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

s
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Hom
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
p
I
Hopf
Figure 5.1: Bifurcation diagram for the FitzHugh-Nagumo equation (3.6).
Shil’nikov homoclinic bifurcations (solid red) and Hopf bifurcations
(solid blue) are shown forǫ= 0.01. The dashed curves
show the singular limit (ǫ= 0) bifurcation curves for the homoclinic
and Hopf bifurcations; see [56] and Section 5.3 for details
on the singular limit part of the diagram.
point at the top end of the C-curve and denote this region by I.
We regardǫ in the FitzHugh-Nagumo equation (5.1) as a small parameter.
In [56], we derived a singular bifurcation diagram which represents several im-
portant bifurcation curves in (p, s)-parameter space in the singular limitǫ= 0.
The singular limits of the Hopf and homoclinic curves are shown in Figure 5.1
as dotted lines. 1 In the singular limit, there is no gap between the Hopf and
homoclinic curves. We demonstrate below in Proposition 2.1 that a gap must
appear forǫ > 0. The main point of this paper is that the termination point
of the C-curve at the end of the gap is due to a fast-slow “bifurcation” where
1 In Section 5.3 we recall the precise meaning of the singular limit bifurcation from [56] and
how they these bifurcations arise whenǫ= 0.
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