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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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The origin is an asymptotically stable equilibrium point if this system. Show that it
attracts all trajectories of the system.
In the notations above,
( , ) =1,
f x y
2 2
( ) =
.
(1 )
x
g x
x
Thus,
2
2
( ) =
,
2(1 )
x
G x
x
*
=1/ 2 < .
G
Take a localizing function
2 def
=
( ).
2
y h
G x
The localizing set
2
is defined by the inequality
( , ) 1/ 2,
h x y
i. e.
2
2
2
1.
1
x y
x
The condition (6) holds and, by Corollary 4, any trajectory enters localizing set
and then never leaves it.
Inside
,
we have
( , ) <1/ 2.
h x y
All sets
2
( , )
: ( , )
,
x y
h x y c
< 1 / 2
c
are
bounded. They are also positively invariant since
2
( , ) = 0.
h x y y
By LaSalle theorem, the equilibrium point
(0,0)
attracts all trajectories in
.
A perturbed system.
Consider the perturbed Levinson — Smith equation
= ;
= ( , )
( ) ( ).
x y
y f x y y g x e t
(7)
We suppose that
f
and
g
are Lipschitz functions. We also assume that
f
is a
function lower bounded by a number
> 0
and
e
is a bounded continuous function.
Set
def
0
( ) = ( ) .
x
G x g t dt
We have
= .
G g
Suppose that
def
*
= sup{ ( )} < .
x
G G x
Since
the function
e
is bounded, its set of values
E
is contained in the segment
[ ; ]
c c
for some
> 0.
c
Take a localizing function
2 def
=
( ).
2
y h
G x
Its derivative with respect to the system
(7) is
= ( )
= ( )
( ( , )
( ) ( )) = ( ( ) ( , ) ).
h G x x yy g x y y f x y y g x e t
y e t f x y y
The universal section
h
s
is contained in the union of the axis
Ox
and the set
2
2
2
( , )
: ( , )
( , )
: ( , )
( , )
:
.
x y
f x y y E x y
f x y y c
x y
y c
Therefore, for each
( , )
,
h
x y s
we have
/ ,
y c
2
*
2
( , )
.
2
c
h x y
G
Hence,
2
def
*
sup
( , )
2
= sup{ ( , )}
,
2
x y s h
c
h
h x y
G
and, thus, the corresponding localizing set is contained in the subset
2
ˆ
defined
by the inequality
2
2
*
2
( )
.
2
2
y
c
G x
G
Consequently,
ˆ
is a localizing set.