Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://matematika.phys.msu.ru/files/stud_gen/26/stationary_points.pdf Äàòà èçìåíåíèÿ: Mon Oct 22 20:14:09 2012 Äàòà èíäåêñèðîâàíèÿ: Sun Feb 3 04:29:00 2013 Êîäèðîâêà:

. .





II

2012


. . / . .: , 2012. 29 . , , , IV , ­ . 1) , , , , 2) , 3) . . . . , . . . , . . . , . . 38.

© , 2012 © . . , 2012


x = Ax, x1 = a11 x1 + a12 x2 , x2 = a21 x1 + a22 x2 . (1)

A -- . (1) A. (1), . . t R; (1), () x1 = f1 (x1 ,x2 ), (2) x2 = f2 (x1 ,x2 ). . , (2) , , (1) x(t) x0 , t. , , ( (1)) Ax0 = 0. (3)

, x0 = 0, , (3), , (1). , , (3) , det A = 0. ? , (3) n - r, n = 2 -- A, r = rank A (« »). , 1) r = 0 (. . A = ) , 2) r = 1 x = C , = 0 -- (3), C R. (. . 1 2 .) 1




. 2. . 1. -- , . . , . , (1) . . , , det A = 0 ( A ). (1) -- x0 0. ; . det(A-E ) = = 0: a11 - a21 a12 = 0 2 - Sp A +det A = 0 a22 -
1 ,2

, 2 Sp A a11 + a22 . det A = 0 1,2 , 1,2 = 0. , . 1. . 2. . 3. 1 . . 1. 1 , 2 . , , (1) x(t; C1 ,C2 ) = C1 1 e1 t + C2 2 e2 t ,
1

=

Sp A ±

(Sp A) - 4det A

2

(4)

, .

2


1 = 11 , 12 2 = 21 22 --

A, 1 , 2 ; C1 , C2 -- . (, C1 = C2 = 0.) 1.1) 2 < 1 < 0. , (4), --- x(t; C1 ,C2 ) - 0.
t+

(5)

, x0 x(t;0, 0) = 0 x(t; C1 ,C2 ) |C1 |
1

+ |C2 | 2 ,

t [0; +);

(6)

. , 2 2 1 11 + 12 . (5) (6) , 2 . , , x(t; C1 ,C2 ) 0 t +. , , . , 2 2 (4) C1 + C2 = 0, , x(t; C1 ,C2 ) t (-;+)3 . , : C1 1 12 e1 t + C2 2 22 e2 t C1 1 12 + C2 2 22 e( dx2 = = dx1 C1 1 11 e1 t + C2 2 21 e2 t C1 1 11 + C2 2 21 e(
2 -1 )t 2 -1 )t

, (7)

2 - 1 < 0 2 < 1 . (7) , t + (. ., (5), ): dx2 12 -- , i) C1 = 0 dx1 11 dx2 12 1 , |1 | -- min ( , C1 = 0, C2 = 0 ); dx1 11
, 10. 3 , « ».
2

3


dx2 22 -- , dx1 21 2 , |2 | -- max. ii) C1 = 0 , t - ( x +) : dx2 22 ( , C1 = 0, C2 = 0 i') C2 = 0 dx1 21 dx2 22 ); dx1 21 dx2 12 ii') C2 = 0 . dx1 11 t ±, t R, . (7) t. a = C1 1 12 , b = C2 2 22 , c = C1 1 11 , = 2 - 1 . (7) a + bet . c + det (7a) d = C2 2 21 ,

, 1 , 2 A, 1 = 2 , bc - ad = 0. (7a), a + bet c + det = bet (c + det ) - (a + bet ) det (c + det )
2

==

et (bc - ad) (c + det )
2

,

t R ( 0). , , -- , « ». ( t , 0, ; t dx1 .) , dx2 4


, (. . , C1 = 0, C2 = 0) . , x(t; C1 ,C2 ) 1 , 2 . (4) C1 = 0, C2 = 0, t R x(t; C1 ,C2 ) 1 2 . x(t; C1 ,C2 ) 1 , 2 . C t, C1 ,C2 (. . 3). 2 2

1

1

. 3.

. 4.

, (2 < 1 < 0) , , . 1.1) 0 < 1 < 2 . , 0 < 1 < 2 t -t, , . 3, (. . 4; . 3). 3 4 1 , 2 , 1 , 2 , |1 | -- min. . ( , .) 1. ( (1)) : ~ 1) x = (x1 , x2 ) (x1 ,x2 ) ~~ -- , x = Ax, ~ x(0) = x; 5 (8)


~ 2) x (x1 ,x2 ) , (8) , t -t. 2. , (x1 ,x2 ) . , , 1), (0, 0) t +, « » . , (0, 0) ( ). , . 3 4 : (0, 0). 3. , , , «» x = Ax, t R,

x(0; C1 ,C2 ) = x(1) , x(1) -- . 4. (1) . , x = (x1 ,x2 ) k x = (kx1 ,k x2 ) «» Ax Ak x = kAx. , (1) ( , ) . 1.2) . , 2 < 0 < 1 . - (4), --- x(t; C1 = 0,C2 ) = C2 2 e2 t - 0,
t+

--- x(t; C1 = 0,C2 ) - +;
t+

(9)
t

x(t; C1 ,C2 ) - x(t; C1 ,C2 = 0) = C2 2 e2 6

- 0, ---
t+


x(t; C1 ,C2 = 0) = C1 1 e1 t - 0 ---
t-

--- x(t; C1 ,C2 = 0) - +;
t-

(9)
t

x(t; C1 ,C2 ) - x(t; C1 = 0,C2 ) = C1 1 e1

- 0. ---
t-

t (7). , 1.1), , 2 < 1 , i)--ii), i')--ii'). , (9), (9) 1.1), 1.1) , (. . 5; 1 , 2 , 1 < 0 < 2 ).

1 2

. 5. . , . 2. , 1 = = 2 = . , : 1) A 2 1 2 , (1) , 1 = 2 = ; 2) A , (1). 2.1) , 7


, A= 0 0
4

.

, A , , (A - E )x = 0, , rank(A - E ) = 0, 00 0 A - E = A= . 00 0 (1) x = C1 1 et + C2 2 et = C1 1 + C2 (C1 12 + C2 22 )et C1 12 + C2 22 x2 = = = const. x1 (C1 11 + C2 21 )et C1 11 + C2 21 . 6. A , . . < 0 > 0.
2

et ,

. 6. () () 2.2) , A , =
4

1 , 2

=

1 , 2

, = 0.

8


(1) x(t; C1 ,C2 ) = C1 + C2 t + et ,

x1 = (C1 1 + C2 1 t + C2 1 ) et , x2 = (C1 2 + C2 2 t + C2 2 ) et ;

dx2 (C1 2 + C2 2 t + C2 2 )+ C2 2 = . dx1 (C1 1 + C2 1 t + C2 1 )+ C2 1 : i) C2 = 0 ii) C2 = 0 dx2 - --- dx1 t± x2 dx2 = x1 dx1 2 ; 1 2 t R. 1 (9)

(10)

, C2 = 0 (9) t 1 ( ). 0 t, dx2 (t) - , dx1 C2 = 0, C1 , . 7.
dx2 dx1 2 1

t

. 7. - , , . 8. ( «» , , . .) . 9










. 8. () () . (10) dx2 (t) (. . 7) , dx1 « » 180 , . . , t = t1 , t = t2 , 180 t1 -, t2 +. 3. -- ¯ , 1 = 2 = = p + iq . , A ( , , ¯ 1 = 2 ), 1 , 2 : = 1 = 2 . , (A - E ) = 0 (A - E ) = 0 A - E = 0 ¯ (A - E ) = 0. , , (1) Re ept (cos qt + i sin qt) , = Im ept (cos qt + i sin qt) 11 + i12 21 + i22 10 , = p + iq .


( , , ij ). x(t; C1 ,C2 ) = cos qt - 12 sin qt cos qt + 11 sin qt = ept C1 11 + C2 12 . (11) 21 cos qt - 22 sin qt 22 cos qt + 21 sin qt 3.1) , , p = 0. (11) ( x(t) 2/q ), ( t /q x(t) ), ( ). (. ) , . «» , (1) «» (. ). ; . 9. , «» , . , , . . t , (. . ), x < C (, C1 ,C2 ).

. 9. 3.2) p = 0. , , t 2/q , , ( p) e2p/q , « », . 10. , p = Re > 0 p = Re < 0. 11


. 10. () () . . 10, « »: / «» (/ ). , p = Re , , , (1)5 . x, . 11, . , . . 12 (. 9).

. 11.
5

.

12


. 12.

, . 13.

( )

( )

( )

( )

(, ) ( )

. 13. 13


1. , , . 2. (, ).

, . (1) ,





2 < 1 < 0 2 = 1 < 0, A 1 2 = 1 < 0, A 2
1,2 1 ,2



0 < 1 < 2 0 < 1 = 2 , A 1 0 < 1 = 2 , A 2 2 < 0 < 1 = ±iq
1 ,2

= p±iq , p < 0

= p±iq , p > 0

14


«» (11): x(t; C1 ,C2 ) =

cos qt - 12 sin qt cos qt + 11 sin qt = ept C1 11 + C2 12 , 21 cos qt - 22 sin qt 22 cos qt + 21 sin qt x1 = (11 C1 + 12 C2 )cos qt +(11 C2 - 12 C1 )sin qt, x2 = (21 C1 + 22 C2 )cos qt +(21 C2 - 22 C1 )sin qt.

(12)

(12) cos qt, sin qt. 2 2 , 0 C1 + C2 > 0: = 11 C1 + 12 C2 21 C1 + 22 C2 11 C2 - 12 C1 2 2 = (C1 + C2 )(12 21 - 11 22 ). 21 C2 - 22 C1

0 , 12 21 - 11 22 = 0. , Re Im , Re = Im ( Im = 0 Im = 0), = (1 + i )Re , , Re , Im = 0 A. , = 0, , (12) cos qt, sin qt: cos qt = b11 x1 + b12 x2 , cos qt = b21 x1 + b22 x2 , (13)

x1 x2 . (13) (b11 x1 + b12 x2 )2 +(b21 x1 + b22 x2 )2 = 1. , , . (. ), ( , ).

15


, . () . . . , «. .. . . » . 0. x1 = 4x1 - 2x2 , x2 = 6x1 - 3x2 . 0= 4- 6 -2 = ( - 4)( + 3) + 12 = 2 - - 12 + 12 = 2 - , -3 -

1 = 0, 2 = 1. , , : . , 1 = 1 , 2

, , x2 = 2x1 . , . ( .) 1. x1 = 2x2 , x2 = -3x1 +5x2 . : 0= - -3 2 = ( - 5) + 6 = 2 - 5 +6, 5- 16


1 = 2, 2 = 3 ( , ). , (0, 0) . , -- . x = C1 1 2t 2 3t e + C2 e, 1 3 x1 = C1 e2t +2C2 e3t , x2 = C1 e2t +3C2 e3t .

2. x1 = -2x1 - x2 , x2 = -3x1 - 4x2 . : 0= -2 - -3 -1 = ( +2)( +4) - 3 = 2 +6 +5, -4 -

. . 1 = -1, 2 = -5. , (0, 0) . , . : x = C1 1 1 -5t e-t + C2 e, -1 3 x1 = C1 e-t + C2 e-5t , x2 = -C1 e-t +3C2 e-5t .

3. x1 = x1 - x2 , x2 = x2 - 4x1 . , «», . : 0= 1- -4 -1 = ( - 1)2 - 4 = 2 +6 +5, 1- 17


. . 1 = 3, 2 = -1. , (0, 0) . x = C1 1 -t -1 3t e + C2 e, 2 2 2 1 x1 = C1 e-t - C2 e3t , x2 = 2C1 e-t +2C2 e3t . 2

1 . 1 1 2 . 2

. 3 4. x1 = -2x1 , x2 = -2x2 .

. 4

, . x1 = C1 e-2t , x2 = C2 e-2t . 5. x1 = x1 + x2 , x2 = -x1 +3x2 . 18


: 0= 1- -1 1 = ( - 1)( - 3) + 1 = 2 - 4 +2 = ( - 2)2 , 3-

1 = 2 = 2, . . 2. , . , rank(A - 2E ) > 0, A ; , , , . . , , (1 1)T , (1 2)T . x = C1 x1 = (C1 + C2 (t +1)) e2t , x2 = (C1 + C2 (t +2)) e2t . ~ ~ : C1 = C1 + C2 , C2 = C2 , x1 = C1 + C2 t e2t , ~ ~ x = C + C (t +1) e2t . ~1 ~2 2 1 2t 1 1 2t e + C2 t+ e, 1 1 2





. 5

. 6

« » , «» x , (0, 1). , x, (1, -1)T (0, 1) , 19


« » « », « » « ». 6. x1 = 2x2 - 3x1 , x2 = x2 - 2x1 . (, «»): 0= -3 - -2 2 = ( - 1)( +3)+4 = 2 +2 +1 = ( +1)2 , 1-

1 = 2 2. < 0. x = C1

= -1 -- , (0, 0) , , 1 -t e, 3
2

1 -t 1 e + C2 t+ 1 1



x = C +2C t e-t , ~1 ~2 1 x = C + C +2C t e-t . ~1 ~2 ~2 2

7. x1 = -x1 - 5x2 , x2 = x1 + x2 . : 0= -1 - 1 -5 = ( - 1)( +1)+5 = 2 - 1+5 = 2 +4, 1-

1,2 = ±2i. , . , x = C1 cos 2t +2 sin 2t - sin 2t +2 cos 2t + C2 . - cos 2t sin 2t 20


t, (x1 ,x2 ), (1; 0). , ( «»). 8. x1 = -2x1 + x2 , x2 = -x1 - 2x2 . : 0= -2 - -1 1 = ( +2)2 +1 = 2 +4 +5, -2 -

1,2 = -2 ± i. -- , (0, 0) . «» x = (1, 0)T -- x = (-2, -1)T . , t +: 1) 0; 2) «» ( ). , cos t sin t + C2 . x = e-2t C1 - sin t cos t 9. x1 = x1 - 2x2 , x2 = 4x1 - 3x2 . : 0= 1- 4 -2 = ( +5)2 +1 = 2 +4 +5, -3 -

1,2 = -1±2i. -- , 21


(0, 0) . «» x = (1, 0)T -- x = (1, 4)T . , t + 1) 0, 2) «» ( ). 8 9 . , , «» « », . x = e-t C1 cos 2t sin 2t . + C2 cos 2t +sin 2t - cos 2t +sin 2t

10. x1 = x1 + x2 , x2 = 3x2 - 2x1 . : 0= 1- -2 1 = 2 - 4 +5, 3-

1,2 = 2 ± i. -- , (0, 0) . «» x = (1, 0)T -- x = (1, -2)T . , t + 1) 0, 2) «» ( ). x = e2t C1 cos t sin t . + C2 cos t - sin t cos t +sin t

22


. 7

. 8

. 9

. 10

23


: 1) , (0, 0) t; 2) . (x1 ,x2 ) (x, y ).

1.

x1 = 2x1 + x2 , x2 = 3x1 +4x2 . x1 = 6x1 +2x2 , x2 = 3x1 +7x2 . x1 = 6x1 , x2 = 6x2 . x1 = -x1 +2x2 , x2 = -2x1 - x2 . x1 = x1 - 3x2 , x2 = 3x1 + x2 . x1 = x1 - 5x2 , x2 = 2x1 - x2 . x1 = 3x1 - 5x2 , x2 = 5x1 - 3x2 . x1 = -x1 + x2 , x2 = -x1 - 3x2 .

2.

x1 = -x1 +8x2 , x2 = x1 + x2 . x1 = 3x1 + x2 , x2 = -x1 +5x2 . x1 = -6x1 , x2 = -3x2 . x1 = 2x1 - x2 , x2 = x1 +2x2 . x1 = x1 +2x2 , x2 = -x1 - x2 . x1 = 5x1 +4x2 , x2 = 20x1 +7x2 . x1 = 13x1 - 8x2 , x2 = 18x1 - 13x2 .

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

24


. 1 5t 1. x = C1 1 et + C2 e , -- 3 -1 . 2. x = C1 3. x = C1 . 4. x = C1 2 3t 4 e + C2 e-3t , -- . 1 -1 1 2 9t e4t + C2 e , -- -1 3

1 4t 1 1 4t e +C2 t+ e , -- 1 1 2

. 1 0 6t 5. x = C1 + C2 e , -- 0 1 . 1 -6t 0 -3t 6. x = C1 e e , -- + C2 0 1 . cos 2t sin 2t -t + C2 7. x = C1 e , -- - sin 2t cos 2t .

cos t - sin t 2t + C2 8. x = C1 e , -- sin t cos t .

- sin 3t cos 3t t + C2 9. x = C1 e , -- cos 3t sin 3t . 10. x = C1 . cos t - sin t - cos t cos t +sin t , -- - sin t

+ C2

25


11. x = C1 -- . 12. x = C1

5 cos 3t cos 3t +3 sin 3t

+ C2

-5sin 3t , 3 cos 3t - sin 3t

1 2 15t e-3t + C2 e , -- . -2 5 3 cos 4t - 4sin 4t 4 cos 4t +3 sin 4t 13. x = C1 + C2 , 5 cos 4t 5sin 4t -- . 14. x = C1 15. x = C1 1 5t 4 -5t e + C2 e , -- . 1 9 1 1 1 -2t e-2t +C2 t+ e , -- -1 -1 0

.

26


2 1

2 1 . 1 2 . 2

1 . 3 . 4

1 2

. 5

. 6

. 7 27

. 8


. 9

. 10

1

2 . 11 . 12

2

1

. 13

. 14

. 15

28


: [1] . ., . ., . . . .: , 1985. [2] . ., . ., . . . . http://matematika.phys.msu.ru/files/stud_gen/26/lekcii.pdf [3] . . . .: , 2011.

29