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

center-unstable manifold W cu (0, 0, s) to (x1, x2)=(0, 0). Similarly we get a two-
dimensional center-stable manifold W cs (1, 0, s) to the second saddle-equilibrium
of the fast subsystem (x1, x2)=(1, 0). We want that the two manifolds intersect
along the heteroclinic connection for s= s ∗ . The situation is abstractly shown in
Figure 2.8.
(0, 0, s ∗ )
W cu (0, 0, s)
W cs (1, 0, s)
(1, 0, s ∗ )
Figure 2.8: Sketch of transversal intersection of the manifolds W cu (0, 0, s)
and W cs (1, 0, s).
The geometric reason for the transversal intersection is that the heteroclinic
connection breaks for s s ∗ . There are many ways to prove this claim but for
now we shall just state it.
Lemma 2.6.2. W cu (0, 0, s) intersects W cs (1, 0, s) transversely in (x1, x2, s) space along
the curve defined by s= s ∗ .
A similar result should hold for the second heteroclinic connection from Cr
to Cl. Now fix s= s ∗ and let y vary in a neighborhood of y ∗ .
Lemma 2.6.3. W cu (x ∗
1,r , 0, y) intersects Wcs (x ∗
1,l , 0, y) transversely in (x1, x2, y) space
along the curve defined by y=y ∗ .
The next step is to define ¯
Cr as the compact part of Cr given by y∈[−δ, y ∗ +δ]
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