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not their boundaries). Thus, the set
Ω
+
∩
S
3
is homeomorphic to the pair
of two balls glued along two disjoint triangles, which is equivalent to the
solid torus.
I
Lemma 3.
Any trajectory in
R
4
generated by a non-trivial solution
to
(1)
either completely lies inside one of the sets
Ω
−
and
Ω
+
(
i.e., in their
interior
)
, or consists of two parts, first inside
Ω
−
and another inside
Ω
+
with a single point in their common boundary.
J
For the trajectories generated by solutions to equation (1), consider
all possible passages between the sets
±
±
±
±
.
Inside
Ω
+
the only possible passages are
+
+
+
+
→
+
+
+
−
→
+
+
−
−
→
+
−
−
−
↑
↓
−
+
+
+
←
−
−
+
+
←
−
−
−
+
←
−
−
−
−
,
(7)
inside
Ω
−
they are
+
−
+
−
←
+
−
+
+
←
+
−
−
+
←
+
+
−
+
↓
↑
−
−
+
−
→
−
+
+
−
→
−
+
−
−
→
−
+
−
+
,
(8)
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2
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