Cl,ǫ). The stable manifold W s (q) was computed by considering a
line segment L⊂E s (q) close to q such that
{φt(L), t→∞}∪{φt(L), t→−∞}≈W s (q)
whereφt denotes the flow of (5.1). Choosing a mesh on L we can use backward
integration to trace out trajectories in W s (q) that intersectΣ0.09. These intersec-
tions have been recorded and give W s (q) in Figure 5.3(a)-5.3(c). The integration
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