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

3.6 Additions
In Section 3.3.3 we considered the re-scaling ¯x2=x2/ǫ giving a system which
formally has two slow and one fast variable (3.14). Note that the slow flow of
(3.14) is another fast-slow system with one fast and one slow variable as shown
in Proposition 3.3.2. This suggests that there should exist a direct re-scaling that
converts the FHN equation into a three time-scale system [80, 81]. Consider the
FitzHugh-Nagumo equation on the fast time scale:
x ′
1
x ′
2
= x2
= 1
5 (sx2− f (x1)+y− p) (3.24)
y ′ = ǫ
s (x1− y)
If we make the general re-scaling ansatz for (3.24) given by
(x1, x2, y, t, s, p)↦→ (ǫ α X1,ǫ β X2,ǫ γ Y,ǫ δ T,ǫ ρ S,ǫ σ P)
it is not difficult to derive algebraic equations for (α,β,γ,δ,ρ,σ) so that (3.24) has
a three-scale structure of the form:
X ′
1
= F1
ǫX ′
2 = F2 (3.25)
Y ′ = ǫG
for functions F1, F2 and G. This yields the next proposition.
Proposition 3.6.1. Consider the re-scaling
(x1, x2, y, t, s, p)↦→ (X1,ǫ 1/2 X2,γY,ǫ −1/2 T,ǫ −1/2 S, P) (3.26)
93