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

We shall not prove this result but refer to [103]. Note that Szmolyan in [103]
uses a technical result contained in [95]. To illustrate how the result works we
will discuss an example. We shall need that the solution of the adjoint varia-
tional equation (2.6) in the case of two fast variables n=2 is given by:
ψ(t)=e − t
0 tr(Dx f (x0(s),y0,0))ds ′
(−x 2 (t), x′ 1 (t))T
(2.9)
The example to be discussed is a toy model to illustrate the necessary computa-
tions in a simple setting.
Example 2.1.3. Consider the (2,1)-fast-slow system
ǫ ˙x1 = 1−(x1) 2
ǫ ˙x2 = y+ x1x2 (2.10)
˙y = y 2 − (x1) 2
The critical manifold C0 is easily found and consists of two lines L±:
C0 = {(x1, x2, y)∈R 3 : x1=−1 and y= x2 or x1= 1 and y=−x2}
= Lx1=−1∪Lx1=+1=: L−∪L+
A projection into the (x1, y)-plane of the situation with the singular flows of the fast and
slow subsystems is shown in Figure 2.3.
In Figure 2.3 we have shown a singular heteroclinic orbit which connects the two
saddle equilibria p1= (−1, 1, 1) and p2= (1, 1,−1). The unstable manifold of the point
p1 is L− and the stable manifold of p2 is L+. Then define the manifolds N 1 and N 2 as two
planes
N1={(x1, x2, x2)∈R 3 : x1∈ [−1, 1]}
N2={(x1, x2,−x2)∈R 3 : x1∈ [−1, 1]}
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