dynamical systems
DESCRIPTION
ctcrgrrjTRANSCRIPT
-
_____ ______
( - 2010-11) _____ ______
: . - .
1 : *
. , - , , - , - . , ..
Pierre Simon Laplace (1749-1827)
. , . - ; , . Heisenberg ; - . Laplace.
* : .
&
-
: 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. :