5 / 23
dg
±
0
dt
q
=
g
±
1
+
4
3
k
+ 1
p
0
g
±
0
k
+1
;
dg
±
1
dt
q
=
g
±
2
+
k
+ 3
3
k
+ 1
p
0
g
±
1
g
±
0
k
sgn
g
±
0
;
dg
±
2
dt
q
= 1 +
2
k
+ 2
3
k
+ 1
p
0
g
±
2
g
±
0
k
sgn
g
±
0
.
Using a partition of unity one can obtain a dynamical system on the
whole sphere
S
3
to describe all trajectories generated by nontrivial solutions
to equation (1).
I
Typical and Non-Typical Solutions.
Now we consider the space
R
4
as the union of its
16 = 2
4
closed subsets defined according to different
combinations of signs of the four coordinates. Denote these sets by
±
±
±
±
⊂
R
4
,
where each sign
±
can be substituted by
+
,
or
−
,
or
0
(for
boundary points). For example,
+
+
0
−
=
y
∈
R
4
:
y
0
≥
0
, y
1
≥
0
, y
2
= 0
, y
3
≤
0
, .
Besides, let
Ω
−
and
Ω
+
denote respectively
+
−
+
−
∪
+
−
+
+
∪
+
−
−
+
∪
+
+
−
+
∪
−
+
−
+
∪
−
+
−
−
∪
−
+
+
−
∪
−
−
+
−
and
+
+
+
+
∪
+
+
+
−
∪
+
+
−
−
∪
+
−
−
−
∪
−
−
−
−
∪
−
−
−
+
∪
−
−
+
+
∪
−
+
+
+
.
Note, that the sets
Ω
−
and
Ω
+
cover the whole space
R
4
,
intersect
only along their common boundary, and can be obtained from each other
using the mapping
(
y
0
, y
1
, y
2
, y
3
)
∈
R
4
→
(
y
0
,
−
y
1
, y
2
,
−
y
3
)
∈
R
4
,
which
corresponds to changing the sign of the independent variable (
x
→ −
x
).
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2
7