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

3.4 The Full System
3.4.1 Hopf Bifurcation
The characteristic polynomial of the linearization of the FitzHugh-Nagumo
equation (3.6) at its unique equilibrium point is
P(λ)= ǫ
5s +
− ǫ
s −λ
− 1
50
+ 11x∗
1
25
3(x∗ 1
− )2
−
5
sλ
5 +λ2
Denoting P(λ)=c0+ c1λ+c2λ 2 + c3λ 3 , a necessary condition for P to have pure
imaginary roots is that c0= c1c2. The solutions of this equation can be expressed
parametrically as a curve (p(x∗ 1 ), s(x∗ 1 )):
s(x ∗
1 )2 =
50ǫ(ǫ− 1)
1+10ǫ− 22x ∗
1
+ 30(x∗
1 )2
p(x ∗
1 ) = (x∗ 1 )3− 1.1(x ∗
1 )2 + 1.1 (3.21)
Proposition 3.4.1. In the singular limitǫ → 0 the U-shaped bifurcation curves of
the FitzHugh-Nagumo equation have vertical asymptotes given by the points p− ≈
0.0510636 and p+ ≈ 0.558418 and a horizontal asymptote given by{(p, s) : p ∈
[p−, p+] and s=0}. Note that at p± the equilibrium point passes through the two
fold points.
Proof. The expression for s(x ∗
1 )2 in (3.21) is positive when 1+10ǫ−22x ∗
1 +30(x∗
1 )2 <
0. For values of x∗ 1 between the roots of 1−22x∗ 1 + 30(x∗ 1 )2 = 0, s(x∗ 1 )2→ 0 in (3.21)
asǫ→ 0. The values of p− and p+ in the proposition are approximations to the
value of p(x∗ 1 ) in (3.21) at the roots of 1−22x∗ 1 + 30(x∗ 1 )2 = 0. Asǫ→ 0, solutions
of the equation s(x∗ 1 )2 = c>0 in (3.21) yield values of x∗ 1 that tend to one of the
two roots of 1−22x∗ 1 + 30(x∗ 1 )2 = 0. The result follows.
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