Chemistry Reference
In-Depth Information
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
θ
Figure 26.
=
4
.
5 and
γ
+
=
2. Dashed lines represent the equator
and the locus of the fixed points of the dynamics. The solid line corresponds to the antipodal parallel.
Numerical values of the parameters are taken to be
ϕ
(0)
=
2
π/
5,
p
θ
=
8,
r
(0)
=
1 and
p
r
(0)
=
0
.
25.
The initial values of
p
ϕ
are -50, -10, 0, 2.637, 3, 5, 10, and 50.
Extremal trajectories for
Numerical Determination of the Conjugate Locus
The goal of this section is to determine numerically the conjugate locus for
the two-input case. We restrict the discussion to the case
<
2. Following
the previous section, numerical computations are undertaken in spherical coordi-
nates for fixed coordinates
p
θ
and
p
ρ
. As already mentioned, the case
|
−
γ
+
|
can
be associated to the Grushin model, the drift vector field
F
0
being purely radial.
In this model, the conjugate locus is known and described in Section II.C. This
result is recalled in Fig. 27, where we represent the projection of the conjugate and
cut loci on the sphere of radius 1 in the coordinates (
θ, ϕ
). The radial dependence
is trivial in this case and depends on the value of
=
γ
+
. Here, the important
point to note is the fact that the projection of the conjugate locus on the sphere is
independent of
p
r
(0) for this model.
The idea is then to start from this model and to deform it by modifying the
parameters
γ
=
γ
+
<
2. A first comparison between
the two models is given by Figs. 27 and 28, where we observe that the global
structure of the extremals is nearly the same. The same conclusion is obtained from
the analysis of the radial coordinate whose evolution is not represented here. Figure
29 displays the projection of the conjugate locus on the sphere in the coordinates
(
θ, ϕ
) for a given value of
p
r
(0). We have added the locus of the Grushin model
and
with the constraint
|
−
γ
+
|
+
