ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 1
15
DOI: 10.18698/1812-3368-2017-1-15-25
THE RESEARCH OF SOLUTION OF LEVINSON — SMITH EQUATION
O.G. Styrt
oleg_styrt@mail.ruA.P. Krishchenko
apkri@bmstu.ruBauman Moscow State Technical University, Moscow, Russian Federation
Abstract
Keywords
We research the behavior of solutions of Levinson — Smith
equation. In the case of an unperturbed system, friction is
supposed to be positive. We consider the behavior of
trajectories with respect to one localizing set that is, subset
containing all compact invariant sets. More exactly, we
show that this set is positively invariant and obtain some
sufficient conditions for any trajectory to enter it. In the
case of a perturbed system, we suggest that friction is lower
bounded by some positive number and perturbation is a
bounded continuous function. Similarly, we consider one
localizing set in terms of non-autonomous systems and
prove that it is positively invariant
Dynamical system, localization,
compact invariant set
Received 20.06.2016
© Bauman Moscow State Technical
University, 2017
The authors were supported by the Ministry of education and science of the Russian
Federation (project 1.644.2014/K and project 736 of the program "Organization of research")
Introduction.
In this work, we consider Levinson — Smith equations of order two [1, 2]
( , )
( ) = ( )
x f x x x g x e t
(1)
researched in different assumptions on the functions
( , ),
f x x
( ),
g x
and
( )
e t
in a
number of papers. First of all, in the monograph [3] devoted to qualitative geometric
analysis of differential equations, especial attention is payed on equations of order
two. Among the rest, the equation (1) and its particular case without perturbation
( )
e t
are studied in detail. Another particular case is researched with many aspects —
the Li'enard equation
( )
( ) = 0.
x f x x g x
(2)
In [4], the qualitative behavior of trajectories of this equation is studied from the
viewpoint of boundedness, oscillation, and periodicity. In [5, 6], upper bounds for the
amplitude of limit cycles of the equation (2) are obtained. In [7], the period function
of the equation (2) is researched in the suggestion that the origin
O
is its equilibrium
point of the "center" type. In this assumption, this function is defined in
\ { },
D O
where
D
is the largest domain containing
O
and consisting completely of cycles
surrounding
O
. Besides, the paper [8] is concerned with a generalization of the
Li'enard equation the perturbed equation
( )
( ) = ( ).
x f x x g x e t