8 / 11
O.G. Styrt, A.P. Krishchenko
22
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 1
0
( ) = ( )
max ( ) .
x
G x g t dt T g x
Thus,
*
=sup{ ( )} < .
x
G G x
The function
G
is periodic and
*
.
G G
Consequently, the integrals
*
0
( )
G G x dx
and
0
*
( )
G G x dx
are divergent. It
remains to apply Corollary 4.
►
Now we give one more corollary of Proposition 1.
Corollary 6.
If
is completely located outside the localizing set
and
*
*
= ,
h G
then
( )
0,
t
y x
i. e.
dist( ( ), )
0.
t
t
Example 1.
Consider the equation
2
1
sin = 0.
x
x
x
x x
In the notations
above, we have
2
1
( , ) =
,
f x y
x y
( )=sin .
g x x
Hence,
( ) = 1 cos ,
G x
x
*
= 2 < .
G
The localizing set
2
is defined by the inequality
2
/2 cos 1.
y
x
Besides,
2
2
= ( ) = ( , 0)
: cos = 1 = ((2 1) , 0)
:
.
Ox x
x
n
n
For each
,
x
we have
( ) 0,
G x
*
( ) 2,
G G x
*
( ) = 2(
( )) 2.
x
G G x
Take
an arbitrary number
> 0.
C
If
( , )
( ),
x y U C
then
| |< ( )
2 ,
y x C C
x
def
2
2
< = 3 / 2 (2 ) ,
y M
C
( , )>1/ .
f x y М
Thus,
inf{ ( ):
( )} 1/ >0.
f p p U C М
Since the number
> 0
C
is chosen arbitrarily, the condition (6) holds and, by
Corollary 5, any trajectory either tends to some equilibrium point
((2 1) ,0) ,
n
P n
,
n
of the "saddle" type or enters the localizing set
and then never leaves it.
The equilibrium points
n
P
divide the set
into the bounded positively invariant
sets
def
= ( , )
:
2 < ,
.
n
x y
x n
n
Each of the sets
n
contains one stable equilibrium point
(2 , 0).
n
By LaSalle
theorem, this equilibrium point attracts all trajectories in
.
n
It follows from above that any trajectory either tends to one of the equilibrium
points
((2 1) , 0)
n
or enters a set
n
and then, not leaving it, tends to the stable
equilibrium point
(2 , 0).
n
Remark.
If the condition (6) holds, the function
g
has period
,
T
and
0
( ) = 0,
T
g t dt
then the system (5) does not have periodic trajectories and each of its trajectories
tends to one of its equilibrium points.
Example 2.
Consider the equation
2 2
= 0
(1 )
x
x x
x
and prove its global asymptotic stability.