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

y
1
0
C
−1
−2 −1 0 1 2
(a)λ=0: Singular solution in black.
x
y
1
0
C
−1
−2 −1 0 1 2
(b)λ=1: Equilibrium point at p+.
Figure 1.1: The critical manifold (blue) C= C VdP for Van der Pol’s equation
(1.3) together with the y-nullcline (dashed red) is shown. Double
arrows indicate the fast flow and single arrows the slow
flow.
Away from fold points the implicit function theorem applied to f (x, y,λ, 0)=
0 locally provides a function h(y)= x so that C can be expressed as a graph.
Hence the slow subsystem (1.5) can be more succinctly expressed as:
˙y=g(h(y), y,λ, 0) (1.7)
We shall also refer to the flow induced by (1.5),(1.7) as the slow flow. In Van der
Pol’s equation we could solve the cubic equation y=c(x) for x on S a,− , S r and S a,+
to obtain (1.7). It is more convenient in this case to use an alternative procedure
to derive the slow flow. We implicitly differentiate f (x, y,λ, 0)=y−c(x)=0 with
respect toτ, then
˙y= ˙xx 2 − ˙x= ˙x(x 2 − 1)
Combining this result with the equation for ˙y we get:
(x 2 − 1) ˙x=λ− x or ˙x=
λ− x
x 2 − 1
x
(1.8)
Note that the slow flow is not well-defined for x=±1 as long asλ±1. In
particular, existence and uniqueness theory for ODEs does not apply in this
scenario. The same idea for deriving the slow flow also works for a general
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