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O.G. Styrt, A.P. Krishchenko
24
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 1
We will further denote by
*
c
G
the number
2
*
2
.
2
c G
The set
ˆ
is given by the inequality
*
( , )
.
c
h x y G
Lemma 4.
The derivative of the function
h
with respect to the system (7) is non-
positive on the set defined by the inequality
*
( , )
c
h x y G
and upper bounded by a
negative number on the set given by the inequality
0
( , )
h x y h
for each
*
0
> .
c
h G
◄
Take an arbitrary number
*
0
> .
c
h G
Set
def
*
0
=
> 0.
c
h G
If
2
( , )
x y
,
t
, and
0
( , ) ,
h x y h
then
2
2
2
*
*
*
0
2
( )
= =
,
2
2
2
c
y
y
c
G G x h G
G
i. e.
2
2
2 ,
y c
2
2
2
2
2
2
( , , ) = ( ( ) ( , ) ) = ( ) ( , )
( )
=
= ( )
= (
)
2
h x y t y e t f x y y ye t f x y y ye t
y
y e t
y c y y
y y c
y c
c
and, therefore,
2
2
2
2
2
2
1
1
( , , )
| |)(
2
2
2
.
h x y t
y c
c
c
c
c
Since
> 0,
then the number
2
2
2
2
1
2
2
c
c
c
is positive.
►
Corollary 7.
The set
ˆ
is positively invariant.
◄
The bound
ˆ
of the set
ˆ
is defined by the equation
*
( , )= .
c
h x y G
By
Lemma 4,
0
h
on
ˆ .
Hence, the bound
ˆ
is semipermeable and trajectories can
intersect it only in the direction of the set
ˆ .
Thus, any trajectory, once entering
ˆ ,
never leaves it. Also, on any trajectory never
entering
ˆ
,
*
>
c
h G
and, by Lemma 4, the function
h
is decreasing and tends to
*
.
c
G
►
Conclusion.
In the paper, Levinson — Smith dynamical system was researched
with the object to localizing compact invariant sets and the behavior of solutions with
respect to one localizing set. In the case of an unperturbed system, with the
assumption of positive friction, it was proved that this set is positively invariant, and
some sufficient conditions for any trajectory to enter it were obtained. In the case of a
perturbed system, with the suggestion that friction is lower bounded by some positive
number and perturbation is a bounded continuous function, it was proved that this
localizing set is positively invariant.
REFERENCES
[1] Levinson N., Smith O.K. A general equation for relaxation oscillations.
Duke Mathematical
Journal
, 1942, vol. 9 (2), pp. 382–403.
[2] Levinson N. On the existence of periodic solutions for second order differential equations
with a forcing term.
J. of Math. and Phys
., 1943, vol. 22, pp. 41–48.