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The Research of Solution of Levinson — Smith Equation
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 1
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A subset
n
M
is called
an invariant set
of the system (4) if, for each point
0
,
z M
there exists a number
0
t
such that the trajectory of the system (4) passing
through the point
0
z
at the time
0
t
is contained in
.
M
In the non-autonomous case,
the localization problem also consists in finding localizing sets (subsets in
n
containing all compact invariant sets of the system (4) [12, 13]).
In the non-autonomous case, we also take a localizing function
1
( ).
n
C
Its
derivative with respect to the system (4) is
=1
( )
( , ) = ( , )
.
n
i
i
i
d
z
z t
q z t
dt
z
The universal section is the subset
def
=
:
( , ) = 0 .
n
d
s
z
t
z t
dt
Like in the autonomous case, the subset
def
inf
sup
( ) =
:
( )
,
n
z
z
where
def
def
sup
inf
= { ( )},
= { ( )},
sup
inf
z s
z s
z
z
is localizing for the system (4) [12, 13].
The case of positive friction.
Let us pass to constructing a localizing set of the
unperturbed Levinson — Smith system
= ;
= ( , )
( )
x y
y f x y y g x
(5)
supposing
>0
f
in
2
.
Take a localizing function
2 def
= ( , ) =
( ).
2
y
h h x y
G x
Its derivative with respect to the system (5) is
2
( , ) = ( ) = ( ( , )
( )) ( ) = ( , ) .
h x y yy G x x y f x y y g x g x y f x y y
The function
2
( , ) = ( , )
h x y f x y y
is non-positive, and the set of its zeros (i. e. the
universal section
h
S
) is
.
Ox
Hence,
def
inf
( , )
def
*
sup
( , )
( , )} = { ( , 0) = { ( )};
inf
inf
inf
=
( , )} = { ( , 0) = { ( )} = .
sup
sup
sup
=
x y S
x
x
h
x y S
x
x
h
h x y
h x
G x
h
h x y
h x
G x G
h