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O.G. Styrt, A.P. Krishchenko
18
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 1
The inequality
2
inf { ( )}
( )
2
x
y
G x
G x
is always true, and, thus, the localizing set is
2
def
2
*
= ( , )
:
( )
.
2
y
x y
G x G
We have
*
( )
;
G x G
so,
2
= ( , )
:
( ) ,
x y
y x
def
*
( ) = 2(
( )).
x
G G x
The
graphs
of the functions
= ( )
y
x
are located in the upper and the lower
semiplains respectively symmetrically to each other across the axis
,
Ox
and the
bound of the localizing set equals
=
.
Also,
def
2
=
= ( , 0)
:
,
x
x K
where the set
def
*
=
: ( ) =
K x G x G
consists of all global maximum points of the
function
( ).
G x
The set
def
2
= \
U
is defined by the inequality
| |> ( )
y x
and has two connected
components
U
and
U
given by the inequalities
> ( )
y x
and
< ( )
y
x
respectively. The set
2
U
is defined by the inequality
( ),
y x
and, thus,
{ } = .
U Ox
The set
*
\ =
: ( ) < ,
K x G x G
being open, is the union of some countable
family
of pairwise disjoint intervals. The connected components of the set
\
are exactly all the subsets of the form
def
2
= {( , )
:
} = ( , )
:
,
( ) ,
.
I
x y
x I
x y
x I y x I
Statement 1.
The function
h
is decreasing on each trajectory of the system (5).
◄
The proof follows from non-positivity of the function
.
h
►
Corollary 1.
The set
is positively invariant. In other words, any trajectory, once
entering
,
never leaves it.
◄
The set
is given by the inequality
*
.
h G
It remains to apply Statement 1.
►
Corollary 2.
The bound
of the set
is semipermeable: trajectories can
intersect it only entering the localizing set
.
The set of all equilibrium points is the set
def
2
0
0
= ( , 0)
:
,
x
x K
def
0
= {
: ( ) = 0}
.
K x g x
If
,
x K
then
x
is a global maximum point of the function
,
G
implying
( ) = ( ) = 0
g x G x
and
0
.
x K
Thus,
0
.
K K
It follows that
0
.
According to