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

One can view this as a projection of the slow flow, which is constrained to the
critical manifold in R 3 , onto the x1-axis. Observe that the slow flow is singu-
lar at the fold points. Direct computation shows that the fixed point problem
x1 = c(x1) has only a single real solution. This implies that the critical mani-
fold intersects the diagonal y= x1 only in a single point x∗ 1 which is the unique
equilibrium of the slow flow (3.8). Observe that q=(x ∗
1 , 0, x∗
1 ) is also the unique
equilibrium of the full system (3.6) and depends on p. Increasing p moves the
equilibrium from left to right on the critical manifold. The easiest practical way
to determine the direction of the slow flow on C0 is to look at the sign of (x1− y).
The situation is illustrated in Figure 3.2.
y
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
eq. pt. q
x 1,−
Slow Flow in the plane x 2 =0
y=x 1
−0.4
−0.4 −0.2 0 0.2 0.4
x
1
0.6 0.8 1 1.2
Figure 3.2: Sketch of the slow flow on the critical manifold C0
3.3.1 The Slow Flow
We are interested in the bifurcations of the slow flow depending on the parame-
ter p. The bifurcations occur when x∗ 1 passes through the fold points. The values
68
x 1,+
C 0