_____ ______

( - 2010-11) _____ ______

: . - .

1 : *

. , - , , - , - . , ..

Pierre Simon Laplace (1749-1827)

. , . - ; , . Heisenberg ; - . Laplace.

* : .

&

spn@eduscience.gr

: 2

. . 2010

, , , , , , , . -, , - . .. , , -, , .

, .. -, - , , . , , , . , - . -, - -. , - Heisenberg, - .

:

Vladimir Arnold, Ordinary Differential Equations, Cambridge MIT Press, 1973.

Lawrence Perko, Differential Equations & Dynamical Systems, Springer-Verlag, 1991.

David Betounes, Differential Equations, Springer-Verlag, 2001.

M. Hirsch, S. Smale, R. Devaney, Differential Equations & Dynamical Systems, Els.Ac.Pr., 2003.

. , , , 1997.

. , . , , , 2003.

. , , 2000.

. , , 2006.

: 3

. . 2010

, 10:1, , , . - , . . ;

. , . 12 . - . ;

: 4

. . 2010

n - . , n :

1( ) ( ( ),..., ( ))n

nx t x t x t= .

- n - n :

: U nif , 1,...,=i n .

- :

1( ,..., )=i

i ndx

f x xdt

, 1,...,=i n .

, .. - , - . , - , ot oUx , , , :

o: I nx U , o o o( )x t x = .

, , :

{ }o o ( ) / I= x x t tU , oUx .

:

o o( ) =x t x , t .

: 5

. . 2010

, - :

: n n U , ( )1( ) ( ),..., ( )nx f x f x= , :

( )dx xdt

= , x U .

, , , , :

o( ) 0x = o o( ) =x t x , t .

:

( )2 1( ) ,x x x=

- . , - , :

: g U Ut , oo

( ) : ( )t xx t= g , t ,

:*

+t t t t

g = g g , , t t .

* 1: ;

: 6

. . 2010

, - , - 0=t :

0o o( )

t= x x=g , ox U .

.*

, :

( ) ( ), ,+ Diff(U) , tt g .

:

( )2 2 2 1 2( ) ( 1)( 2), ( )x x x x sin x x= + .

, - , :

: g U U , o o( , ) : ( )tt x x=g g .

:

( )o o( , ) ( )ot tot

t t x x= =g g , ox U .

* - . 2: ;

: 7

. . 2010

:

{ }o o

( ) /= x t,x tg U , ox U ,

:

o( ) 0x = o o( , )t x x=g , t .

- .

, - :

{ }o o o

/ ( , ) x t t x x= =T g , ox U .

( , )+ , :

{ }o

0 x =T , o x = T , { }o oo o T 0T : T / x k k >= = T .

- : *

o x= T

ox ,

o oT x = T ox ,

{ }o

0 x =T ox .

. :

{ }A :t t g U U { }B :

t

t

g U U

:

:h U U

* 3: ( , )+ ;

: 8

. . 2010

: A Bo o( ) ( )t th x h x= g g , ox U , t ,

: ( ) ( )A Bo o( , ) , ( )=g gh t x t h x , ox U , t .

:

B

At

t

h h

g

g

U U

U U

t

:

( )o o( )= x h xh , o x U .

, , -, , , nU . , - : : ( )nh Hom n ,

: ( )nh Diff n ,

: GL( )nh n . *

GL( )nh

( )nh Diff ( )nh Hom .

. - , - . , - .

_______________ . _______________

* 4: .

: 9

. . 2010

11..

:

( )dx f xdt

= , x U .

1. . *

:

= x x , x = U , ,

ox U :

o:x U , o o( )

= tx t x e .

0 < 0 >

= x x , x .

:

:t g U U , o o( )=gt tx x e , t ,

:

: g U U , o o( , )=g tt x x e .

* (>0) - (

: 10

. . 2010

:

o o( ) ( )

g = g gt+t t tx x , o x U , , t t . :

{ }o o

/= tx x e tU

0 o 0x = . :

{ }o 0

/ ( ,0) 0= = = = T gx t t { } { }o 0 o o/ ( , ) 0x t t x x = = =T g .

0 = . - :

{ }o o o

/ ( , )x t t x x= = = T g .

0 - - . . , .

: .

, :

= ix x , x U = , i , i = 1,2 , :

: g U Ui , o o( , )=g i ti t x x e , i = 1,2 ,

:

:h U U :

( ) ( )A Bo o( , ) , ( )=g gh t x t h x , ox U , t . - 1 2 0 > .

: 11

. . 2010

, 1 2 0 < : h U U : | (0) |= +h . 1 2 0 > , - :

:h U U , 2 1

2 1

/

/

| | , 0( )

| | , 0x x

h xx x

+ =

:

0 x ( ) ( ) ( )2 11 2 1 21 2/ /

o o o o( , ) , ( ) = = =t th t x x e x e t h xg g ,

0 x ( ) ( ) ( )2 11 2 1 21 2/ /

o o o o( , ) | | , ( ) = = =t th t x - | x | e - x e t h xg g .

, :

( )o o( )= x h xh , o x U .

1 2 0 < < 1 20 < <

2 1 0 < < 2 10 < <

. *

* 5: ;

. h 2 10 < < 2 1 0 < <

1h 1 20 < < 1 2 0 < < , -

2 1 = .

: 12

. . 2010

2. . U :

2/3x x= , xU .

ox U :

o:x U , ( )o

31/3o( ) / 3x t x t = + ,

o 0x = ( ) 0x t = , t .

2/3x x= , x .

o 0x , , , :

( ){ }o 31/3o / 3 /x x t t= + U .

o 0x = , , , - 2( ) / 9x t t = ; , , ( ) 0x t , t ,

3( ) / 27x t t= , t , . ;*

* 6: , :

=x x , 0 1< < .

: 13

. . 2010

( , )t x U . , , . , , :

2/3( )f x x= , x .

.

: .

U :

( )x f x= , xU , :

:f U . ot ox U , o o( , )t x U :

: I U , o o( ) =t x .

, :

o( ) 0f x = o( ) =t x , It ,

o( ) 0f x o

( )

o ( )

=

t

x

dt tf

, It .

: 14

. . 2010

1. : *

o( ) 0f x = o( )t x o o( ) =t x . o( ) 0f x ( )x t=

o o( )t x = , : o( ) 0t , ot , - ( )x t= . ox U

( )t x= o o( )x t = ( ) ( ) 1x f x = , 1/ ( )f x -, :

oo( ) ( ) ( )

x

x

dx xf

= + .

o( ) 0x , o o( )t x= , ( )t o o( )t x = . , , ot , :

o

( )

o( )t

x

d t tf

=

.

, ( )x t= , ( )x , o o( )t x = . :

( )1( ) ( ( )) ( ( ))t t f t = = , o o( )t x = .

. ;

- . , , , . , -. ;

. , - , , . , - - . , - , . * 7: : 2.3x x= , x .

: 15

. . 2010

: .

:

( )x f x= , x U ,

:

:f U .

ox U , .. o 0x = , (0) 0f = , :

= x x , x .

, > 0 , 0x , o 0x = , :

( )f x x . ox U :

o o( ) ( )f x f x x x . Lipschitz . - . - . - , - .

Lipschitz Lipschitz x=0. x=0.

: 16

. . 2010

3. .

U :

2=x x , xU . :

1( )c

x tt

=

, c ,

ct . , - 0t = ox U , :

o o: Ix x U , o

o

o

( )1x

xt

x t =

,

o 0x = :

o( ) 0x t = , t . *

2x x= , x .

. , ! :

:t g U U ,

oo( ) : ( )t xx t= g , t .

* .

8: ;

: 17

. . 2010

: . = U :

( )x f x= , xU ,

. :

( )dx t k

f x= + , k .

, 1,..., pc c , 1] , [j j jc c = , 1,..., 1j p= + , oc = ,

1pc + =+ , . j jm , 1,..., 1j p= + , :

:j jH , ( ) ( )jx

j m

dH xf

= , 1,..., 1j p= + ,

:

( ) 1/ ( )jH x f x = , jx , 1,..., 1j p= + . -, :

1: { ,..., }pH c c U , ( ) ( )jH x H x= , jx , 1,..., 1j p= + .

1 / ( )y f= 1: { ,..., }pH c c U .

: 18

. . 2010

. o(0)x x= , o ix c= , 1,..., 1j p= + , o o( )x t x . ,

1,..., 1j p= + , o jx c= . :

:c jH , ( ) ( ) ( )c j jH x H x H c= ,

ox c= , - jH . 1jH ] , [j ja b :

{ }( ) /j j ja inf H x x= { }( ) /j j jb sup H x x= ,

1cH ] , [j ja b :

( )j ja a H c = ( )j jb b H c = .

:j jH :c jH .

:

( )j

j

c

m

df

cH , jH , jx c= jb = , jb cH jx c= .

: 19

. . 2010

0t = o jx c= :

o: ] , [x a b U , o

1( ) ( )x ct H t = .

, *

:

o

1o(0) (0)x cH x

= = .

:

o( )x t = ( ) ( ) ( ) ( )o

111 1 1

1

1( ) ( ) ( ) ( )( )c c c c xc

H t H H t f H t f tf H t

= = = = , ] , [t a b .

, . ] , [ ] , [a b a b - 0t = ox c= :

o: ] , [x a b U , o o(0)x x = , o o( ) ( ( ))x xt f t = ,

o o o o

o

1( ( )) ( ( )) ( ) ( ) 1( ( ))c x c x x xx

d H t H t t tdt f t

= = =

, ] , [t a b .

, k :

o( ( ))c xH t t k = + , ] , [t a b : o o( (0)) ( )c x cH H x k = =

o o

( ( )) ( )c xH t H x t = , ] , [t a b .

] , [t a b { }o( ) ( ) / jt H x H x x .

{ }o o( ) ( ) / ( )j jinf H x H x x a H x = { }o o( ) ( ) / ( )j jsup H x H x x b H x =

o o] ( ), ( )[ ] , [j jt a H x b H x a b =

,a a b b = = .

* 9: , , . , - -. ;

: 20

. . 2010

4. .

U :

(1 )x x - x= , xU .*

ox U :

o o: Ix x U , o

o

o o

( )1

t

x t

x et

x x e =

+,

:

o 0=x o ( ) 0 =x t , t , ( ),

o 1=x o ( ) 1 =x t , t , ( ).

.

, :

o o1 0tx x e + = .

, :

=D { }oo o( , ) / , Ixt x x t U U

o [0, 1]x oIx = ,

o 0x < ( )o o oI ] , ( 1) / [x ln x x= ,

o 1x > ( )o o oI ] ( 1) / , [x ln x x= + .

* , ( Verhulst)

: 21

. . 2010

(1 )x x - x= , x = U . ox U , 0t = , :

oo

o o

( , )1

t

t

x et x

x x e=

+g , I

oxt .

- :

oo( ) : ( )t xx t= g , t .*

* 10: :

o o( ) ( )t+t t tx x g = g g , t,t ,

oo

o o

( )1

tt

tx ex

x x e=

+g , ox U

:

( ) ooo o

( )( )1 ( ) ( )

t tt t

t t tx ex

x x e

= = +g

g gg g

o

o o

o o

o o o o

1

11 1

tt

t

t tt

t t

x e ex x e

x e x e ex x e x x e

+= =

+ + +

o oo

o o o o

( )1 1

t t t+tt+t

t t t+tx e e x e x

x x e e x x e

= = = + +g .

: 22

. . 2010

: . :

( )x f x= , x U .

ox U : :

oo0, 0 : , | | | ( ) ( ) | , 0x xx x x t t t > > < < U .

:*

&

o o| | ( )xtx x lim t x+ < = .

: . . ( )y f x= ox U o( ) 0f x , :

o( ) 0f x < ox ,

o( ) 0f x > ox .

4 :

( ) 1 2f x x = (0) 1 0(1) 1 0

ff

= > = < U ,

(2) o o0 : , 0 | | ( ) ( ) 0x x x x x f x > < < < < < < >U .

. ox U : ( ) 0of x = .

(1) (1) :

( ) ( )2( ) 2 ( ) 2 ( ) ( )o o od dxx t x x t x x t x f xdt dt

= =

(0)x 0 (0) ox x< < :

2( ) 0od x t xdt

< .

0t = , t , t | ( ) |ox t x - t . , , | ( ) |ox t x < 0t . , = , - :

oo| | | ( ) ( ) | , 0x xx x t t t < < ..

(2) | ( ) |ox t x t ,

o( )tlim x t x+ = .

0> . , :

( ){ }max 2 ( ) ( )oxM x t x f x= { }: | |ox x x = .

(2) :

( )2( ) 2 ( ) ( ) 0o od x t x x t x f x Mdt

= <

20 0

( )t t

od x s x ds M dsdt

2 2( ) (0)o ox s x x x M t +

t , 0= .

: 24

. . 2010

(3) . :

o( ) ( ) 0x x f x > , o0 x - x< < .

. , : 0 | (0) |ox x< < , / 2 = , :

0 (0) ox x< < ( ) , 0ox t x t < .

< :

( )2( ) 2 ( ) ( ) 0o od x t x x t x f xdt

= > | ( ) | | (0) |o ox t x x x .

: ( ){ }min 2 ( ) ( )oxm x t x f x= { }: (0) ( )o ox x x x t x =

0m > ( )x t 0t . :

2

0 0( )

t t

od x s x ds mdsdt

2 2( ) (0)o ox s x x x mt +

t + . o 0- x - x < < . . ( )y f x= ox U o( ) 0f x , :

o( ) 0f x < ox ,

o( ) 0f x > ox . . ox U ( ) 0of x < . ox U :

o o] , [ ( ) 0x x x f x + <

x x ox :

o o( ) ( ) ( ) ( )f x f x f x x x = .

, o o] , [x x x + ( ) 0f x < , :

o o] , [x x x + ( ) 2o o o( ) ( ) ( ) ( ) ( ) 0x - x f x f x f x x - x = <

o o] , [x x x + o( ) ( ) 0x x f x < .

: 25

. . 2010

.

2. :

( )x f x= , x U ,

3 1/ 0( )

0 0

x sin x xf x

x

==

. 0x = .*

* . . 0ox = 1 /kx k= , k , , 0ox = , 1/kx k= , k . 0ox = , >0,

on 1/ on < 1/ on = , :

| | ( ) [ 1 / ,1 / ] , 0 | ( ) | , 0t to ox g x n n t g x t< < ,

[ 1 / ,1 / ]o on n .

]0, [x , on 1/ on

< | ( ) | 1 /t og x n> , 0t , :

o( ) 0t

tlim x x+

=g .

: 26

. . 2010

5. . = U :

x sin x= , xU .

, ox U , :

o:x U , o ( )x t = .

*

sinx x= , x .

t , :

: tg , oo

( ) ( )= t xx tg .

. o ,= x k k , , - , - . - . - :

{ }o o

( ) /tx x t= g U , o Ux ,

:

{ }o

/ ( )tx k t k k= = = = T g , { } { }o o o/ ( ) 0tx k t x x = = =T g . * 11. ;

: 27

. . 2010

: . :

( )x f x= , x U ,

ox U :

o( )x f x x= , x U .

. :

= x x , = x U ,

o 0x = 0< 0> . , ( )y f x= ox U o( ) 0f x , :

o( ) 0f x < ox ,

o( ) 0f x > ox ,

:

o( )x f x x= oo( )

o( )f x t

x t x e = o( )o o( )

f x tt x x e =g , ox U .

:

x sin x= , xU ,

:

ox = 2k x x= ,

ox = (2 1)k+ x x= + .

3. - :

( )x f x= , x U ,

3 1/ 0( )

0 0

x sin x xf x

x

==

: 28

. . 2010

6. . = U :

2x x= + , xU ,*

:

2( )f x x = + .

. 1 0x x= < & 2 0x x= >

0< , 0x= 0= 0> . , 0< ,

o 1x x= o 2x x= , - 1( ) 0f x < 2( ) 0f x > . 0< - 0= 0> . 0= -, , , , . o 0x = - 0= -:

0 o( ) 0f x= = .

: 0,

: 2x x= + , .

* 12. ;

: 29

. . 2010

: .

:

( )x f x= , x U ,

:

( )x f x= + , .

:

= x x , = x U ,

. , :

x x= + , x U ,

( 0)= o 0x = - ox = , . , :

x x= + , x U ,

( 0)= o 0x = ox = , - . , , . , , - .

.

: 30

. . 2010

, -, o ox U :

o o( ) 0f x = o o( ) 0f x = .

, . , :

3x x= + , x = U ,

:

3 0x + = 3x =

0 = o 0x = .

4. , - :

( )x f x= , x U ,

2( ) (1 )f x x x = + .*

o 0 = , 1ox = o 4 / 27 = , 1 / 3ox =

.

* . :

2

2

( ) 0 (1 ) 0 1 0( ) 0 1/ 3 4 / 273 4 1 0

o

o

o o o o o

o o oo o

f x x x xf x xx x

= + = = = = = = + =

: 31

. . 2010

7. . = U :

3x x x= + , xU . :

3x x x= , x = U ,

, ox U , :

o:x U , o ( )x t =

*

.

- :

o 0, 1x = : o 0 ( ) 0x t= , o 1( ) 1x t= , o 1( ) 1x t= ,

: (0) 0f > o 0x = ,

( 1) 0f + < o 1x = + ,

( 1) 0f < o 1x = ,

: 3x x- x= , x .

.

* 13. ;

: 32

. . 2010

: .

:

3( )f x x x = + .

, :

3x x = .

:

1 12

2 2

3 / 3 2 3 / 9( ) 1 3 0

3 / 3 2 3 / 9

xf x x

x

= + = = =

= = +

, :

1 1

2 2

1 1 1 1

2 2 2 2 .

2 3 / 9 : 3 / 3 ( ) ( ) 0

2 3 / 9 : 3 / 3 ( ) ( ) 0

x f x f x

x f x f x

= = + = =

= + = = =

.

. :

2 3 / 9- < , 2 3 / 9- = & ,

2 3 / 9 2 3 / 9- < < , 2 3 / 9+ = & , 2 3 / 9 > + .

: 33

. . 2010

.

5. , - , , 0t = (0)x .*

.

* . 2 2 3 / 9 = , -, , - ( )x t x+ . , - 2 . , - 1 2 3 / 9 = , - ( )x t x -. , - 1 2 3 / 9 = , , . 1 2 . , - (hysteresis loop), . .

: 34

. . 2010

8. .

= U :

(1 ) +x x - x= , xU , 0> .*

0 = , , ox U :

o o: Ix x U , o

o

o o

( )1

t

x t

x et

x x e =

+,

:

o 0=x o ( ) 0 =x t , t , ( ),

o 1=x o ( ) 1 =x t , t , ( ).

:

(1 )= x x - x , xU , 0> . :

:f U , ( ) (1 )f x x - x = , + , ( 1) = .

* , ( Verhulst). 14. :

(1 )x x - x= , x = U , 1, 0 = > ;

: 35

. . 2010

1x x= 2x x= . , 2( ) 0f x < .

: .

- = >. = , , -, , .

.

, , , - - :

:f U , ( )( ) (1 ) 1 2f t,x x - x sin t = + , + , ( 1) = .

: 36

. . 2010

: . - , , - , :

:f U , ( )y = f t,x , :

( , )x = f t x , x U, t .*

o o( , )t x U o o( , )f t x - . , , - :

( )(1 ) - 1 2x x - x sin t= + , x U , 0, 0 > > .

0 > 1 = .

:

( , ) ( 1, ),f t x f t + x x = U, t , + ,

, ox U :

( )x x t= , o(0)x x= : ( )( ) , ( )x t f t x t= , oIxt , * . .

: 37

. . 2010

, , :

1 1( )x x t= , 1 o(0)x x= : ( )1 1( ) , ( )x t f t x t= , 0 1t .

, - :

2 2 ( )x x t= , 2 1(0) (1)x x= : ( )2 2( ) , ( )x t f t x t= , 0 1t . :

1 2( 1) ( )x t x t+ = , 0 1t , :

( ) ( )1 2 2 1( 1) ( ) , ( ) 1, ( 1)x t x t f t x t f t x t + = = = + + , 0 1t .

. 0 1t ;

1t = ; Poincar:

o: , ( ) : (1)x xp U U p = .

k t k= , - ox U :

o( ) ( (1)) (2)x x x= =p p p , o( ) ( (1)) ( (2)) (3)x x x x= = = p p p p p p , *

Poincar : ( )(1 ) - 1 sin 2x x - x t= + , xU , 5, 0.8 = = .

* .

: 38

. . 2010

, - Poincar:

o o o( )x x x U : p = .

, ox U o(0)x x= :

o o( )x x=p o o( )k x x =p , k o( )x k x = , k ,

0t = 1t = , :

( ) ( 1) ... ( )x t x t x t k= + = = + , k ,

o(0)x x= - 1T = . , Poincar - :

( )( , ) (1 ) 1 sin 2x f t x x - x t= = + ;

:

( )(1 ) - 1 sin 2x x - x t= + , xU , 5, 0.8 = = . *

:

: g U U , oo

( , ) : ( )xt x t= g .

, ox U , :

o o( , ) ( , ( , ))t t x f t t x =g g , o o(0, )x x=g ,

* .

: 39

. . 2010

Poincar :

o o: , ( ) : (1, )x x =p U U p g .

:

o o o0( , ) ( , ( , ))

tt x x f s s x ds= + g g

o oo o o0( , ) 1 ( , ( , )) ( , )

t

x xt x f s s x s x ds = + gg g g .

o o( ) ( , )xw t t x= g

:

o o(0) (0, ) 1xw x= =g .

:

oo o o( ) ( , ( , )) ( , ) ( , ( , )) ( )xw t f t t x t x f t t x w t = = g gg g g

:

o( ) ( , ( , )) ( )w t f t t x w t = g g , (0) 1w = , :

o0( ) ( , ( , ))

tw t exp f s s x ds= g g .

o o

1

o o0(1, ) ( , ( , ))x xx exp f s s x ds = g g

o

1

o o0( ) ( , ( , )) 0xx exp f s s x ds = >p g .

Poincar - 2 :

( )o o o1 1o o o0 0( ) ( , ( , )) ( , ( , ))x x xx f s s x ds exp f s s x ds = = p g g ( )( )o o1 2o o o0 0( ) ( , ( , )) ( , ( , )) 0sx xx f s s x exp f u u x du ds=

: 40

. . 2010

, Poincar U U . , :

o o o( )x x x U : p = o o( 1, ) ( , )t+ x t x=g g o o( 1) ( )x xt + t = ,

o o o( )x x x U : p = o o( 1, ) ( , )t+ x t x =g g o o( 1) ( )x xt + t = .

( 5, 0.8 = = ).

:

( )( , ) (1 ) 1 2f t x x - x sin t = + o( , ) (1 2 )f t x sin t = +

o( , ) 0f t x < 3 / 4t = . , Poincar - . o = - . - .

( 5, 0.8 = = ).

: 41

. . 2010

1. :

2( )f x x x =

. ( )x t :

( )x f x= .

20 / 4< , ox : - (0) ox x< , - (0) ox x> .

2 / 4 > , (0)x .

:

2( )f x x x x =

< , . = , > ;

:

2( ) (1 2 )f x x x sin t = + ;

: 42

. . 2010

22..

:

11 1 2( , )

dx f x xdt

= , 2 2 1 2( , )dx f x xdt

= .

1. - . - :

1 12 2

= =

x xx x

1 2( , )x x x= = U .

:

o:x U , ( )o 01 02( ) , = t tx t x e x e , o 01 02( , )x x x= U .

:*

:t g U U , ( )o 01 02( ) , =gt t tx x e x e , t ,

:

01o

02 .

0( )

0

tt

t

xex

xe

=

g

:

o o( ) ( )

g = g gt+t t tx x , o x U , , t t .

:

: g U U , ( )o 01 02( , ) ,t tt x x e x e=g , :

{ }o o

( , ) /x t x t= g U , o Ux .

* te te .

: 43

. . 2010

:

2 1| | | |x c x= .

=0 . :

0< : . . .

0 <

0= : . .

0 =

: 44

. . 2010

0> : . 1 =1 . .

0 1< < 1 = 1<

0 1< < 1 = 1<

.*

6. . ;

* 15. ;

: 45

. . 2010

2. - .

- :

1 22 1 .

x xx x=

=

1 2( , )x x x= = U .

- :

1 1 2=y x + x , 2 1 2= y x x ,

:

1 1

2 2

yy =

=

yy

1 2( , )y y y= U .

:

1 1( ) (0)ety t = y , 2 2( ) (0)e

ty t = y ,

:

1 1 2( ) (0) (0)x t = x cht + x sht , 2 1 2( ) (0)s (0)cx t = x ht + x ht .

1 2( , )y y 1 2( , )x x

:

:t g U U , ( )o 01 02 01 02( ) ,t x x cht + x sht x sht + x cht=g

:

10o

20 .

( )txch t sh t

xxsht cht

=

g

:

{ }o o

( ) /= g Utx x t , o Ux .

: 46

. . 2010

3. . - :

1 2

2 1 .

x xx x=

=

1 2( , )x x x= = U .

:

1x r cos= , 2x r sin= , > 0, ( 2 )r mod ,

:

( ) 0

( ) 1

r t

t

=

=

o

o

( )( )

r t rt t=

=

:

1 o o( ) (x t r cos t= ) , 2 o o( ) (x t r sin t= ) ,

1 1 2(0) (0)( )x t x cos t x sin t= + , 2 2 1(0) (0)( )x t x cos t x sin t= . - :

10o

20 .

( )txcos t sin t

xxsin t cos t

= g

:

{ } { }o 0 o o

/ ( ) 2 /tx t x x k k = = = T g , { }o 0 / (0) 0tx t= = = = T g .

.

: 47

. . 2010

4. . - :

1 1 2

2 1 2 .

x x xx x x= +

= +

1 2( , )x x x= = U .

:

1x r cos= , 2x r sin= , > 0, ( 2 )r mod ,

:

( ) ( )

( ) 1

r t r tt

=

=

o

o

( )( )

tr t r et t=

=

:

1 o o( ) (tx t r e cos t= ) , 2 o o( ) (

tx t r e sin t= ) .

1 1 2(0) (0)( )t tx t x e cos t x e sin t= + , 2 2 1(0) (0)( )

t tx t x e cos t x e sin t= .

te :

10o

20 .

cos sin( )

sin cost t xt tx e

xt t

= g

:

{ } { }o o o

/ ( ) 0tx k t x x = = =T g , { }o 0 / (0) 0tx t= = = = T g .

.

: 48

. . 2010

: . 2= U :

1 11 1 21 2

2 12 1 22 2 .

x a x a xx a x a x= +

= +

:

2:x , ( )1 2( ) ( ), ( )x t x t x t= .

- - :

2: ix , 1,2=i .

. :

1 11 21 1

2 12 22 2

x a a xx a a x

=

:

X( ) A X( )t t= .

2 2:

- - :

2: iy , , 1= i iy x , 1,2=i .

2 2

i ix y

: 49

. . 2010

.

:

( )3 0 =

det I .

:*

1 2 1 2a b b a = det 1 2a b = +tr ,

:

( )2 0 + =tr det

:

2(A) ( A) 4 A = tr det . :

(A) 0 > , , ,

(A) 0 = , , = ,

(A) 0 < , , ,i i = + = . , . -. .

* , , .

: 50

. . 2010

(A) 0 > 1 2 1 2, , .

:

{ }2E / Ai i = =

, 1,2i = .

:

1 11

22 2 .

00

y yy y

=

:

1 1

2 2

( ) ( )( ) ( )

y t y ty t y t

= =

1 1

2 2

( )

( )

t

t

y t c e

y t c e

=

= 1 21 1 2 2( )

t tx t c e c e = +

.

: 51

. . 2010

(A) 0 = , , = .

, :

{ }2E / A = =

.

- E 2dim = :

1 1

2 2

00

x xx x

=

:

1 1

2 2

( ) ( )( ) ( )

x t x tx t x t

= =

1 1

2 2

( )

( )

to

to

x t x e

x t x e

=

= ( ) tox t x e

= .

,

.

- E 1dim =

:

A = +

.

:

1 1

2 2 .

10

y yy y

=

: 52

. . 2010

:

1 1 2

2 2

( ) ( ) ( )( ) ( )

y t y t y ty t y t

= + =

1 1 2

2 2

( ) ( )

( )

t

t

y t c c t e

y t c e

= +

= 1

2

( ) ...( ) ...

x tx t

= =

(A) 0 < , , i , i = + = .

- *

:

1 2 = +

i , 1 2 =

i .

:

1 12 = + =

, 2 2( ) 2 = =

i ,

:

1 1

2 2 .

y yy y

=

* :

1 1 1 1

2 2 2 2

( ) (0) (cos sin )00 ( ) (0) (cos sin )

t

t

z z z t z e t i tz z z t z e t i t

= + = =

: 53

. . 2010

:

1 1 2

2 1 2 .

y y yy y x=

= +

:*

1 cosy r= , 2 siny r= , > 0, ( 2 )r mod ,

:

r r=

=

( )( )

to

o

r t r et t

=

= +

:

1

2

( ) cos( )

( ) sin( )

to o

to o

y t r e t

y t r e t

= +

= +

1

2

( ) ...( ) ...

x tx t

= =

7. - , . * .

: 54

. . 2010

: 55

. . 2010

8. :

[1] 1 1 2

2 1 24 2

x x x

x x x

= +

=

[2] 1 1 2

2 1 2

2 3

3 2

x x x

x x x

= +

= +

[3] 1 1 2

2 1 2

2 3

3 4

x x x

x x x

=

= +

[4] 1 1 2

2 1 2

3 3

3

x x x

x x x

=

=

.

[1] : 1 23, 2 = = , : 1 2(1, 4), (1,1) = =

. :

1 1

2 2

3

2

y y

y y

=

=

3

1 12

2 2

( )

( )

t

t

y t c e

y t c e

=

=

3 21 1 2

3 22 1 2

( )

( ) 4

t t

t t

x t c e c e

x t c e c e

= +

= +

[2] : 1 21/ 2, 3 = = , : 1 2(1, 2), (1, 3) = =

. :

1 1

2 2

/ 2

3

y y

y y

=

=

/2

1 13

2 2

( )

( )

t

t

y t c e

y t c e

=

=

/2 31 1 2

/2 32 1 2

( )

( ) 2 3

t t

t t

x t c e c e

x t c e c e

= +

= +

[3] : 1 = , : (1, 1) =

, : ( 1/ 3, 0) =

. :

1 1 2

2 2

y y y

y y

= +

=

1 1 22 2

( ) ( )

( )

t

t

y t c c t e

y t c e

= +

= 1 2 1 2

2 2 1

( ) 2( / 3)

( ) 2( )

t

t

x t c t c c e

x t c t c e

= +

= +

[4] : 1 5, 1 5 = + = i i , : (2 5, 3), (2 5, 3) = + =

i i .

1 2(2, 3), ( 5, 0) = =

, :

1 1 2

2 1 2

5

5

y y y

y y y

=

= +

1 2sin , cosy r y r= = :

( ) ( )

( ) 5

r t r t

t

=

=

( )

( ) 5

to

o

r t r e

t t

=

= + / 5r c e=

1

2

( ) cos( 5 )

( ) sin( 5 )o o

o o

t

t

y t r e t

y t r e t

= +

= +

( )1

2

( ) 2cos( 5 ) 5 cos( 5 )

( ) 2 sin( 5 )

o o o

o o

t

t

x t r e t t

x t r e t

= + + = +

: 56

. . 2010

[1] [2]

[3] [4]

9. :

1 1 1 2

2 2 1 2

( , )( , )

x f x xx f x x=

=

:

1 2 2 1 2 1 1 21 2

( , ) ( , ) ( , )x x = f x x f x xx x +

.

- :

;=div

- . , .., ; , ;

: 57

. . 2010

10. . .

: 58

. . 2010

- : 3F : : . ; - . , , ... . , - , , ... : . : . .14F : :. . & , .... : 0, : , . :. . . 5. , - , , ... ., , , . . , . = , , -, , ... ., , , ., , : ., , Poincar :, - Poincar:., :, , , , :, , - . , Poincar - : : 23F : ( ). ( ). - : - : - :. . - :. . : .. 9. :