. , , ,

,,,

, . . , . , - , . .

, .. ., . .. , .. , .. , .. .

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 , , .

, .

 

 

 

! , , , .
. , :