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

curve of homoclinic bifurcations on the right side of the U-shaped Hopf curve.
Since (3.6) has the symmetry
x1→ 11
15 − x1, x2→ 11
15 − x2, y→−y, p→ 11
15
1− 33
− p (3.7)
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we shall examine only the left side of the U-curve. The homoclinic C-curve is
difficult to compute numerically by continuation methods using AUTO [32, 33]
or MatCont [46]. The computations seem infeasible for small values ofǫ≤ 10 −3 .
Furthermore multi-pulse homoclinic orbits can exist very close to single pulse
ones and distinguishing between them must necessarily encounter problems
with numerical precision [18]. The Hopf curve and the bifurcations of limit cy-
cles shown in Figure 3.1 have been computed using MatCont. The curve of
homoclinic bifurcations has been computed by a new method to be described in
Section 3.4.2.
Since the bifurcation structure shown in Figure 3.1 was also observed for
other excitable systems, Champneys et al. [18] introduced the term CU-system.
Bifurcation analysis from the viewpoint of geometric singular perturbation the-
ory has been carried out for examples with one fast and two slow variables
[60, 11, 50, 92]. Since the FitzHugh-Nagumo equation has one slow and two
fast variables, the situation is quite different and new techniques have to be
developed. Our main goal is to show that many features of the complicated 2-
parameter bifurcation diagram shown in Figure 3.1 can be derived with a com-
bination of techniques from singular perturbation theory, bifurcation theory and
robust numerical methods. We accurately locate where the system has canards
and determine the orbit structure of the homoclinic and periodic orbits associ-
ated to the C-shaped and U-shaped bifurcation curves, without computing the
canards themselves. We demonstrate that the basic CU-structure of the system
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