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4.
.

,
, -,
W ( s ) ; -,
;
-,
.
..

.
1938 . .
.


D (
l ) = l n + a1
l n-1 + a2 l n-2 + ... + an = 0. (13)
l 1 , l 2 , ... , l n ,
(13)
D (
l ) = ( l - l 1 ) ( l - l 2 ) ... ( l - l n ). (14) Im Im 0 Re 0 Re ) )
.12. -
- :
-
l l i ;
-
l 1 , l 1 , l 2
, l 2

l .
, , (14)
( l - l i ),
.12,. , l =
j w ; w (.12,). w - ¥ + ¥ j w - l 1 j w - l 1 l l 1
, + p , j w - l 2 j w - l 2 ,
- p . ,
arg( j w - l i ) l i ,
, + p , ,
, - p .
D
arg D( j w )
. 1 n m
,
D
arg D( j w ) = (
n - m ) p - m p = ( n - 2m ) p . (15)
- ¥ < w < ¥


,

D ( j w ) = ( j w )n
+ a1 ( j w )n-1
+ a2 ( j w )n-2
+ ... + an (16)

w , -
,
arg D ( j w ) = - arg D ( -j w ), (17)

w 0 ¥ .

D
arg D( j w ) = ( n - 2m ) p / 2
.
(18)
0 £ w < ¥

, m = 0, (18) ,

D
arg D( j w ) = n p / 2 . (19)
0 £ w < ¥

:
,
,
( )
,
, n
( n -
). j V j V 0 U 0 U ) )
.13.
:
-
n = 1 - 6 ; -
n = 4


. 13,. . 13,
n = 4.




, .

.

, .
,
- ,
.
.

, ..
.,
.
.. , .. , .. , ..
.
1. ,
.

x = f ( t , x )

(1)
蠠 x ( t0 ) = x0 (2)
堠 x
= ( x1, x2,
... , xn ) - n -
; t Î I = [t0, + ¥ [ -
, ;
f ( t, x ) = ( f1
( t , x ) , f2 ( t , x ) , ... , fn ( t , x ) ) - n -
- .
(1),
(2).
ࠠ x= f ( t , x ) x ( t0 ) = x0.
. 10
, , n = 1. x 0 t .1

, t
, - x ( t ) -
Rn+1 (.1)
(1),
(2)
. ( t0 , x0
)
. ( t0 , x0 ) , .
, , : x ( t ) = x ( t ; t0 , x0 ).

( t0 , x0 )
x ( t ; t0
, x0 ) , ,
, ( t0 , x0 ) ,
. ,
,
.

, .
x ( t ) = x ( t ; t0 , x0 ) ,
D x0
x0 , :
| x ( t ; t0 , x0
+ D x0 ) - x ( t ) |
= | x ( t ; t0 , x0 + D x0
) - x ( t ; t0 , x0 ) |.
1. x ( t ) = x ( t ; t0 , x0 ) (1)
( ),
x0 堠 I = = [ t0, + ¥ [ , .. " e > 0 $ d
> 0 ,  " D x0
| D x0 | £ d Þ | x ( t ; t0 , x0 + D x0
) - x ( t ) | £ e " t ³ t0.
,
, x ( t ) t
+ ¥ D x0 , .. $ D > 0 " D x0.
| D x0 | £ D Þ | x ( t ; t0 , x0 + D x0
) - x ( t ) | 0
, t
+ ¥ . (3)
x ( t ) (1)
(
).

.

1. 1) ( t )
( .1 ) : x ( t ; t0 , x0 + D x0
) , t0 x ( t ) (.. d -
) , e -
t ³ t0 . x 0 t .2
2)
(3) : x1
( t ) , t0 D - ,
x ( t ) (.2).
D
x ( t ). x2 (
t ), t = t0 , d -
, e -
, x(t).
2. x ( t ) = x ( t ; t0 , x0 ) (1)
( ),
.

.
2.
, ,
t0 ( t ) , ,
t1 ( )
e - (.3).

,
, .. n = 1.

, m,
l (.4). I, a ;
, , I.
,
I ,
, - .
,
I -
. II,
II -
.

x
0 t
.3 .4

x ( t ) (1)
. ,
(1) y
( t ) = x - x (t).
y = F ( t, y ). (4)
F ( t , y ) = f ( t , y ( t ) + x ( t ) ) -
f ( t , x ( t ) ) , F (t, 0) º 0 " t ³ t0.
x ( t ) (1)
y (t) º 0 (4).

, (1) , .. f ( t , 0 ) = 0 " t ³ t0,
.
x ( t ) º 0 (1).
3. x ( t ) º 0 (1)
( ), " e > 0 $ d = d
( e ) > 0 , " x0
|
D x0 | £ d Þ | x ( t ; t0 , x0 ) | £ e " t ³ t0.
,
$ D > 0 " x0
| D x0 | £ D Þ | x ( t ; t0
, x0 ) | 0
, t
+ ¥ ,
x ( t )
º 0 (1)
( ) .
4. 堠 x ( t
) º 0 (1)
(
), , ..
$ e
> 0 $ t1
> t0 " d > 0 x0 ¹ 0 | x0 | £ d Þ | x ( t ; t0 , x0 ) | > e .

,
x ( t ) º 0 (1)
.5-7.
x t 0 .5 x t 0 .6 x t 0 .7
2.
.
.

(
, , ),
.

n - :
dx / dt = f ( x ). (5)

(5) (2).
, (5), (2)
.
x = x ( t ) -
(5).
g , xi = xi
( t ) ( i = 1, ... , n ),
( ) (5) x
= x ( t ). Rn ( x1 , ... , xn
), (5),
(5). ,
(5)
t = t , x1 = x1 ( t ), ... , xn = xn
( t ). , Rn+1
( t , x1 , x2 , ... , xn ) ,
Rn
t. n = 2 , .. Rn+1
- , Rn - . .8,
, t = t, x1
= x1 ( t ) , x2 = x2 ( t ), .8, - , .. ,
x1 = x1 ( t ) , x2
= x2 ( t ). t. x2 x2 0 t 0 x1 x1 ) .8 )
5. ( a1, a2 , ... , an )
( ) (5),
f1 , f2 , ... , fn (5) ,
.. f (a) = 0, a = ( a1
, a2 , ... , an ) , 0 = ( 0 , 0 , ... , 0 ) .
( a1 , ... ,
an ) - , (5) x ( t )
= a. , , ,
a .
, (5) x ( t ) º 0 , .. f ( 0 ) = 0,
Rn.
Rn+1 .
.8 n = 2.
,
(5)
(5), .

,
, .. n = 2. .5-7
R2, e -
d -
e
d . x = 0 , ,
d -
, e -
" t ³ t0 (.9) ; ,
, D , (.10) ; , 頠 e - d
> 0
, (.11).

,
dx / dt
= A x, (6)
A - n ´ n , (5). ,
. x2 0 x1 .9 x2 0 x1 .10 x2 0 x1 .11
3. .


æ dx / dt = P ( x , y ),
í (A)
î dy / dt = Q ( x , y ).
( x0 , y0
) (A), P ( x0
, y0 ) = 0 , Q ( x0 , y0 ) = 0.


æ dx / dt = a11 x + a12 y,
í (7)
î dy / dt = a21 x + a22 y.
aij ( i , j = 1 , 2 ) -
. ( 0 , 0 ) (7).
(7) .

x
= a 1 e k
t , y = a 2
e k t . (8)
k

a11 - k a12
= 0. (9)
a21 a22 - k
.
I.
. :
1) k1
< 0, k2 < 0. (
).
2) k1 >
0, k2 > 0.
( ).
3) k1 > 0, k2 <
0. ().
4) k1 = 0,
k2 > 0. .
5) k1 = 0,
k2 < 0. , .
II.
:
k1 = p + q i, k2 = p - q i. :
1) p < 0 ,
q ¹ 0.
( ).
2) p > 0 ,
q ¹ 0.
( ).
3) p = 0, q ¹ 0.
(). .
III. : k1 = k2 . :
1) k1
= k2 < 0. ( ).
2) k1
= k2 > 0. ( ).
3) k1
= k2 = 0. . ,
.


dxi n
= å ai j xj (
i = 1 , 2 , ... , n ) (10)
dt i=1


a11 - k a12 a13 ... a1n
a21 a22 - k a23 ... a2n = 0. (11)
. . . . . . . .
an1 an2 an3 ... ann
- k
1)
(11)
(10) , xi ( t ) º 0 ( i = 1 , 2 , ... , n
) .
2)
(11)
, Re k i = p i > 0, xi
( t ) º 0 ( i = 1, 2, ... n ) (10) .
3)
(11)
(.. ), xi ( t
) º 0 ( i = 1, 2, ... n ) (10) ,
.


.
æ x = a11 x + a12
y,
í . (12)
î y = a21 x + a22
y

(9)
k2 + a1
k + a2 = 0.
1) a1
> 0 , a2 > 0, (12)
.
2) 1
> 0 , a2 = 0, a1 = 0 , a2 > 0 , ,
.
3)
; a1 = a2
= 0 , ,
.


:
1. . ., . ., . . .
. . .: , 1981.
2. . .,
. ., . . . .: ,
1987.
3. . .
. .: , 1973.
4. . .,
. ., . . .
. .
.: , 1968.
5. ..

. .: ,1977.
. .

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