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

The return map of the Shilnikov model is constructed from two components:
the flow map past an equilibrium point, approximated by the flow map of a lin-
ear vector field, composed with a regular map that gives a “global return” of
the unstable manifold of the equilibrium to its stable manifold [54]. Place two
cross-sectionsΣ1 andΣ2 moderately close to the equilibrium point and model
the flow map fromΣ1 toΣ2 via the linearization of the vector field at the equilib-
rium.
x2
y
x1
R
C0
Figure 5.6: Sketch of the geometric model for the homoclinic bifurcations.
Only parts of the sectionsΣi for i=1, 2 are shown.
The degree one coefficient of the characteristic polynomial at the equilibrium
has order O(ǫ), so the imaginary eigenvalues at the Hopf bifurcation point have
magnitude O(ǫ 1/2 ). The real part of these eigenvalues scales linearly with the
distance from the Hopf curve. Furthermore we note that the real eigenvalue of
the equilibrium point remains bounded away from 0 asǫ→ 0.
Letψ(x1, x2, y)=(u, v, w) be a coordinate change near q so thatψ(q)=0 and
the vector field is in Jordan normal form up to higher order terms. We denote
the sections obtained from the coordinate change into Jordan form coordinates
135
ψ
u
v
0
Σ1
w
F12
Σ2
F21