) (

1392

(122)

- : . : / :

. : : . :

- - - - : : . :

. :

. : . :

.( ( : .( ( :

.( ( : .( ( :

. : .TA / :

/ : :

:

: :

: ()

:

: :

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . SFT .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

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. . .

.

. .

.

: ut(x, t) = f(u(x, t), ux(x, t)) : -

u(x, t+ ) = f(u(x, t)) : -: -

d

dtu(n, t) = f(u(n , t), u(n, t), u(n+ , t)),

: -

u(n, t+ ) = f(u(n , t), u(n, t), u(n+ , t)),

.u(n, t+ ) = f(u(n , t), u(n, t), u(n+ , t)) : - u(x, t)

. . . - -

. - - .

. . .

. . .

. .

. .

. . . .

. :

. .

. . . . . [] [] [] [] [] .

.

. .

.

. (-) . (-) : .

.

. (-) .

David HilbertFelix Klein

Erlangen Program Georg Cantor

. .

. . . . . ... Z .

R+ R Z+ .

.

X P (X) X ...: X P (X) . X

, X - -

. -

. (X, ) X

. X . B B f : X Y (Y, Y ) (X, X)

: Y Y

f(Y) = {x X : f(x) Y} X .

. f

.

. X . X {, X}

. X = P (X) ... . X

. X

A = {Ai : i I} (X, ) A .X =

iI Ai I X

.|I| < A X . Ai

Aj = i, j I

X .

.

X d : X X R ... : x, y, z X

d(x, y) d(x, y) = x = y -d(x, y) = d(y, x) -

.d(x, y) d(x, z) + d(z, y) - . (X, d)

. X

y = (y, , yn) x = (x, , xn) X = Rn ... (Rn, d) d .d(x, y) := (

ni=(xiyi))

. Rn

. Rn d .

d .A X (X, d) (A, d) d A A

x X . (X, d) B .B(x, r) = {x X : d(x, x) < r} r > P (X) X

(X, d) B. d

x X f : X Y (Y, dY ) (X, dX) > > x f f .dY (f(x), f(x)) < dX(x, x) <

.

. f : X Y Y X ...

. f

n N > (X, d) {xn} x .d(xn, xm) < m,n n n N > limn xn = x A X x .d(xn, x) < n n A A .xn x A {xn} A x X . A := A A A A .x U A U (X, d) . .

. (X, )

.( [] (. .. (Rn, d) ...

. ...

diam : P (X) {} R {} (X, d) :

diam(A) := sup{d(x, y) : x, y A}.

.

X A (X, d) ... . diam(A) <

.

(X, ) ... . X . X d

.diam(A) diam(B) A B ...

. ...

X {An}n= j k .An+ An n N

.k

n=j An = Aj k

n=j An = Ak

{Cn} (X, d) ... C :=

n= Cn .diam(Cn) X

.

C Cn n N .. . .diam(C) = diam(Cn) diam(C) diam(Cn). diam(C) d(x, y) > x = y CCcn X =

n= C

cn C =

n= Cn =

n n nr X Cn Cn Cnr = X = Ccn C

cnr

. Cn Cnr = Cnr =

.diam(A) diam(A) diam(A) = diam(A) ...

A X A ... .A A A A

{xn} (X, d) ... .

X . .Bn B {xn} . /n

n > C = B Cn diam(Cn) diam(Bn) /n .Cn = B Bn C =

n= Cn .. .Cn+ Cn

.xnk Ck {xn} {xnk} . x .limk xnk = x ..

. {x} .. {xnk} ...

... xnk x {xnl} {xnk} {xn}

.x = x xnl x

{xn} (X, d) ... .xn x x

. {xn} (X, d) ...

x {xn} .. ..xn x .. .. .

T = {(X, ) : A} A ... B = {B : A} X =

AX .

A Y B P (X) Y B

.Y = X

: X =

AX X ... :

. - f f : Z X Z -

. f : Z X -

.

.

Rn ... . Rn

n N X ...

i= X . Xn

.

.

: G G G G ... : e G

.(g, e) = (e, g) = g g G -. (g, (h, k)) = ((g, h), k) g, h, k G -

G . g h gh (g, h) :

.g h = h g = e h G g G - . g g h i : G G G H G .H G G .

.g h H g, h H H G

X A(X) X (X, X) .

X Home(X) .Home(X) A(X)

f : X X . (X, dX) dX(f(x), f(x)) = dX(x, x) x, x X . Iso(X) X .

.Iso(X) Home(X)

.Home(X) A(X) X ...

.Iso(X) Home(X) X ...

(G, G) . G G (G, G) . : G G G i : G G G

.

(X, X) (X,G, ) ... : GX X () G

: .(e, x) = x x X -

.(g, (h, x)) = (gh, x) x X g, h G -

G X gx g x (g, x) . (X,G, ) X .

. X

. (X,G, ) G ... .g(x) = (g, x) : g : X X g G

.g Home(X) - (g) = g : G Home(X) -

.(gh) = (g) (h) g, h G

x X (X,G, ) . Gx G x x .X O(x) := {g x : g G}

(X,G, ) x X ... x . .O(x) = {x}

: x G

Gx := {g G : g x = x},

.ker :=

xGGx

.gGxg = Ggx g G x X ...

.

.Gx G x X kerG ...

x X ker = {e} Y X .X = O(x)

G Y = {g y : g G, y Y } Y.

. X - ... . O(x) -

. X -.Y =

yY O(y) Y X -

... .

(Y,G, ) (X,G, ) ... g x X h : X Y h = h . h((g, x)) = (g, h(x))

.

Aut(X,G, ) X X . Aut(X)

h : X Y ... : (Y,G, ) (X,G, )

.Gx = Gh(x) h(O(x)) = O(h(x)) x X - X X -

. h(X) Y

Aut(X,G, ) Home(X) - ... .

.g Aut(X) g Z(G) g G -: Z(G)

Z(G) := {h G | k G hk = kh}.

Y X (Y,G, ) (X,G, ) (Xi, Gi, i) . G Y : G Y Y . i = , , n

: GX X G =n

i= Gi X =n

i= Xi (X,G, ):

((g, , gn), (x, , xn)) = (gx, , gnxn).

(Xi, Gi, i) ... (X,G, )

.

(X,G, ) . X G

. . . (M,dM ) (X, dX) . ) ( : M GX X : G X X M X . (g, x) = (, g, x)

, B(, ) B(, ) . =

). ( =

(X,G, ) (Y, dY ) (X, dX) ... I : X Y X

. I(g(x)) = I(x) g G x X

.

(X,G, ) .G = Z Z+

.

: n

: - . .

.. - .

.

-. ...

- . ( . )

.

(X, dX) (X, X) ... (X,G, ) G = Z+ G = Z

.

f(x) := (, x) f : X X (G = Z ( G = Z+

n Z+ ) ( f .f () (n, x) = fn(x) (n Z () ( .f(x) = x x . x X f : X X x X f .O+(x) = {fn(x) : n }

.O(x) = {fn(x) : n }

f : X X x ... n .fn(x) = x n N m > n x . x x m,n .fm(x) = fn(x) .k = mn (n, k)

. k = m n O(x)

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

T (x) = |x | x = / ... . ) (

x . T (x) = T (x) = x . x = p/q

: x

T (x) = x

T (x) = x

Tn(x) = nx.

x = (k l)/(mn) knx = lmx Tn(x) = Tm(x) x = p/q . x k/q T (x) .T (x) = p/q T (x) = p/q

.

k/q Tn(x) .: x [, ] k/q

: x = / -

x = /, x = /, x = /.

: x = / -

x = /, x = /, x = /, x = /, x = /.

p x = p/q ... . q

{xn} . (X, ) U limn xn = x x .xn U n n n N x{an} A x X . A X X A .limn an = x A .A

X A A A .A = X

.a U A U A a (A) = A . A A

. A

R m,n Z m/n ... .

... . .

... .

.. ... . [, ]

h : X Y g : Y Y f : X X ... :

n x X -. n y = h(x) Y

y = h(x) f (n, k) x X -. g (n, k)

T (x) = |x /| U(x) = x( x) ... . S(x) = / arcsin( x)/

. S(U(x)) = T (S(x)) .x [/, ] x < / x > /

:

S(U(x)) =

arcsin( x( x))

, T (S(x)) = +

arcsin( x)

,

x = / . :

d

dxarcsin( x+ x) = d

dx arcsin( x).

. d/ dx(arcsin(x)) = /

x . [, /]

T . U T U T US(x) = (cosx)/ . x = / . x = S(/) = ( cos(/))/

:

x = /

Sy =

x = /

Sy =

x = /

Sy =

x = /

Sy =

x = /

Sy = .

Stanislaw Ulam

.

. .

.

n x X ... W s(x) x . f : X X

: x

W s(x) := {y X : limk+

fnk(y) = x}. (.)

W u(x) x f : X X : x

W u(x) := {y X : limk

fnk(y) = x}. (.)

X = R . x n = x |x|, |f(x)|, |f(x)|, . W s() x R .

. W u() f : R R

.

W s() = (,)(,+) f(x) = x f : R R ... .W s() = {} W s() = {} W s() = (, )

g(x) = |x | g : R R ... . g {, } g() = g() = {, } W s() = W s() = Z

. Z+

m Z k N x R ... .gk(x) = m x

(X, dX) ... f : R R q, p X

.W s(p) W s(q) = q = p

k k p, q X . .p = q W s(p) W s(q) = . f : R R N N > x W s(p) W s(q).dX(q, fnk(x)) < / n N dX(p, fnk(x)) < / n N

n M .M := max{N, N}

dX(p, q) dX(p, fnkk(x)) + dX(fnkk(x), q) < / + / = .

: x X .. ...

h(W s(x)) = W s(h(x)).

f X A ( ( f : X X .(f(A) A)f(A) A ( (

. ... A X f : X X f .fn(A) = A n Z

.

. .

: . Rn . -. Rn -

Luitzen Brouwer

.

. -.

. -.

: V R+ . (V,+, ) ( C ( R x, y V V

:

.x = x = -

.x = || x -

.x+ y x +y - (V, d) . (V, ) d(x, y) := x y (V, ) .

.

|x| := d(x, ) Rn d ...

. (Rn, | |) .

. Rn L(Rn) ... | | T = sup{|Tx| : |x| = } : L(Rn)

: . Rn

. -

: R -

T = sup{|Tx| : |x| = }

= sup{|| |Tx| : |x| = }

= || sup{|Tx| : |x| = }

= || T.

Juliusz Schauder Stefan Banach

: y = Rn -

T = sup{|Tx| : |x| = }

= sup

{T ( x|x|) : x = } T ( y|y|

) = |Ty||y| ,

|Ty| T |y|. (.)

: S L(Rn)

T + S = sup{|Tx+ Sx|

|x|: |x| =

} sup{|Tx|+ |Sx| : |x| = }

sup{|Tx| : |x| = }+ sup{|Sx| : |x| = }

= T+ S.

. ...

... .

. [] .

(X, d) f : X X .dX(f(x), f(x)) kd(x, x) x, x X < k <

f : X X (X, d) ) ( ... x X f

.f(x) = x

.f(xi) = xi i = , x, x X . k <

d(f(x), f(x)) = d(x, x) kd(x, x).

.

x X . d(x, x) = xn .xn+ = f(xn) n

. f n N :

d(xn, xn) = d(f(xn), f(xn)) kd(xn, xn),

.d(xn, xn) knd(x, x) n > m :

d(xn, xm) d(xn, xn) + d(xn, xn) + + d(xm+, xm)

(kn + kn + + km)d(x, x)

. x {xn}

: xn+ = f(xn) f : .x = f(x)

f : X X (X, d) ... x X f .d(f(x), f(y)) < d(x, y) x = y

.limn fn(x) = x x X

(X, d) ... f : X X

d(f(x), f(y)) < d(x, y) (/)(d(f(x), f(y))).

.limn fn(x) = x x X f

|A| := maxij |aij | n n A = (aij) ... .|A| A n|A| | |

.tr(A) < n Rn x 7 Ax ...

detA = / A ... . A .tr(A) = /

.A = (aij)i,jn A

i,j aij ...

.A | detA|/n ...

. ...

. f : X X (X, ) x U f x X x .O+(x) U x V V x U x U U .limn fn(x) = x( ( f m x . x. ( ( fm x O(x) = {x, f(x), , fm(x)}

. ( ( ( ( x U x

: x

B :=

n=

fn(U).

m x

B =

n=

fnm(U).

f j(x) f i(x) i = j (mod m) O(x) .

.

... .

.

.x X f : X X (X, d) {nk} x - y X - .limk fnk(x) = y limk nk = + - f . (x) f (x) x y X x X - .limk+ fnk(x) = y {nk}

. (x) f (x) x f .x (x) x U x X . R(f) .fn(U) U = n N x . NW (f) f Y X .R(f) NW (f) . f : X X

. X f

x X f : X X ... limn fn(x) = b limn fn(x) = a a, b X . a x a = b . b a x

- - ... .

f ... . (x) (x) x X

.R(f) NW (f) ...

: ...

(x) =nN

in

fi(x), (x) =nN

in

f i(x).

.NW (f) R(f) ...

... .

: f : X X X ...

.(y) = Y y Y Y X -

. Y y Y Y -

. X ... U X f : X X

.n

k=n fk(U) = X n N

O+(x) X ... X . (x)

.

. f : X X ... .p X

.O+(p) = O+(p) (p) -

. X O+(p) (p) = -

.O+(p) = (p) . X O+(p) -

. G ...

{gn : n Z},

G . G .

G ... g, , gn G U X X

.n

i= gi(U) = X

.

n .. . .

.

.

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

:

x = f(x) = x( x) = , x = f(x) = , .

n : T . T (n) = n + : n T (n) = n/ . n, T (n), T (T (n)),

. n, T (n), T (n), , T k(n) = n n+ ) ( . .

. n T (x) = x (mod ) . n+ : f(Tn()) [x] f(x) = [x]

Lothar Collatz

. (.) ..

. .

. xn = f(Tn()) .: .

T (x, y) =

(xy

x+ y,x+ y

), (.)

. F (T (x, y)) = : T F (x, y) = xy . .F (x, y) y = x . . . (

,

) (xn, yn) (, ):

(, ), (

, ), (

,

), (

,

),

/ = . [xn, yn]

n N T .

=

. . y = T (x) x . . xn, xn, xn+ yn

.

. . . . ,xnk] , xn+k+] = [, , , , ] k n x

. . :

x = y

y = x (x )y,(.)

.[] ((, ) ( . : .

x = (y x)y = xz + x y

z = xy z,

(.)

(x(t), y(t), z(t)) . . ( ) .

.[] R T (x, y) = (x y + c sinx, x) . .

:

T (x+ , y) = T (x, y)+ (, ) T (x, + y) = T (x, y)+ (, ).

. c

Van der Pol Edward Lorenz

. ..

c = . . c . xn+ = xn + c sinxn xn

. .k Z x = y = k .[] c =

.[] .

H(x, y) = (ax + by, x),

. |b| = |b| < b = a = . . xn+ = axn + bxn xn+ = xn+xn . xn

H = (b + ) a . x, y = ((b + )

)/a

Boris ChirikovMichel Hnon

Fibonacci

Dynamical_Systems_and_Chaos_3-2-92.pdf

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