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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.ru

A.P. Krishchenko

apkri@bmstu.ru

Bauman 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