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We assume that points
(
y
0
, y
1
, y
2
, y
3
)
and
(
z
0
, z
1
, z
2
, z
3
)
in
R
4
\{
0
}
are
equivalent if there exists a positive constant
λ
such that
z
j
=
λ
4+
j
(
k
−
1)
y
j
,
j
= 0
,
1
,
2
,
3
.
The factor space obtained is homeomorphic to the three-dimensional
sphere
S
3
=
{
y
∈
R
4
:
y
2
0
+
y
2
1
+
y
2
2
+
y
2
3
= 1
}
.
On this sphere there is
exactly one representative of each equivalence class because for any
point
(
y
0
, y
1
, y
2
, y
3
)
∈
R
4
\{
0
}
the equation
λ
8
y
2
0
+
λ
2
k
+6
y
2
1
+
λ
4
k
+4
y
2
2
+
+
λ
6
k
+2
y
2
3
= 1
has exactly one positive root
λ
.
It is possible to construct another hyper-surface in
R
4
with a single
representative of each equivalence class, namely,
E
=
(
y
∈
R
4
:
3
X
j
=0
|
y
j
|
1
j
(
k
−
1)+4
= 1
)
.
(6)
We define
Φ
S
:
R
4
\{
0
} →
S
3
and
Φ
E
:
R
4
\{
0
} →
E
as mappings taking
each point in
R
4
\{
0
}
to the equivalent point in
S
3
or
E
. Note that the
restrictions
Φ
S
|
E
and
Φ
E
|
S
3
are inverse homeomorphisms.
Lemma 1.
There is a dynamical system on the sphere
S
3
such that
all its trajectories can be obtained by the mapping
Φ
S
from the curves
generated in
R
4
\{
0
}
by nontrivial solutions to equation
(1)
. Conversely,
any nontrivial solution to equation
(1)
generates in
R
4
\{
0
}
a curve whose
image under
Φ
S
is a trajectory of the above dynamical system.
J
First we define on the sphere
S
3
a smooth structure using an atlas
consisting of eight charts.
The two semi-spheres defined by the inequalities
y
0
>
0
and
y
0
<
0
are
covered by the charts with the coordinate functions (respectively
u
+
1
, u
+
2
,
u
+
3
and
u
−
1
, u
−
2
, u
−
3
) defined by the formulae
u
±
j
=
y
j
|
y
0
|
−
4+
j
(
k
−
1)
4
sgn
y
0
,
j
= 1
,
2
,
3
.
The semi-spheres defined by the inequalities
y
1
>
0
and
y
1
<
0
are
covered by the charts with the coordinate functions (respectively
v
+
0
, v
+
2
,
v
+
3
and
v
−
0
, v
−
2
, v
−
3
) defined as
v
±
j
=
y
j
|
y
1
|
−
4+
j
(
k
−
1)
k
+3
sgn
y
1
,
j
= 0
,
2
,
3
.
The semi-spheres defined by the inequalities
y
2
>
0
and
y
2
<
0
are
covered by the charts with the coordinate functions (respectively
w
+
0
, w
+
1
,
w
+
3
and
w
−
0
, w
−
1
, w
−
3
) defined as
w
±
j
=
y
j
|
y
2
|
−
4+
j
(
k
−
1)
2
k
+2
sgn
y
2
,
j
= 0
,
1
,
3
.
Finally, the semi-spheres defined by the inequalities
y
3
>
0
and
y
3
<
0
are covered by the charts with the coordinate functions (respectively
g
+
0
,
g
+
1
, g
+
2
and
g
−
0
, g
−
1
, g
−
2
) defined as
g
±
j
=
y
j
|
y
3
|
−
4+
j
(
k
−
1)
3
k
+1
sgn
y
3
,
j
= 0
,
1
,
2
.
Note that each of these coordinate functions can be defined by its
own formula on the whole corresponding semi-space (
y
j
≷
0
) and it takes
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2
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