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Дата индексирования: Mon Oct 1 22:44:40 2012
Кодировка:

..

, . , , ,

-

.

. , , , . . , . . : m Gm n 1 n a , n , , G G
0 1

, .

-

, ,

. -

-

n m 1, n

a

a

n (1)

a

m :

( 1) 5

,1 1

2

,1 1

3

,

0

4

,

0

,

(2)

39

5. Part 5. Mathematical Theories and Calculation Methods
i

--
,
0

, . (1)

(

0

)

m n na

G

1

mn 1
3

2

na ,
(3)
5

nm 1
a 0 4

, m
0 4

na

.
a

0 m 0 , n 0 , na ,

(3)
n 0 , na ( )
0 na .

(1) m, n , n

m ,

m( ),

n n( ), n

a

na ( ) , m( ), n( ), na ( )
m 0 , n( )

0 , m( )

x1
dx dt Ax B( ) x

m m ,x (3),
f ( x, ) .

2

nn

,x

3

n

a

n

a

. (3 )

(3 )
G0 n A
B( ) --
0

-

1n n 0
0

0

a

G0 m (m
0

0

G0 m 1)
0

0

0
0

,
m
0

0
m0

G0 q( ) r ( ) B( ) q( ) 0

2

1

(G0 ( p( ) 0

1

) p( ) m0
3

1

(G0
5

1

) p( )

)
4

0 (m0

,
p( )

3-

-

f ( x, ) --
0 1

f ( x, )

G

x1 x

2

x3 ,

x 2 x3 ,

x3 x1 (

0

5

).
-

(3 ) . (3 ) :

40

. . -- 2009, . 2, . 3945 Gorbunova Ju. A. -- MCE 2009, v. 2, pp. 3945

dx dt

Ax B ( ) x

f ( x, ) ,

(4) ,A
B( ) --

x -- nf ( x, ) -- n-

n n,

-

. : En , x 0. (
0

D(

0

)

t , x,

: t [0, ], x
0 0

0

,

Em ,

0

,

( 0) W( 0)

Em : En :
,

, ,
0

)
q ( ),

, 0, B ( ) 0 . D( 0 ) f (t , x, ) -- nf ( x, ) , f (t ,0, ) 0 lim x0 x ,

B( ) r ( ),

-

p( )
.

-

,

0

D( 0 )

. x(t , , ), x(0, , ) (4),
1

(4) (4)

. ,

x ,

0
-

0,

0

,

,
t x (t , , ) ,

W ( 1 ), 0,

( 1) 0, W ( 1) ( 1)
(4) (

(4)

t

0,

x(t , , )

0

.

, (3 )).

-

41

5. Part 5. Mathematical Theories and Calculation Methods

.

(4)
f (t , x (t , , ), ) . : t x (t , , ) . t y (t ) (4) t 0, . t x(t , , )

(5)
y Ay B ( ) x (t , , )

(5) (5)

1.

(4) y ( 0) y (t ) x (t , , )

. y (t )

. (4), t 0, x(t , , ) Ax(t , , ) B( ) x(t , , ) f ( x(t , , ), ) . t x(t , , ) (4). , t y (t ), y (0) (5), (5) t x(t , , ) . , (5) , t 0, , . . (5)

x(t , , ) , . . t
t

y (t )
X
0 1

(4).
f ,x , , , B( ) x(t, , ) d , y(0)

y(t ) X t

Xt

,

y (t )

x(t , , ) , x(t , , )
t

:
1

x(t ) X t

Xt
0

X

f

,x , ,

,

B( ) x(t, , ) d . (6)

,

(4).

, (4).

(6),

-

2.

(4)

,

x(t , , )

X (t )

o( ) O

42

. . -- 2009, . 2, . 3945 Gorbunova Ju. A. -- MCE 2009, v. 2, pp. 3945

X (t ) --

x

Ax ,

X (0)

E, E

--

, lim
0

o( )

0, lim O( ) 0 .
0

.
t

x t, ,

Xt

Xt
0

X

1

B

x,,

fx , ,

,

d.

,
0; W
1 1

x t, ,

.
t

,
x t, ,
x t, ,
t

Xt
0

X

1

B

x,,

d

O

,
.

t, ,

, ,

t, ,

--

t, , X
0 1

t

Xt

B
t

x,,
1

d
,, d
t

Xt
0

X

1

B

,,

d,

lim X t
0 0 t

X B

B x,,

d O
1

0,

,

Xt
0

X

1

.

,

Xt X
0

fx , , , d

O

,
W

..
,

lim
x0

f x t, , ,
.
t

0

t,

0,

,

1

1

t
X
0 1

0;

,

W
,

1

,
t

1

:
, 1 d,

1

Xt

fx , ,

d

K

2 0

fx , ,

43

5. Part 5. Mathematical Theories and Calculation Methods
t

Xt

K,

X
1

1

t
t

K, K
X
0 1

2 0

fx , , , x,,
, d

x,,

d

0,

,

Xt

fx , ,

o

. :

,
x t, , x Xt o O

,

Xt --

Ax , x 0

3. ( , ) 0, (3a) x 0.

E. -- det X ( ) E

.

0,
(4), -

m0 , n0 , n0 , .
.

,

-

. . , . , , , , . .

-

-

-

44

. . -- 2009, . 2, . 3945 Gorbunova Ju. A. -- MCE 2009, v. 2, pp. 3945

. ., .. .-- .

.. .-- . , 1981. .-- . , 1970. . , 1980. -

MATHEMATICAL MODEL OF LASER WITH SATURABLE ABSORBER OPERATION Gorbunova Ju. A.

Operation of an optical quantum generator (laser) with saturable absorber was studied by means of mathematical modeling. Nonhomogeneous systems of the differential equations with coefficients depending on a parameter and nonlinear vector-function are studied. The aim of the research is to develop procedure for discovering of periodic operation modes of the optical quantum generator.

45