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

x2
0.04
0.03
0.02
0.01
0
−0.01
−0.02
W u (Cl,ǫ)
W s (Cl,ǫ)
−0.03
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
x2
0.06
0.04
0.02
0
−0.02
x1
(a) p=0.059, s=1.37
W u (Cl,ǫ)
W s (Cl,ǫ)
W s (q)
−0.04
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
x1
(c) p=0.06, s=1.37
W s (q)
x2
0.04
0.03
0.02
0.01
0
−0.01
−0.02
W u (Cl,ǫ)
W s (q)
W s (Cl,ǫ)
−0.03
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1
s
1.39
1.385
1.38
1.375
1.37
1.365
x1
(b) p=0.0595, s=1.37
p
0.059 0.0592 0.0594 0.0596 0.0598 0.06
(d) Parameter space, region I.
Figure 5.3: Figures (a)-(c) show the movement of the stable manifold W s (q)
(cyan) with respect to E u (Cl,ǫ) (red) and E s (Cl,ǫ) (green) in phase
space on the section y=0.09 forǫ= 0.01. The parameter space
diagram (d) shows the homoclinic C-curve (solid red), an extension
of the C-curve of parameters where W u (q)∩W s (Cr,ǫ) is
nonempty, a curve that marks the tangency of W s (q) to E u (Cl,ǫ)
(blue) and a curve that marks a distance between Cl,ǫ and W s (q)
(dashed blue) of 0.01 where the arrows indicate the direction in
which the distance is bigger than 0.01. The solid black squares
in (d) show the parameter values for (a)-(c).
Figure 5.3(d). On this curve we verified that W s (Cl,ǫ) and W u (Cr,ǫ) still intersect
transversally by computing those manifolds; see [56, 55] for details.
The blue curves in Figure 5.3(d) have been obtained by measuring the Eu-
clidean distances between W s (q) and Cl,ǫ in the sectionΣ0.09. Along the dashed
blue curve the distance between Cl,ǫ and W s (q) is 0.01. The arrows indicate the
direction in which this distance increases. The solid blue curve marks a tan-
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