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

of p can simply be found by solving the equations c ′ (x1)=0 and c(x1)− x1= 0
simultaneously. The result is:
p−≈ 0.0511 and p+≈ 0.5584
where the subscripts indicate the fold point at which each equilibrium is located.
The singular time-rescaling ¯τ= sc ′ (x1)/τ of the slow flow yields the desingu-
larized slow flow
dx1
d¯τ = x1− c(x1)= x1+ x1
10 (x1− 1) (10x1− 1)− p (3.9)
Time is reversed by this rescaling on Cl and Cr since s>0 and c ′ (x1) is negative
on these branches. The desingularized slow flow (3.9) is smooth and has no
bifurcations as p is varied.
3.3.2 The Fast Subsystem
The key component of the fast-slow analysis for the FitzHugh-Nagumo equa-
tion is the two-dimensional fast subsystem
x ′
1
x ′
2
= x2
= 1
5 (sx2−x1(x1− 1)( 1
10 − x1)+y− p) (3.10)
where p≥0, s≥0 are parameters and y is fixed. Since y and p have the same
effect as bifurcation parameters we set p−y= ¯p. We consider several fixed y-
values and the effect of varying p (cf. Section 3.4.2) in each case. There are either
one, two or three equilibrium points for (3.10). Equilibrium points satisfy x2= 0
and lie on the critical manifold, i.e. we have to solve
0= x1(x1− 1)( 1
10 − x1)+ ¯p (3.11)
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