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O.G. Styrt, A.P. Krishchenko
16
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 1
We will further suppose that
f
and
g
in (1) are continuous and Lipschitz
functions, the function
e
is continuous and bounded, and, unlike the most of famous
works, we will also suggest that the function
def
0
( ) = ( )
x
G x g x dx
is upper bounded. Thus,
def
*
= sup{ ( )} < .
x
G G x
In these and some other assumptions, the paper makes qualitative research of
behavior of solutions of the system (1) based on the method of localizing compact
invariant sets of dynamical systems [9–13].
The general form of the localization problem.
Let us briefly describe this
localizing method applied in analysis of different nonlinear systems [14–16].
Consider a dynamical system
= ( ),
z q z
(3)
where
,
n
z
т
1
( ) = ( ( ), , ( )) ,
n
q z q z q z
and
( )
q z
is a Lipschitz function.
A subset
n
M
is called
an invariant set
of the system (3) if, for each point
0
,
z M
the trajectory
0
( , )
z t z
of the system (3) passing through the point
0
z
is
contained in
.
M
The localization problem consists in finding sets (localizing sets) in
n
containing all compact invariant sets of the system (3) [9–11].
Let
be an arbitrary function in
1
( )
n
C
and
def
=1
( )
( ) = ( )
n
i
i
i
z
z
q z
z
the derivative of
with respect to the system (3). The subset
def
= {
: ( ) = 0}
n
S z
z
is called
the universal section
. Set
def
def
sup
inf
= { ( )},
= { ( )}.
sup
inf
z S
z S
z
z
Then all
compact invariant sets of the system (3) are contained in the subset [9–11]
def
inf
sup
( ) =
:
( )
.
n
z
z
In other words, this subset is localizing for the system (3). The above-mentioned
function
is called
localizing
.
This method of localizing compact invariant sets is also applicable in a more
general case of non-autonomous systems
= ( , ),
z q z t
(4)
where
,
n
z
,
t
т
1
( , )=( ( , ), , ( , )) ,
n
q z t q z t
q z t
and
( , )
q z t
is a continuous and
z
-Lipschitz function.