289.Введение в динамику одномерных отображений учебное...
TRANSCRIPT
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2006
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517.925
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91 , 2006. 104 .
ISBN 5-8397-0491-1 (978-5-8397-0491-6)
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517.925
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ISBN 5-8397-0491-1
c (978-5-8397-0491-6)
. .. , 2006
c .., 2006
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5
1. 7
1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1. . . . . . . . . . . . . 8
1.2. . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.1. . . . . . . . . . . . . . . . . 18
1.2.2. . . . . . . . . . . . . . . . . . . . . . . 20
1.3. . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4. . . . . . . . . . . . . . . . . . . . . . . . 31
1.5.
. . . . . . . . . . . . . . . . . . . . . . . 38
2. 45
2.1. . . . . . . . . . . . . . . . . . . . . . 45
2.2. 3
. . . . . . . . . . . . . . . . . . . . . 53
2.3. f(x) = 4x(1 x) . . . . . . . . . . . . . . 582.4. f(x, r) = rx(1 x) r > 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.4.1. . . . 73
2.4.2. 2 f(x, r) r > 2 +
5 . . . . . . . . . . . . . . . . . . . . . . . . . 76
79
1. . . . . . . . . 79
2.
. . . . . . . . . . . . . . . . . . . . . . . 84
3.
f(x, r) = rx(1 x) r > 4 . . . . . . . . . . . . 883
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4. - . 93
5. 3 . . . . . . . . . . . . . . . 95
6. . . . . . . . . 95
7. . . . . . . . . . . . . . . . . 99
102
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f(x, r) = rx(1 x) r 0 r > 4. , ,
. f(x) = 4x(1x) . -
, -
. f(x) = rx(1x) r > 4. - .
f(x) = rx(1 x) r > 2 +5. , -
,
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f(x) = rx(1x) 4 < r < 2+5,
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1.
1.1.
-
xn+1 = f(xn), n = 0, 1, . . . ,
f(x) , R. x0 R, f(x), O(x0)
x0, f(x0), f(f(x0)), f(f(f(x0))), . . . (1.1)
,
f(x) = kx(k > 0). x0 R
x0, kx0, k2x0, . . . , k
nx0, . . .
k < 1, x0 6= 1 n . k > 1, . n = 1, O(x0) x0:
x0, x0, . . .
, , ,
, O(x0) n, .. (1.1) x0. , f(x) = sin x, ( )
0 n.7
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1.1.1.
. -
x, - R () I . f(x) , f(x) 6= f(y), x 6= y. f : I J , f1(y)(y = f(x), x = f1(y)). , f(x) = x3, f1(x) = 3
x. R R. f(x) = tg x, f1(x) = arctg x. f(x) : (pi/2, pi/2) R, f1(x) : R (pi/2, pi/2). f(x) , - . , f(x) = tg x (pi/2, pi/2) R. f(x) f(x) - , f(x) . ,f(x) = tg x , f(x) = x3 , , f1(x) = 3
x - x = 0. C f(x) g(x) f(g(x)) = f(x) g(x), n f(x)
f (n)(x) = f(x) f(x) f(x) n
.
f1(x),
f (n)(x) = f1(x) f1(x) f1(x) n
.
(
):
[f(g(x))] = [f g] = f (g(x))g(x).
, , g(x) = f (n1)(x),
[f (n)(x)] = f (f (n1)(x))f (f (n2)(x)) . . . f (f(x))f (x).
.
. f(x) : [a, b] R - , c [a, b],
f(b) f(a) = f (c)(b a).
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1. f(x) [a, b],
|f(x) f(y)| K|x y|, x, y [a, b].
, f (x) [a, b] : |f (x)| K. . f : [a, b] R ., f(a) = u, f(b) = v. u < z < v c, a c b , f(c) = z. 2. I = [a, b] f : I I -. a c b , f(c) = c. ( f(x) f(x).)
. g(x) = f(x) x. g(x) I. f(a) > a, f(b) < b ( a b f(x)). g(a) > 0, g(b) < 0. c, g(c) = 0. , f(c) = c.
. I = [a, b],f : I I. f(x) I,
|f(x) f(y)| q|x y|, x, y I, q < 1. - , I , .. .
( ). f : I I f(x) I. f(x) I.
, -
() f(x).. x0 -
x0, x1 = f(x0), x2 = f(x1), . . . , xn = f(xn1), . . . (1.2)
, (1.2) x I. ,
|x2 x1| = |f(x1) f(x0)| q|x1 x0| = q|f(x0) x0|,
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|x3 x2| = |f(x2) f(x1)| q|x2 x1| q2|f(x0) x0|,
|xn+1 xn| = |f(xn) f(xn1)| qn|f(x0) x0|.,
|xn+p xn| |xn+p xn+p1|+ |xn+p1 xn+p2|+ + |xn+2 xn+1|++|xn+1 xn| (qn+p + + qn)|f(x0) x0| =
=qn qn+p1
1 q |f(x0) x0|. q < 1,
|xn+p xn| < qn
1 q |f(x0) x0|, , |xn+p xn| 0 n (1.2) x I. , f(x) = x. ,
|x f(x)| |xn x|+ |xn f(x)| = |xn x|+ |f(xn1) f(x)| |xn x|+ q|xn1 x|. > 0 n
|xn x| < 2, |xn1 x| <
2.
,
|x f(x)| < . > 0 ,
|x f(x)| = 0 x = f(x). . -
x, y, f(x) = x, f(y) = y.
|x y| = |f(x) f(y)| q|x y|. q < 1. |x y| = 0. , x = y. (1.2)
. ,
|x1 x| = |f(x0) f(x)| q|x0 x|, |x2 x| q2|x0 x|.
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,
|xn x| qn|x0 x|., (1.2) x co c - q., , -
f (x) |f (x)| q < 1 x I. x f(x).
( ). -
f(x) x
|f (x)| = q < 1. xn = f(xn1) (n = 0, 1, . . . ) c x, x0 x
.
|xn x| (q + )n|x0 x|, (1.3) (q + < 1).
. . - > 0, |x0 x| <
|f(x0) f(x) f (x)(x0 x)| |x0 x|. |x0 x| <
|f(x0) x| |f(x0) f(x) f (x)(x0 x)|++ |f (x)(x0 x)| (q + )|x0 x|.
(1.4)
q+ < 1, x1 = f(x0) x, x0 |x1 x| < . (1.4) x1, x2, . . . , xn, - (1.3).
. (1.3) , (1.2)
x , f (x) = 0.
F (x, z), z . , F (x, z) |x x0| , |z z0| . F (x, z), z, ( ), z |z z0|
|F (x, z) F (y, z)| q|x y|, |x x0| , |y x0| ,
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q < 1 q z. F (x, z) z |x x0| . -
|x x0| x = x(z) x = F (x, z).
f(x, z), |x x0| b, |z z0| a.
f(x0, z0) = 0,
.. x0 f(x, z) = 0 (1.5)
z = z0. z, z0, x0 x(z) (1.4), , (1.4)
x(z). -
, .
. f(x, z) - :
1) f(x, z) x, z |x x0| b,|z z0| a f(x0, z0) = 0,2) fx(x, z), - (x0, z0),3) fx(x0, z0) 6= 0. , > 0, z
|z z0| (1.5) |x x0| x(z). x(z) |z z0| ..
x = x [fx(x0, z0)]1f(x, z) (1.5). , -
F (x, z) = x [fx(x0, z0)]1f(x, z) , > 0 |x x0| |zz0| . x(z). ,
Fx(x, z) = 1 [fx(x0, z0)]1fx(x, z)
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. Fx(x0, z0) = 0, > 0, |x x0| , |z z0|
|Fx(x, z)| q < 1. F (x0, z0) = x0, , |z z0|
|F (x0, z) x0| (1 q).
|F (x, z) x0| |F (x, z) F (x0, z)|+ |F (x0, z) x0| q|x x0|+ (1 q) q + (1 q) = ., F (x, z) |xx0| |zz0| . z
|F (x, z) F (y, z)| q|x y| (|x x0| , |y x0| ). x(z) . - x(z) z1 |z z0| . ,
|x(z1) x(z)| = |F (x(z1), z1) F (x(z), z)|
|F (x(z1), z1) F (x(z1), z)|+ |F (x(z1), z) F (x(z), z)| |F (x(z1), z1) F (x(z1), z)|+ q|x(z1) x(z)|,
|x(z1) x(z)| 11 q |F (x
(z1), z1) F (x(z1), z)|.
F (x, z) . . f(x, z) = x2 + z2 1. -
f(x, z) = x2 + z2 1 = 0
x, z. f(x0, z0) = 0 z0 > 0 ( (x0, z0) ), fz(x0, z0) = 2z0 6= 0. , z(x) ,
f(x, z(x)) = 0
x, x0. z(x) z =
1 x2. f(x, y) = x5y4xy5 yx2+1. ,
f(1, 1) = 0, fy(1, 1) = 2 6= 0.
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y = p(x) , f(x, p(x)) = 0, - 1. y = p(x) .
. -
fz(x, z) (x0, z0), x(z) -,
dx(z)dz
= fz(x, z)fx(x, z)
. (1.6)
f(x(z), z) 0, - (1.6).
x(z) . - fz(x, z) fx(x, z), f(x, z)
f(x+ h, z + k) = f(x, z) + hfx(x, z) + kfz(x, z) + 1h+ 2k, (1.7)
1 2 0 h k. (x, z), (x+h, z+k), , x = x(z), x + h = x(z + k). f(x, z) = 0,f(x+ h, z + k) = 0 (1.7)
0 = hfx(x, z) + kfz(x, z) + 1h+ 2k. (1.8)
x(z) , k 0 h 0 1 0 2 0. (1.8) kfx(x, z) 6= 0, (
1 +1fx
)h
k+fzfx
+2fx
= 0.
k 0,
limk0
h
k+fzfx
= 0.
limk0
h
k= lim
k0x(z + k) x(z)
k=dx(z)dz
.
x(z) .
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1.2. 15
1.2.
.
1.1. x O(x) - l,
f (l)(x) = x, f (j)(x) 6= x 0 < j < l. 2 x0, x1 = f(x0)
(f (2)(x0) = x0, f(2)(x1) = x1). 3 x0, x1 = f(x0), x2 = f(x1) = f
(2)(x0). l l . . f(x) = x3 0, 1 1 . f(x) = x2 1
152 , 0,1 2.
1.2. x , f (i)(x) i f(x)., f(x) = x2 x = 1 , x = 1 , f(1) = 1. f(x) == x2 1 1 , f(1) = 0, ( ) 2. 1.3. x0 f(x) -, U , fU U limn f
(n)(x) = x0 x U . 1.4. x0 f(x) -, U , U \{x0} , .. x U \{x0} n = n(x), f (n)(x) / U . 1.1. x0 f(x)
|f (x0)| < 1, x0 . |f (x0)| > 1, x0 .. |f (x0)| < 1, > 0, |f (x)| q < 1 x [x0 , x0 + ].
|f(x) x0| = |f(x) f(x0)| q|x x0|.
, f(x) [x0 , x0 + ] f(x) x0, x. ,
|f (n)(x) x0| qn|x x0|.
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f (n)(x) x0 n. x0 . |f (x0)| > 1, > 0, |f (x)| q > 1 x [x0, x0+].
|f(x) x0| = |f(x) f(x0)| q|x x0| > |x x0|. f(x) [x0 , x0 + ], , ,
|f (n)(x) x0| qn|x x0|. , n, f (n)(x) / [x0 , x0 + ]. . f(x) = x3 f (x) = 3x2. , x0 = 0 f(x) , 1, 1 . , f(x) = x21 x0,1 =
152 .
, , -
|f (x)| = 1. , f(x) = xa(x)3 x = a > 0 , f () = 1. 1.5. x0 l f(x) , x0 f
(l)(x) . ,
x0 f(l)(x) .
1.1 .
1.2. x0 l f(x) , ddxf (l)(x0)
< 1, , ddxf (l)(x0)
> 1. = ddxf
(l)(x0) - (). ,
d
dxf (l)(x0) = f
(f (l1)(x0))f (f (l2)(x0)) f (x0).
x0, x1 = f(x0), . . . ,xl = f
(l1)(x0). - , 1.2,
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.
, .
,
.
. f(x) = x2 1 0,1 , d
dxf (2)(0) = f (0)f (1) = 0.
f(x), ,
.
1.
) f(x) = 2x x2,) f(x) = 2x 2x2,) f(x) = 2, 44x x3,) f(x) = arctan x,
) f(x) =ax2 + bx(1 x)
ax2 + 2bx(1 x) + c(1 x)2 , 0 x 1, b > a, b > c.e) f(x) = 1 +
3 + x, x+ 3 0.2.
f(x) =2x
1 + x3.
2?
3. , R - 2. ,
2.
4. 2 -
) f(x) = 3, 2x 3, 2x2,) f(x) = 2, 2x3 + 1, 2x.)
f(x) =
{2x, 0 x 1/2,2 2x, 1/2 x 1.
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) F (x) = x+tf(x), t > 0,
f(x) =
{ x, x 0,x, x > 0.) f(x) = (c 1/2)x2 + (1/2 2c)x+ c, 0 < c 1.5. 2 3 - f(x) = 1 2|x|.6. -
f(x) =
{ 13x+
23 +
13p , 0 x 1 1p ,
p px, 1 1p x 1, p > 1. 2 3.
1.2.1.
xn+1 = f(xn), n = 0, 1, . . . , (1.9)
f(x) : A A . h(x) , - B A. (1.9) :
xn = h(yn).
h(yn+1) = f(h(yn)),
yn+1 = h1(f(h(yn))) = g(yn), n = 0, 1, . . . (1.10)
, g = h1 f h B B.
x0, f(x0), f(2)(x0), . . .
(1.9) (1.10):
y0 = h1(x0), y1 = h1(f(h(x0))), . . . .
x0 f(x), y0 = h1(x0) g(x). x0 l f(x)(f (l)(x0) = x0),
(h1f(h(y0))(l) = h1f (l)(h(y0)) = y0,
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.. y0 l g(y). , y0 g(y), x0 = h
1(y0) f(x).
1.6. (1.9) (1.10) -
, f(x) g(y) = h1(f(h(y)) .
, f : A A g : B B -, h : A B , h g = f h. -
. x0 f(x) , y0 g(y) . h(y) ,
h g(l) = f (l) h
,
f(x) g(y) . x0 , f (x0) = 0 ( f(x)), y0 g(y).
. f(x) = x21 x = h(y) = y+c (c > 0). y = h1(x) == x c g(y) = h1[(y + c)2 1] = y2 + 2cy + c2 c 1 - f(x). x = h(y) = y3, g(y) = (y6 1)1/3 f(x) = x2 1.
1. f g, f(x) g(x) -. ,
, ..
) f f ,) f g , g f ,) f1 f2, f2 f3, f1 f3.2. f(x) = x2, g(x) = x2+ax+ b(x R).
(a, b), f g.3. a R b R, f(x) = 1 ax2 g(x) = bx(1 x)(x R) ? .
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1.2.2.
, ,
.
( )
, . -
(-
).
1.7. f(x) g(x) , R = (,+). C0- f(x) g(x)
d0(f, g) = supxR
|f(x) g(x)|.
Cr-
dr(f, g) = supxR
{|f(x) g(x)|, |f (x) g(x)|, . . . , | dr
dxrf(x) d
r
dxrg(x)|}.
f(x) g(x) - J = [a, b]. , Cr- , . , C0- f1(x) = 2x g2(x) = (2 + )x , f2(x) = 2x g2(x) = 2x + C
r- r. J = [0, 5] C0- f1(x) g1(x) 5||. 1.8. f : J J . , f(x) - Cr- (Cr- ) J , > 0, g(x), J , dr(f, g) < , f(x).
.
. -
. f(x) = 12x. C1- R. - . > 0, , d1(f, g) < , f(x) g(x) . , < 1/2. d1(f, g) < 1/2, 0 < g
(x) < 1 x R. , g(x) . , g(x) p R x R p . , |g(x)| < 1 ,, g(x) .
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h(x). -
. 5 < |x| 10. f(x) ( 0) - . g(x) . , g(10) < x 10 10 x < g(10) g(x),.. , , g(x) - . h(x) - [5, 10] [10,5] h : [5, 10] [g(10), 10], h :[10,5] [10, g(10)]. , h(x) , h(10) = 10. h(x) - . x 6= 0. n, f (n)(x) - f(x). , hg(x) . h(x) = g(n)hf (n)(x). , g(n) h(x) = h f (n)(x). f(x), g h(x) = hf(x). , h(0) = p. , h(x) ., , , -
, .
. f0(x) = xx2. x = 0 - x0 (0, 1), f (0) = 1. f(x) = xx2+., f(x) C
r - f0(x). f(x) , > 0, -, < 0. , f(x) f0(x). f0(x) . .
1.9. p k f(x) - ,
0 6= ddxf (n)(p)
6= 1. f(x) Cr-, - f(x) (, , f(x) ). -
f(x) C1-, f(x) . - f(x) Cr- (r 2), f(x) ( x0 f(x) , f (x0) 6= 0). .
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f(x) C1- . ,
> 0 , g(x) C1 - f(x) , f(x) g(x) . , , .
. p - f(x). U p V 0 - h : U R, f(x) U l(x) = f (p)x V .
, ,
.
1.3.
, -
. , -
()
, .
f(x, c), c, - . c = c0 , f(x, c) - f(x, c0) c c0. c -, . ,
.
, ..
.
, -
f(x, c). f(x, c) c = c0 k, .. f (k)(p, c0) = p. (p) , |(p)| < 1 , |(p)| > 1 . (p) 6= 1, c0 p(c)(p(c0) = p) - k f(x, c), c. - -
(x, c) = f (k)(x, c)x = 0, (p, c0) = 0, d(x,c)dx(p,c0)
= (p)1 6= 0. , k .
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, -
k c, p k, (p) = 1. - |(p)| = 1. , , -
1. , f (k)(x, c) x c.
1.1. f(x, c) -
1) f (k)(x0, c0) = x0,
2)
f (k)
x(x0, c0) = (x0) = 1,
3)
2f (k)
x2(x0, c0) =
(x0) > 0,
4)
f (k)
c(x0, c0) > 0.
(c1, c0) (c0, c2) > 0 , c (c1, c0), f (k)(x, c) (x0 , x0 + ), , . c (c0, c2), f (k)(x, c) (x0 , x0 + ).. g(x, c) = f (k)(x, c) x.
g(x0, c0) = 0,gx(x0, c0) = 0,
gc(x0, c0) > 0, c = h(x) , c0 = h(x0)
g(x, h(x)) 0 (1.11) (x0, c0). (1.11), -
gx(x, h(x)) + gc(x, h(x))dh
dx= 0. (1.12)
dh
dx(x0) = 0,
gx(x0, c0) = 0. (1.12),
gxx(x, h(x)) + 2gxc(x, h(x))dh
dx+ gcc(x, h(x))
(dh
dx
)2+ gc(x, h(x))
d2h
dx2= 0.
x = x0
gxx(x0, c0) + gc(x0, c0)d2h
dx2(x0) = 0.
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d2h
dx2(x0) = gxx(x0, c0)
gc(x0, c0)< 0.
, x0 - c = h(x), f (k)(x, c) :f (k)(x, h(x)) = x. , c > c0, , c < c0 ( k). , -, , ,
xf
(k)(x, h(x)) x (x0, c0), fxx(x0, c0) 6= 0. (. 1.1)
( ), (-
) .
. 1.1.
. 3) 4),
(c1, c0) (c0, c2). , x0, (x0) = 1, c k k. c < c0 k, c = c0 c > c0 . c < c0 k, c = c0 k -, 1, c > c0 k, , .
1.2 ( ). -
f(x, c) 1) f (k)(x0, c0) = x0,
2)
f (k)
x(x0, c0) = (x0) = 1. x(c) k c c0 x(c0) = x0.
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(c) =f (k)
x(x(c), c)
3)
d
dc(c0) > 0
, ,
4)
3f (2k)
x3(x0, c0) < 0,
(c1, c0) (c0, c2) > 0 , i) c (c1, c0), f (k)(x, c) , f (2k)(x, c) (x0, x0+);ii) c (c0, c2), f (2k)(x, c) (x0 , x0+ ), f (k)(x, c).
.
f (k)(x, c) g(x, c) = f (k)(x, c) x, 3) 1.2 , - f (k)(x, c) (c1, c0) (c0, c2). 2k, -
h(x, c) = f (2k)(x, c) x. h(x0, c0) = f
(2k)(x0, c0) x0 = 0. , 2)
hx(x0, c0) =
x(f (k)(f (k)(x, c), c))
x=x0,c=c0
1 =
= f (k)x (f(k)(x0, c0), c0)f
(k)x (x0, c0) 1 = [f (k)x (x0, c0)]2 1 = 0,
hxx(x0, c0) =2
x2(f (k)(f (k)(x, c), c))
x=x0,c=c0
=
= f (k)xx (f(k)(x0, c0), c0)(f
(k)(x0, c0))2 + f (k)x (f
(k)(x0, c0), c0)f(k)xx (x0, c0) =
= f (k)xx (x0, c0)[f(k)x (x0, c0)]
2 + f (k)x (x0, c0)f(k)xx (x0, c0) = 0
4)
hxxx(x0, c0) =3
x3f (2k)(x0, c0) < 0.
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x(c) h(x, c) = 0, h(x, c)
h(x, c) = (x x(c))g(x, c).
hx(x, c) = g(x, c) + (x x(c))gx(x, c)
g(x0, c0) = 0.
,
hxx(x, c) = 2gx(x, c) + (x x(c))gxx(x, c),hxxx(x, c) = 3gxx(x, c) + (x x(c))gxxx(x, c) hxx(x, c),hxxx(x, c) (x0, c0) ,
gx(x0, c0) = 0, gxx(x0, c0) < 0.
gc(x0, c0).
hxc(x, c) = gc(x, c) dxdcgx(x, c) + (x x(c))gxc(x, c).
hxc(x0, c0) = gc(x0, c0).
hxc(x, c).
hx(x, c) = f(k)x (f
(k)(x, c), c)f (k)x (x, c) 1,
hxc(x, c) = f(k)xx (f
(k)(x, c), c)f (k)c (x, c)f(k)x (x, c) + f
(k)xc (f
(k)(x, c), c)f (k)x (x, c)+
+f (k)x (f(k)(x, c), c)f (k)xc (x, c).
hxc(x0, c0) = f(k)xx (x0, c0)f
(k)c (x0, c0)f
(k)x (x0, c0) + f
(k)xc (x0, c0)f
(k)x (x0, c0)+
+f (k)x (x0, c0)f(k)xc (x0, c0) = f (k)xx (x0, c0)f (k)c (x0, c0) 2f (k)xc (x0, c0).
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f (k)(x(c), c) x(c),
f (k)c (x(c), c) + f(k)x (x(c), c)
dx
dc=dx
dc. (1.13)
(1.13)
f (k)c (x0, c0) = 2x(c0).
, 3)
d
dc(f (k)x (x(c), c))
c=c0
= f (k)xx (x0, c0)x(c0) + f (k)xc (x0, c0) > 0.
gc(x0, c0) = hxc(x0, c0) = f (k)xx (x0, c0)f (k)c (x0, c0) 2f (k)xc (x0, c0) == 2f (k)xx (x0, c0)x(c0) 2f (k)xc (x0, c0) < 0.,
g(x0, c0) = 0, gx(x0, c0) = 0, gxx(x0, c0) < 0, gc(x0, c0) < 0. (1.14)
x x0 - c(x), c(x0) = c0
g(x, c(x)) 0. ,
gx(x, c(x)) + gc(x, c(x))dc
dx= 0. (1.15)
(1.14) (1.15) ,
c(x0) = 0.
(1.15),
gxx(x, c(x)) + 2gcx(x, c(x))dc
dx+ gc(x, c(x))
d2c
dx2+ gcc(x, c(x))
(dc
dx
)2= 0.
c(x0) = gxx(x0, c0)gc(x0, c0)
< 0.
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, c(x) x = x0 . -, c < c0 2k, c > c0 - . , 2k . 2f (2k)(x0,c0)
x2 = 0 4), df (2k)(x,c(x))
dx
x = x0. df (2k)(x,c(x))
dx
x=x0
= 1 c < c0 ,
2k . . 1.2 c(x) 2k, x(c) k.
. 1.2.
. 3) 1.2 -
, k -, 4) 1.2
, 2k . 3), 4) ,
2k. f(x)
Sf(x) =f (x)f (x)
32
(f (x)f (x)
)2.
, -
. ,
3f (2k)(x, c)
x3=3f (k)(f (k)(x, c), c)
x3= f (k)xxx(f
(k)(x, c), c)[f (k)(x, c)]3+
+3f (k)xx (f(k)(x, c), c)f (k)xx (x, c)f
(k)x (x, c) + f
(k)xxx(x, c)f
(k)x (f
(k)(x, c), c).
f (k)(x0, c0) = x0, f(k)x (x0, c0) = 1
3f (2k)
x3(x0, c0) = 2f (k)xxx(x0, c0) 3[f (k)xx (x0, c0)]2,
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, ,
3f (2k)
x3(x0, c0) = 2Sf
(k)(x0, c0),
.. 4) -
1.2.
, , -
, :
k k 2k. (. 1.3) . -
. -
, ,
1.1 1.2. -
, x = 0 .
. 1.3.
1.3. f(x, c) -,
1) f(0, c) = 0,
2)
f
x(0, c) = (c), (0) = 1
d
dc(0) > 0,
3)
2f
x2(0, 0) > 0.
f(x, c) x(c) c 0, x(0) = 0 x(c) 6= 0, c 6= 0. x = 0 , c < 0, , c > 0, c < 0 c > 0.
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f(x, c) = f(x, c), .. f(x, c) x, f(0, c) = 0, ,
2fx2 (0, 0) = 0. .
1.4. f(x, c) -,
1) f(x, c) = f(x, c),2)
f
x(0, c) = (c), (0) = 1 ddc (0) > 0,
3)
3f
x3(0, 0) < 0.
(c1, 0) (0, c2) > 0 , i) c (c1, 0), x = 0 f(x, c) (, );ii) c (0, c2), f(x, c) (, ). x = 0 , .
. 1.1 - 1.4. -
f(x, c) = c x2. f(x, c) = c x2 = x , x1,2 =
11+4c2 c > 1/4. c = 1/4 -. c = 1/4 x = 1/2. 1. 4) 1.1 , 3) . -
c < 1/4 c > 1/4. x1,2 1,2 = 2x = 1
1 + 4c. -
, x1 =1+1+4c
2 ,
1 < 11 + 4c < 1. , 1/4 < c < 3/4. c = 3/4 - 1. 1.2. 1) 2) . (c) = 11 + 4c,
d
dc
c=3/4
= (
21 + 4c
)c=3/4
= 1 < 0.
, f(x, c) = cx2 . 4) . , c > 3/4 -
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2.
f (2)(x, c) = c c2 + 2cx2 x4 x = 0,, cx2x. f(x, r) = rx(1 x), - , 1.3. -
, f(0, r) = 0 r = 1
1.3 (fx(0, r) = r, (1) = 1, (1) = 1
2fx2 (0, 1) = 2 < 0).
r > 1 x(r) = r1r . f(x, c) = cxx3 1.4. , -
c 1. c < 1 x = 0 . c > 1 x = 0, x1,2 =
c 1. x = 0 , x1,2 -.
1. , -
:
1) f(x, r) = rx(1 x), r = 3,2) f(x, c) = cex, c = e1, c = e,3) f(x, r) = xer(1x), r = 2,4) f(x, c) = cx x3, c = 1.2. F (x) = x+tf(x),
f(x) =
{(x)1/3, x 0,(x)1/2, x > 0, t > 0, 2 - t 0 t.
1.4.
, -
. -
, . -
. .
1.1. f(x) k > 1, ( 1).
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. a k f(x)(f (k)(a) = a). f(a) > a.
a, f(a), f (2)(a), . . . , f (k1)(a), f (k)(a) = a. (1.16)
b = f (i)(a), (i < k) (1.16), f(b) < b. (1.16) a. , (x) = f(x) x, , c (a < c < b), f(c) = c. f(a) < a. ,
.
1.2. f(x) , - J . I J ( -) . I f(I). p I, f(p) = p, .. p f(x).
. I = [0, 1]. i, i = 0, 1 I , f(i) = i. (x) = f(x)x (0) = f(0)0 = 00 < 0,(1) = f(1) 1 = 1 1 > 0. p I, (p) = 0, .. f(p) = p. 1.3. f : I R I. I1 f(I) Q1 I, f(Q1) = I1.. I1 = [f(p), f(q)], p, q I. p < q,
r I, f(r) = f(p). s r I, f(s) = f(q). f([r, s]) = I1, .. Q1 = [r, s]. p > q. . f : R R x1, x2, . . . , xk - k . a xi(i = 1, . . . , k). a, , - k 1 . I1, I2, . . . , Ik1 -. , f(x) 5
a < f (3)(a) < f(a) < f (2)(a) < f (4)(a),
I1 = [a, f(3)(a)], I2 = [f
(3)(a), f(a)], I3 = [f(a), f(2)(a)], I4 = [f
(2)(a), f (4)(a)]. f(I1). I1 - a f (3)(a), f(x) f(a) f (4)(a)
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. f(I1) - , f(a) f (4)(a). ,f(I1) I3 I4. f(I2) I4, f(I3) I2 I3 f(I4) I1. ,
(directed graph).
I1, I2, . . . , Ik1. si ti Ii. Ij f(si) f(sj) (f(Ii) Ij), , - Ij Ij, Ii Ij. , . 1.4.
. 1.4. 5
, k, k . -
IkIl . . . ImIk, f(Ik) Il, . . . , f(Im) Ik. , -
, . ,
I1I3I2I4I1I3I2I4I1 . 1.4 8, I1I3I3I3I3I3I2I4I1 8.
k (k > 2) 2: I1I2I1.
I1f f(I1)
I2
.
1.3 , Q1, -
I1f f(I1)
Q1
f I2.
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1.3 , f(I2) I1,
I1f f(I1)
Q1f I2
Q2
f (2) I1 f f(I1) .
, f (2)(Q2) = I1 Q2 1.1 , f(x) Q2 x Q2 2: f (2)(x) = x. x f(x), f(x) = x I2. x I1I2 ( I1 I2) x k > 2, . .
1.3. k , - f(x), l, f(x) l.
. I0I1 . . . Il = I0 -
. 1.5. 1.3
, f(Ii) Ii+1(i = 0, 1, . . . , l 1). , f(x) l. 1.2 , (. 1.5).
I0 Q1 Q2 Ql1 Ql,
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Qlf (l) Il = I0. , Ql I0 = f (l)(Ql). 1.2 , f (l)(x) x Ql. , x k < l f(x). 1.1. f(x) 3, .
.
3:a < f(a) < f (2)(a), a < f (2)(a) < f(a).
I1 = [a, f(a)], I2 = [f(a), f(2)(a)]. , f(I1) = I2,
f(I2) I1 I2. I1 = [a, f (2)(a)], I2 = [f (2)(a), f(a)], f(I1) I1I2, f(I2) = I1. 3 (. 1.6). -
. , 7 I1I2I2I2I2I2I2I1. 5 (. 1.4), , , 3.
. 1.6. 3
4. 4:
a < f(a) < f (2)(a) < f (3)(a), (1.17)
a < f(a) < f (3)(a) < f (2)(a), (1.18)
a < f (2)(a) < f (3)(a) < f(a), (1.19)
a < f (2)(a) < f(a) < f (3)(a), (1.20)
a < f (3)(a) < f (2)(a) < f(a), (1.21)
a < f (3)(a) < f(a) < f (2)(a). (1.22)
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(1.17)
I1 = [a, f(a)], I2 = [f(a), f(2)(a)], I3 = [f
(2)(a), f (3)(a)]
, ,
f(I1) I2, f(I2) I3, f(I3) I1 I2 I3. I3 - I2I3I3I2 3. 1.1 4 (1.17) -. ,
(1.18), (1.21), (1.22) 4, (1.18), (1.19) 4 2. , .
1.4. f : R R k > 1, , k 1. ,
.
1.4. k- - k, -.
k, 1.4, , .
, ,
, .
1.5. f(x) k > 1, 2.
. k = 2. - , k 3. 1.4 k- f(x) k, - . , , -
1. .
, -
, .
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. .. ,
()
. ,
.
1.5. -
:
3C 5C 7C C 2 3C 2 5C 2 7C C 2n 3C 2n 5C 2n 7C . . .C2n C C 23 C 22 C 2C 1. f : R R , n, f(x) m m N , nCm. 1.1 1.1, ,
1.5.
f(x), I. [n] - , f(x) n. 1.5 .
1.6.
[1] [2] [4] [5 2] [3 2] . . . [5] [3].
1. 5- .
2. 8,
.
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1.5.
.
, -
1. -,
. , -
,
,
.
. -
f(x) x
Sf(x) =f (x)f (x)
32
(f (x)f (x)
)2.
f(x) = ax2 + bx + c, f (x) 0, Sf(x) < 0 x (Sf(x) = x = b2a). -,
. , S(ex) = 1/2, S(sinx) = 1 32(tan x)2. ,
.
1.7. f(x) . f (x) - , Sf(x) < 0.
. ,
f (x) 1.
f (x) =Nk=1
(x ak),
ak .
f (x) =Nj=1
f (x)x aj =
Nj=1
Nk=1(x ak)x aj ,
f (x) =Nj=1
Nk=1,k 6=j
Ni=1(x ai)
(x aj)(x ak) .
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,
Sf(x) =j 6=k
1
(x aj)(x ak) 3
2
(Nj=1
1
(x aj)
)2=
= 12
(Nj=1
1
(x aj)
)2
Nj=1
(1
(x aj))2
< 0.
1. :
) f(x) = x3 + 1,) f(x) = x3 + 10x,) f(x) = ln x.
2. ,
.
3. , f(x)
, f(x) =ax+ b
cx+ d, a, b, c, d .
4. h(x) =ax+ b
cx+ d, a, b, c, d g = h f, Sg(x) = Sf(x).
.
1.8. Sg(x) < 0 Sf(x) < 0, S(f g) < 0..
(f g)(x) = f (g(x))g(x),(f g)(x) = f (g(x))g2(x) + f (g(x))g(x),
(f g)(x) = f (g(x))g3(x) + 3f (g(x))g(x)g(x) + f (g(x))g(x). ,
S(f g)(x) = Sf(g(x))g2(x) + Sg(x). , S(f g)(x) < 0. 1.8 .
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1.2. Sf(x) < 0, Sf (n)(x) < 0 n > 1.
, ,
.
.
.
1.9. x0 f(x) , f (x0) = 0.
1.9. Sf(x) < 0 (Sf(x) = ). f(x) n . f(x) n + 2 .
1.9 .
1.6. Sf(x) < 0, f (x) - , -
.
., x0 f(x),.. f (x0) = 0. Sf(x0) < 0,
f (x0)f (x0)
< 0.
, f (x0) f (x0) . . , -
f (x) - x. , f (x) f(x).
1.7. f(x) , f (m)(x)(m > 1) .
. c f1(c) . , f(x1) = c f(x2) = c, p, x1 < p < x2 f (p) = 0. f (2)(x)
[f (2)(x)] = f (f(x))f (x) = 0.
, f (2)(x) - xc f(x) f
1(xc) . - f1(xc) ,
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f (2)(x) . -. g(x) = f (m1)(x) , , ,
, , f (m)(x) .
1.8. Sf(x) < 0 f(x) - , f(x) m m.
. g(x) = f (m)(x). 1.2 Sg(x) < 0., g(x) , .. g(x) = x . - g(x) = 1 . g(x) = 1 x, Sg(x) = 0, - Sg(x) < 0. x1 < x2 < x3 , g(x) = 1. g(x) (x1, x2) (x2, x3) g
(x) > 1, x2 - ,
1.6. , (x1, x3) x0 , g(x0) < 1. g(x) , , g(x) = 0. , g(x) - , .
1.9.
p m f(x). W (p) , p, - p f (m)(x), .. W (p) () {x|f (mj)(x) p j }, p. -, W (p) f (m)(W (p)) W (p). W (p) = (r, s). , p . f(W (p)) W (p) (r, s) , f(x) (r, s), r s . , , :
1. f(r) = r f(s) = s.2. f(r) = s f(s) = r.3. f(r) = f(s). . , r < p < s.
f(r) = r, f(p) = p , u(r < u < p), f (u) = 1. , v(p < v < s), f (v) = 1. f (p) < 1 f (x)
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, f(x) (r, s). , f(x) f (2)(x). f(x) (r, s), (r, s). r s , . .
p , - W (p) g(x) = f (m)(x).
f(x).
1. ,
-
. , ,
, , . f(x) = arctan x > 1. Sf(x) = 2/(1 + x2)2 < 0. f(x) , -
, f(x) = arctan x .
. 1.7. f(x) = arctan x, > 1
2. 1.9 ,
1. - .
, f(x) Sf(x) < 0. f(x) c f (c)= 1 f (c)=1. c - W (c). , f (c) = 1 ( f (2)(x)). 1.8 f(x) . , , f(x)
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, c. , c - , .. x < c c f(x) < x, y > c c f(y) > y. f (x) 1. 1.6 , f(x) > x a < x < c, f(x) < x c < x < b. - , c , , ., W (c) , , 1.9, W (c) .
m > 1. g(x) == f (m)(x). , cW (c)c. f(x) = ex1. f(x) - x = 1. . , f(x) = ex1 .
3. , f(x) x R. f : I I, I = [a, b] . 1.6 .
1.9 ( ). I = [a, b] f(x) I. Sf(x) < 0 (a, b), |f (x)| > min{|f (a)|, |f (b)|} x (a, b).. |f (x)| I, x0 I. x0 (a, b), f (x0) 6= 0, Sf(x) < 0 (a, b). f (x0) > 0, f (x) (a, b), 1.6. f (x0) < 0, f (x) (a, b), 1.6. , x0 = a x0 = b.
-
1.9 .
1.10. f : I I - Sf(x) < 0, W (p) ( p)
f(x), I.
.
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1.10. f(x), I= [0, 1] I, , :1. f(0) = f(1) = 0.2. f(x) I c, - I. f(x) .
, -
[0, c] [c, 1]. f(x, r) = rx(1 x) 0 < r 4 -. -
g(x, p) =
{px, 0 x 12 ,
p px, 12 x 1, p, 0 < p 2. -
h(x, l, p) =
{lx, 0 x 1p ,
lp1 lp1x, 1p x 1,
l/p 1, p > 1. 1.10 -
.
1.3. f : I I , , Sf(x) < 0 I \ {c}. x = 0 I -. f(x) .
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2.
2.1.
xn+1 = rxn(1 xn), (2.1)
r,
xn+1 = fxn x2n. (2.2)
(2.2) (2.1),
xn =f
zn.
(2.2) ,
zn+1 = fzn(1 zn).
, f(x, r) = rx(1 x) [0, 1] :
0 rx(1 x) 1 (2.3)( 0 x 1). , (2.3) r > 0, , r2 4r 0.
rx2n rxn + 1 0, x [0, 1].45
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, 0 < r 4 f(x, r) [0, 1] . , -
[0, 1] Hx = fx x2,
0 1. x < 0, f (n)(x, r) n ., x > 1, f (n)(x, r) n.. x < 0, f(x, r) = rx(1 x) < x. ,
f (n)(x, r) . p. f (n+1)(x, r)f(p, r) < p, f (n)(x, r) p. , f (n)(x, r) . x > 1, f(x, r) < 0, f (n)(x, r) , .
2.2. f(x, r) - .
. , Sf(x, r) < 0 |x| , |f (n)(x, r)| n . , - .
1.9.
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f(x, r) x0=0. , fx(0, r)= r, 0 < r < 1. , fx(0, 1) = 1. r > 1 x0 x1 = 1 1/r. (r) = fx(1 1/r, r) = r+ 2, x1
1 < r + 2 < 1,..
1 < r < 3. (2.4)
(2.4) x1. 1 < r < 2 O(x0), x0 (0, 1) x1 , (r) > 0. 2 < r < 3 O(x0), x1, x1, x1. r = 3 fx(1 1/3, 3) = 1. r > 3 x1 2, 1.2. . ,
f (2)(x, r) = r3x4 + 2r3x3 r2(1 + r)x2 + r2x. f (2)(x, r) = x:
r3x4 + 2r3x3 r2(1 + r)x2 + r2x = x. (2.5) (2.5) : x0 x1. (2.5) -
x
(x r 1
r
)(r3x2 r2(1 + r)x+ r(1 + r)) = 0,
r3x
(x r 1
r
)(x 1 + r +
(r + 1)(r 3)2r
)
(x 1 + r
(r + 1)(r 3)2r
)= 0.
, 2:
x =1 + r +
(r + 1)(r 3)2r
,
= 1. , x+, x 2 f(x, r). ,
= fx(x+, r) fx(x, r) = r2 + 2r + 4 = f2(r).
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,
1 < f2(r) < 1.
, 2 :
3 < r < 1 +6.
r x+, x . - x0 (0, 1) \ {f (n)(x1)(n = 0, 1, . . . )}, O(x0) (x+, x), f (2n)(x0), (n = 0, 1, . . . ) , f (2n+1)(x0), (n = 0, 1, . . . ) -. , (0, 1), , (.. x1 -), . r = 1+
6 1. r, x = 1/2 f(x, r). r . -, r = 1+
5 x = 1/2. f2(r) : f2(3) = 1 r = 3 ( - 2), f2(1+
5) = 0 2 , f2(1+
6) = 1 2 4 1.2.
, H(x) = fx x2 2
x(n,2) =f + 1 + (n, 2)
(f + 1)(f 3)
2, (2.6)
(n, 2) = cos(n 1)pi, n = 1, 2. 4 f(x, r), H(x). , x(n,2) -
H(2)(x):
H(2)(x(n,2)) = x(n,2).
2 H(2)(x) 4 - H(x). x = x(n,2) + y, x 4 H(x).
H(2)(x) = H(2)(x(n,2) + y) = H(2)(x(n,2)) +H
(2)x (x(n,2))y+
+1
2H(2)xx (x(n,2))y
2 +O(y3).
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H(2)x (x) = Hx(H(x))Hx(x), H(2)xx (x) = Hxx(H(x))(Hx(x))
2 +Hx(H(x))Hxx(x),
H(2)x (x(n,2)) = Hx(x(1,2)) Hx(x(2,2))
H(2)xx (x(n,2)) = (Hx(H(x(n,2))))x Hx(x(n,2)) = Hxx(H(x(n,2)))[Hx(x(n,2))]2 +Hx(H(x(n,2))) Hxx((x(n,2))). x(n,2) (2.6),
H(2)(x(n,2) + y) = x(n,2) + fy y2 +O(y3),
f = f 2 + 2f + 4, = [(f + 1)(f 3) + 3(n, 2)
(f + 1)(f 3)]. f , , (n, 2) (+1 1).
H(2)(y) = fy y2 (2.7) , H(x).
2 H(2)(y):
y =f + 1 +
(f + 1)(f 3)2
, = 1. (2.8)
4 H(x) ( O(y3))
x, = x + y,
.. x1,1, x1,1, x1,1, x1,1. f = = r, .. - f(x, r), 4 f(x, r):
a(4)n =r + 1 + (n, 2)
(r + 1)(r 3)
2r+
+f2(r) + 1 + (n, 4)
(f2 + 1)(f2 3)
2(n, 2), n = 1, 2, 3, 4,
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f2 = r2 + 2r + 4, (n, 2) = r[(r + 1)(r 3) + 3(n, 2)
(r + 1)(r 3)
],
(n, 2) = cos(n 1)pi, n = 1, 2, 3, 4
(n, 4) =
{+1, n = 1, 2,1, n = 3, 4. , 8. - y = y+ y (2.7) , y = H(4)(y). , H(4)(y+ y) y. -, (2.8). ,
p = 2k:
a(p)n =k
=1
(fm + 1) + [(fm + 1)(fm 3)]1/2(n, 2m)2(n,m)
(2.9)
[m = 21, p = 2k
],
fm (n,m) r,
f2m = f 2m + 2fm + 4 (f1 = r), (2.10)
(n, 2m) = (n,m){(fm + 1)(fm 3) + 3[(fm + 1)(fm 3)]1/2(n, 2m)
}(2.11)
((n, 1) = r),
(n, 2m) +1,1 , . (2.9) - (2.11) -
p . rp < r < r2p p. p/2 p p = 2k (2.9):
a(p)n = a(p/2)n +
(fp/2 + 1) + [(fp/2 + 1)(fp/2 3)]1/2(n, p)2(n, p/2)
.
0 < r < rp r = rp, fp/2(rp) = 1. m = p/2 fm, fp(rp) = 1. rp < r < r2p r = rp, p . y ,
. , fp(rp) = 0. , :
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fp/2(rp) = 1, p/2 ,fp(rp) = 1, p ,fp(r
p) = 0, p ,
fp(r2p) = 1, p ,f2p(r2p) = 1, 2p .
,
fp(rp) = f2p(r2p) = 1,
.
:
r sp s2p , fp(sp) p - f2p(s2p) 2p
fp(sp) = f2p(s2p) = (), p = 2, 4, . . .
rp r2p. - f2(r2) = 1 , f1 = r2, (2.10) r2 = 3.,
f4(r4) = f 22 (r4) + 2f2(r4) + 4 = f2(r2) = r22 + 2r2 + 4 = 1.
f 22 (r4) 2f2(r4) + 2r2 r22 = 0
f2(r4) = r24 + 2r4 + 4 = 2 r2.
r4 = 1 +3 + r2.
1 = f4(r8) = f 22 (r8) + 2f2(r8) + 4 = f2(r4) = r24 + 2r4 + 4,f 22 (r8) 2f2(r8) + 2r4 r24 = 0.
f2(r8) = 1 +1 2r4 + r24 = 2 r4, r28 + 2r8 + 4 = 2 r4.
, ,
r8 = 1 +3 + r4.
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,
f2(r2p) = r22p + 2r2p + 4 = 2 rp.
r2p = 1 +3 + rp. (2.12)
, r2 = 3, r4 = 1 +6 3.4495. r4. :
r8 = 1 +4 +
6 3.5396, r16 = 1 +
4 +
4 +
6 3.5573,
r32 = 1 +
4 +
4 +
4 +
6 3.5607.
(
, y) :
r8 = 3.5441, r16 = 3.5644, r32 = 3.5688.
, 2 3 < r < 3.4495, 4 3.4495 < r < 3.5396, 8 3.5396 < r < 3.5573, 16 3.5573 < r < 3.5607. . , -
(2.12), . ,
. p (2.12), r = 1 +
3 + r,
r =3 +
17
2 3.5615. r = 3.569. r > r .
limp
r2p rpr4p r2p = limp
r2p rp1 +
3 + r2p r2p
= limp
r2p rp3 + r2p
3 + rp
=
= limp
(r2p rp)(3 + r2p +
3 + rp)
r2p rp = limp(3 + r2p +
3 + rp) =
= 23 + r = 2(r 1) = 2
(3 +
17
2 1)= 1 +
17 5.1231.
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, ,
= 4.6692. -. f(x)= 1+
3+x, r2p= f(rp),
r, f (r) =1
23 + r
=1
.
1 1
|r2np r| Cn, n,
C . , -
, ,
, , -
. .
p r = rp(rp < rp < r2p), p. a = 1/2. ap p, - a. , rp
r2p = 1 +3 + rp (r
1 = 2),
r2 = 1 +5 = 3.23606 . . . .
= limp
[ap aa2p a
]= {(r + 1)(r 3) 3
(r + 1)(r 3)}
r
=
= 2.2399 . . . (2.50290 . . . ),
.
2.2. 3
xn+1 = rxn(1 xn) r < r . -. r < r,
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2m ( m r). - (0, 1), , 2i, i = 0, 1, . . . ,m 1. r > r . r = 3.57 , . -, x0 (0, 1) , - x0. , [0, 1] ( A [0, 1], (, ) [0, 1] , A). , - [0, 1] , (. 2). -
x0 = 1/2 f(x, r).
. 2.2. f(x, r) r = 3.832
r = 3.832. r 3 ,, ,
1/2. ,
f (3)(x, r) = r3x(1 x)[r4x3(1 x)3 + 2r3x2(1 x)2 r(1 + r)x(1 x) + 1].
x = 1/2
f (3)(1/2, r) =r3
256(4 r)(r3 4r2 + 16) = 1
2,
r = r = 3.832. , r 3 c. , - (0, 1). 3 x = 0.15, y = 0.5, z = 0.95.
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[0, x] = , [x, y] = , [y, z] = , [z, 1] = . f(x, r)(. 2.2)
f(, r) = + ,f(, r) = ,f(, r) = + ,f(, r) = .
, , , . . 2.3.
. 2.3.
, -
f(x, r), . f(x, r) {, , , }
1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0
T =
1 1 0 00 0 1 00 1 1 01 0 0 0
,
tij =
{1, i j,0, .
-
k. , + , .. , , , + . , x1 = 1 1/r f(x, r) > x x < x1. +, x = 0.
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, x , f(x, r) > x f (n)(x, r) 3, n . x , x - + , , . , f(x, r) - . ,
0 1 1 1
A =
(0 11 1
).
A , f(x, r). ,
A2 =
(1 11 2
),
f (2)(, r) , f (2)(, r) , . , f (2)(, r), , - f (2)(x, r) , 2. x1 2. f
(2)(x, r) , , . , f (2)(x, r) + 3 A2 (, ).
, A2 2, . A3
A3 =
(1 22 3
).
3, :
,
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3, :
, , . A3 4. x1 - 3. f (k)(x, r) k - , k, . A4 7. x1, - 2
4. Nk f (k)(x, r),
Nk = tr (Ak) (2.13)
(tr (A) A). - k, Nk -, k, k. , tr (A6) = 18. , 2 ( ) 3 ( ). , 6. , k- f(x, r) , - ,
.
. -
f (k)(x, r) . , , , . , (2.13). Ak, A:
det |A E| = (1 ) 1 = 0, 1,2 = 15
2.
Nk = tr (Ak) =
(1 +
5
2
)k+
(15
2
)k.
, f(x, r), 0,
Nk + 1 = 1 +
(1 +
5
2
)k+
(15
2
)k= 1 + Fk+1 + Fk1,
Fk
F0 = 0, F1 = 1, Fk = Fk1 + Fk2, k = 2, 3, . . .
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Nk + 1 = 1 + Fk+1 + Fk1
Fk =15
(1 +52
)k(15
2
)k (
1 +5
2
)k= Fk
1 +5
2+ Fk1,
(15
2
)k= Fk
152
+ Fk1.
k = 5, N5 + 1 = 12, .., , -
5. k = 8 N8 = F9 + F7 = 47. , 2,
4 8.
, , + , , . , , ,
:
. . . .
2.3. f (x) = 4x(1 x)
xn+1 = 4xn(1 xn), (2.14).. r = 4. (2.14).
xn+1 = rxn(1 xn) (2.15) r = 2.
xn =1
2(1 yn).
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f(x) = 4x(1 x) 59
(2.15)
yn+1 = y2n
(yn+1 =
r
2y2n + 1
r
2
).
yn = y2n0 .
xn =1
2[1 (1 2x0)2n]. r = 4
yn = cos pizn
yn+1 =r
2y2n + 1
r
2(r = 4 : yn+1 = 2y
2n 1).
cos pizn+1 = 2 cos2 pizn 1 = cos 2pizn = 2 cos2 2pizn1 1 =
= cos 22pizn1.
,
cos pizn = cos 2npiz0.
zn : 0 zn 1. cos pizn+1 = cos 2pizn
cos u = cos v
u = v + 2kpi (2.16)
zn+1 =
{2zn, 0 zn 12 ,
2(1 zn), 12 zn 1(2.17)
z0 =1
piarccos y0,
0 arccos y pi arccos y, (2.14):
xn =1
2(1 cos pizn) = 1
2(1 cos 2npiz0) = 1
2[1 cos{2n arccos(1 2x0)}],
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:
xn = sin2(2n arcsin
x0).
(2.17)
zn =1
piarccos cos(2npiz0).
, (2.14) (2.17), -
.
f(x) = 4x(1 x),
g(x) =
{2x, 0 x 12 ,
2(1 x), 12 < x 1,
h1(g(h(x))) = f(x),
h(x) =2
piarcsin
x.
, 0 x 12
0 h1(x) 12
h1(g(h(x))) = sin2(2 arcsinx) = (2
x(1 x))2 = f(x).
12 x 1,
1
2 h1(x) 1
h1(g(h(x))) = sin2[2(pi2 arcsinx
)]= sin2(2 arcsin
x) = f(x).
(2.17) (2.14).
(2.14)
f(x) = 4x(1 x), . , -
g(x) (|g(x)| = 2 x 6= 1/2). , p
cos(2ppizp) = cos pizp.
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f(x) = 4x(1 x) 61
, q < p
cos(2qpizp) 6= cos pizp.
, (2.16)
zp =2k
2p 1 , (2.18)
k = 0, 1, 2, . . . , 2p1 12 12,
zp 6= 2k
2q 1 ,
k = 0, 1, 2, . . . , 2q1 12 12 q < p. p. - (p = 1)
z1 = 0 2
3.
(2.14) x0 = 0 x1 = 3/4. 2:
z2 =2
5
4
5.
2 (2.14):
x2 =(55)
8
(5 +5)
8.
3
z3 =2k
8 1 , k = 1, 2, 3, 41
2 12,
3:
z3 =2
7,4
7,6
7 z3 =
2
9,4
9,8
9.
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p = 4
z4 =2
15,4
15,8
15,14
15; z4 =
2
17,4
17,8
17,16
17; z4 =
6
17,12
17,10
17,14
17.
, 5 .. ,, p, - (2.18) ,
.
p (2.18) () z(p)+(z(p)). z(p)+
z(p)+ =2k
2p + 1=
2k
22p 1 ,
k = k(2p 1)., z(p)+ z(2p), .. z(p). , (2.17), -
.
cos(2jpiz0) = cos(pizp), (2.19)
z0, j - p, , (2.19) z0, (2.18) , zp - p. , - , .
, z0 , - l/m, l < m l,m . , -
z0 =2k
2p 1 ,
k = 0, 1, 2, . . . , 2p1 1.
1: l /m . .
r m .
r(m) = 1 (mod m), (r(m) 1 m),
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f(x) = 4x(1 x) 63
(m) , 1, 2, . . . ,m1, m (m , (m) == m 1 (1) = 1). 1 z0 z0 = 2l
/m. m , r = 2. ,
2(m)1m . , z0
z0 =2k
2(m) 1 ,.. 1
(m) (, , q < (m), (m)) (2.17).
2: l /m . z1, ,
2z0 = 2l/m l/m 1/2, 2(1 z0) = 2l/m 1, l = m l 1/2 l/m 1. , 2 .
3: l /m . m = 2sm, s , m . l
arccos cos(2spiz0) =l
mpi (0 l
m 1).
, s 3 - 1, 2. ,
3 .
,
(2.17). -
,
(2.17) [0, 1] ( [0, 1], ,.. , [0, 1]). , [0, 1], l/m, l , m . ,
(2.14). .
2.3. (2.14) -
[0, 1].
, [0,1] -
.
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(2.17). z [0, 1] :
z =a12+a222
+ + ak2k
+ = .a1a2 . . . ak . . . ,
ak 0, 1. z < 1/2, g(z) 2z, ..
g(z) = g(.a1a2 . . . ) = .a2a3 . . . .
z > 1/2, g(z) 1 2z -
g(z) = g(.a1a1 . . . ) = .a2a3 . . . ,
ak = 1 ak k. g(z) :
g(.a1a2 . . . ) =
{.a2a3 . . . a1 = 0,.a2a
3 . . . a1 = 1,
.. g(z) a1a2a3 . . . - , ( a1 = 0) 1 ai ( a1 = 1). , :
g(.0111 . . . ) = .111 . . . , g(.111 . . . ) = .000 . . . .
:
z0 = .000 . . . , z1 =2
3= .101010 . . . .
z = 1/3 = .010101 . . . z1 = 2/3. . , g(n)(z) z z1 = 2/3, - , z = .a1a2 . . . 010101 . . . . g(n)(z) z
, z 000 . . . , .. z .
z = .100100 . . . g(z) = .110110 . . . , g(2)(z) = .010010 . . . , g(3)(z) = .100100 . . . . z, , . ,
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f(x) = 4x(1 x) 65
g(n)(z) z - , z - p.
p 2, - z = 0, 2/3. z = .011011 . . . , O(z) z .
g(z) = .110110 . . . , g(2)(z) = .01001001 . . . , g(3)(z) = .1001001 . . . ,g(4)(z) = .110110 = g(z). , g(n)(z) n .
z = .11001100 . . . , g(z) = .01100110 . . . , g(2)(z) = .11001100 = z. , g(n)(z) n . , g(z) ()
( ). -
1 ak, ak1 6= ak. p , ak+p = ak k. ak, ak+p, ak 1 . ak 1 , ak+p. :
O(.101101 . . . ) = {.100100 . . . , .110110 . . . , .010010 . . . }.,
. -
p , g(n)(z) p. , a1 ak+p, , , , -
1. ak = ak+p, ak ak+p, , . - , ak ak+p g(n)(z) (.. - ) p. ,
g(n)(z) n p. . g(n)(z) , - z .
.
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2.4. z p, g(n)(z) n p. , g(n)(z) , z .
O(z), -, . -
. O(z), z = .1010010001 . . . . , O(z) 1, 1/2, . . . , 1/2n, . . . , 0. , O(z) - [0, 1].
2.5. z [0, 1] z z, O(z) [0, 1].
. s0 , - z , s0 z, . S :
s1 = .0, s2 = .1, s3 = .00, s4 = .01, s5 = .10, s6 = .11, . . . ,
n 1 , s2n2. z . - z s0. z sn S , s
n
. ,
n , L z. - z sn, L . z sn, , , ( 1) sn. , , z. z z0, . , z0 [0, 1] - , sN S M , 1/2M < /2 |sN z0| < /2. , s O(z) sN . ,
|s z0| |s sN |+ |sN z0| < 2+
2= .
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f(x) = 4x(1 x) 67
, z0 O(z) O(z) [0, 1].
2.5 , -
z, , z O(z). , - [0, 1] , 0 2/3. .
2.6. [0, 1]:1) z , g(n)(z) 0 2/3;2) z , g(n)(z) p;3) z , O(z) [0, 1].
, 2.6 f(x) ( 0 3/4). (2.17) , .
, .
2.1. f : J J , - > 0 , x, y J n, |f (n)(x) f (n)(y) > . , [0, 1] J [0, 1] n > 0, g(n)(J) = [0, 1]. , 1/2 / J , l(g(J)) = 2l(J), l() ; 1/2 J , > 0, [0, ] g(2)(J), 1 g(J). g(m+2)([0, ]) = [0, 2m] 2m < 1 g(m+2)([0, ]) = [0, 1] 2m > 1. ,
(2.17). ,
(2.14).
2.7. [0, 1] J [0, 1] - m , f (m)(J) = [0, 1].
(2.14) (2.17), , [0, 1] - .
2.2. f : J J , > 0 ,
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x J N x y N n 0, |f (n)(x) f (n)(y)| > . ,
x, x, - , f . , , - x, x . , , x. - .
, , -
.
.
-
, x . (2.17) ,
- 1/2.
2.3. f : J J , U, V J k > 0, f (k)(U) V .
, , -
.
.
,
.
2.4. V . f :V V V , :1) f ,2) f ,3) f V .
(2.14) (2.17),
.
2.8. (2.14) (2.17) [0, 1].
. -
.
O(x0),
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f(x, r) = rx(1 x) r > 4 69
, n = 100, ,
. (2.17)
m log 1log 2 [0, 1]. = 1020, , [0, 1] m > 20 log2 10( 70), - . (2.14) (2.17) (x0), n : -
J [0, 1]? , J = (, ), (2.17) , (2.14)
1
pi
dxx(1 x) =
2
pi(arcsin
arcsin) = h() h(),
h(x) , f(x) g(x).
2.4. f (x, r) = rx(1 x) r > 4
f(x, r) = rx(1 x) r > 4. - . 2.4.
. 2.4. f(x, r) r > 4
f(x,r) r/4, x = 1/2 - f(x, r) I = [0, 1]., , x = 1/2
I . , r = 4.002, [0.49,0.51] I. A0 I, . ,
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A0 x = 1/2. x A0, f(x, r) > 1 f (2)(x, r) < 0. f (n)(x, r) n. I, A0, I f(x, r). A1 = {x I : f(x, r) A0}. x A1, f (2)(x, r) > 1,
f (3)(x, r) < 0 , , , f (n)(x, r) n. - An = {x An : f (n)(x, r) A0}. ,An = {x I : f (i)(x, r) I i n, f (n+1)(x, r) / I}. ,An , I (n+1)- . x An, O(x) . , , An. - I, . r. ,
r = I \( n=0
An
)=
n=1
f (n)(x, r).
, .
r. A0 1/2, I1 = I \ A0 : B0 = [0, q0] B1 = [q1, 1], q0 < q1 f(q0, r) = f(q1, r) = 1. , f
(2)(q0, r) = f(2)(q1, r) = 0. - , f(x, r) B1 B2 I. f(x, r) B1 B2. f(B0, r) = f(B1, r) = I, ( B0, B1), f(x, r) A0. , A1. f (2)(x, r) x = 1.
. 2.5. f (2)(x, r) r > 4
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f(x, r) = rx(1 x) r > 4 71
I2 = I \ (A0 A1). - [0, q00], [q01, q0], [q1, q10], [q11, 1], f(x, r) B1, B2. -, f (2)(x, r) I. , I2 = I \ (A0 A1) , f (2)(x, r) A0. - I . , A2. , f
(2)(x, r) . , f (2)(x, r) . An, . -, An 2n . , In+1 == I \ (A0 An) 2n+1 ,
1 + 2 + + 2n = 2n+1 1.-, f (n+1)(x, r) - I. f (n+1)(x, r) . f (n+1)(x, r) 2n I., In = f
(1)(In1, r) = f (n)(I, r) In In1 r
r =n=0
In (I0 = I). (2.20)
, -
, .. .
2.1. r .
. r (2.20) ( - ). , . -
, n In ( An) r. , , 0. I - f(x, r). x r In,jx n In, x. A |A| -. |In,jx| 0, n , In,jx x n , , x
r \ x. |In,jx| 0 n, n=0
In,jx
x n=0
In,jx r. , x r \ x. , r .
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B1 B2 q0 =rr24r
2r , q1 =r+r24r2r -
. f (q0, r) =r2 4r. r > 2+5
f (q0, r) > 1. r > 2+5 f (q1, r) < 1. f (x, r) (f (x, r) < 0), |f (x, r)| > 1 x I1., |f (x, r)| > 1 x r, r > 2 +
5.
2.5. f : R R . ( ) f (.. f() = ). f , n0 1 , |(f (n0))(x)| > 1 x .
, r - f(x, r) r > 2 +
5 n0 = 1. ,
, .. -
, . -
. ,
r . r r > 2 +
5 .
2.2. r r > 2 +5 .
., r . [a, b]r. n 1 f (n)(x, r). , [a, b],
f (n)(b, r) f (n)(a, r) = f (n)(cn, r)(b a),
cn [a, b]. f(q0, r) =
r2 4r = . , |f (x, r)| > x I1. r > 2 +
5, > 1. cn [a, b] r, f (i)(cn, r) r I1 0 i n 1. |f (n)(cn, r)| n
|f (n)(b, r) f (n)(a, r)| = |f (n)(cn, r)||b a| n|b a|.
, n |f (n)(b, r) f (n)(a, r)| > 1, > 1. r ,, {f (n)(a, r), |f (n)(b, r)} r [0, 1] n. . r . , .
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f(x, r) = rx(1 x) r > 4 73
2.4.1.
f(x, r) - r. - f(x) = 4x(1 x) , f(x) - - (2.17). -
,
f(x, r) r. ,
0 1.
2.6. 2 = {s = (s0s1s2 . . . )|sj = 0 1}.
2 0 1. - 2 . - s = (s0s1s2 . . . ) t = (t0t1t2 . . . )
d[s, t] =t=0
|si ti|2i
.
|si ti| 1, d[s, t] i=0
12i = 2, ..
d[s, t] . , s = (111 . . . ), t = (1010 . . . ),
d[s, t] =i=1
1
22i1=
2
3.
2. d[s, t] 2.
. , d[s, t] 0, d[s, t] = d[t, s] d[s, t] , si = ti i. |ri ti| |risi|++|si ti| , d[r, t] d[r, s] + d[s, t].
, 2 , , , .
, s, t 2 si = ti i = 0, 1, . . . , n,
d[s, t] =ni=0
|si ti|2i
+
i=n+1
|si ti|2i
i=n+1
1
2i=
1
2n.
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, d[s, t] < 1/2n, si = ti i n, sj 6= tj j n
d[s, t] 12j 1
2n.
, -
. 2, . -
.
2. s, t 2 si = ti i = 0, 1, . . . , n. d[s, t] 1/2n. , d[s, t] 1/2n, si = ti i n.
2.
2.7. 2
(s0s1s2 . . . ) = (s1s2s3 . . . ).
. ,
, s0 0 1. , .
3. : 2 2 .
. > 0 s = (s0, s1s2 . . . ) 2. n , 1/2n < . = 1/2n+1. t = (t0t1t2 . . . ) 2 , d[s, t] < , 2 - si = ti i n+ 1. , d[(s), (t)] 1/2n < .
. (1.17) . -
.
.
. (0, 0, 0 . . . ) (1, 1, 1, . . . ). 2 -, ,
(010101 . . . ). n
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f(x, r) = rx(1 x) r > 4 75
(s0s1 . . . sn1s0s1 . . . sn1 . . . ). 2n - n. 2n - 0 1 n. -, 0 1 0 1., 2. - s = (s0s1s2 . . . ) t, d[s, t] 1/2n., 2 , .. , 2. -. s :
s0 = 0, s1 = 1, s2 = 00, s3 = 01, s4 = 10, s5 = 11, . . . ,
0 1 3 , 4 .. , , s, ,
. , -
,
t 2. -
t. , .
2.9. 2n n. 2. 2 .
1. s = (001001001 . . . ), t = (010101 . . . ), r = (101010 . . . ). d[s, t], d[t, r], d[s, r].2. 2, 3 4 .3. 2 2, - : sj = 0, sj+1 = 1, ..
2 ,
. ,
a) 2 2 2;b) 2;c) 2 .
4. s 2. W s(s) s : W s(s) t, d[i(s), i(t)] 0, i. , W s(s).
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2.4.2. 2 f (x, r)
r > 2 +5
f(x, r), ., r B0 = [0, q0],
q0 < 1/2 B1 = [q1, 1], q1 > 1/2. x r, , x.
2.8. x S(x) = s0s1s2 . . . , sj = 0, f
(j)(x, r) B0; sj = 1, f (j)(x, r) B1., x r 0 1. S(x) r 2.
2.10. S(x) .
. , S(x) -. a, b r a 6= b. , S(a) = S(b). n f (n)(a, r) f (n)(b, r) 1/2. - , f(x, r) f (n)(a, r) f (n)(b, r). - B1 B2 n. , r .
, S(x) r 2. s = s0s1s2 . . . . x r, S(x) = s. . J I, J , f1(J, r) = {x I : f(x, r) J} , B0, B1. Bs0s1...sn
Bs0s1...sn = {x I : x Bs0, f(x, r) Bs1, . . . , f (n)(x, r) Bsn} =
= Bs0 f1(Bs1, r) f (n)(Bsn, r)., Bs0s1...sn - n. ,
Bs0s1...sn = Bs0 f1(Bs1...sn). , Bs1...sn . -, f1(Bs1...sn) ,
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f(x, r) = rx(1 x) r > 4 77
B0, B1. Bs0 f1(Bs1...sn) -. ,
Bs0s1...sn = Bs0...sn1 f (n)(Bsn, r) Bs0...sn1. ,
n0Bs0s1...sn
. x n0
Bs0s1...sn, x Bs0, f(x, r) Bs1 .. ,S(x) = (s0s1 . . . ). , S(x) r 2.,
n0
Bs0s1...sn . -
S(x). , , diamBs0s1...sn 0 n. , S(x) . x r , S(x) = s0s1s2 . . . . > 0 n , 1/2n < . Bt0t1...tn, , t0t1 . . . tn. , r. 2
n+1
. < 1/2n+1 , |x y| < y r y Bs0s1...sn. C, S(x) S(y) n+1. 2
d[S(x), S(y)] 2 +
5 r . ,
. n 2n , 2, 2 . -, .
2.12. f(x, r) r > 2 +5 - .
. , f B, > 0, x B U(x) x - y U(x) n 0 , |f (n)(x) f (n)(y)| > . , A0, .. - [0,1], I . x, y r. x 6= y, S(x) 6= S(y). x y , , , n-. - f (n)(x, r) f (n)(y, r) A0. ,|f (n)(x, r) f (n)(y, r)| > . , , f(x, r) r > 2 +
5 r :1) r;2) r ;3) f(x, r) - .
1.4 .
2.13. f(x, r) = rx(1 x) r > 2 +5 r.
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1.
f(x) (0, X) - 0 < f(x) < x . xn+1 =f(xn) (n = 0, 1, 2, . . . , x0 (0, X)) , (0, X). - (0, X). , = 0, f(x), , f(x). , xn . - n ? .
1 [5]. f(x) = x + O((x)), 0 < < 1, (x) > 0, x1(x)
0
x2(x)dx
-
80
(., , [9]), -
,
k=0
(xk)
xk.
, (x), , x1(x) 0 x 0.
limk
xkxk1
= limk
xk1 +O((xk1))xk1
= .
, x1(x) -,
xk1xk
x2(x)dx x1k (xk) lnxk1xk
cx1k (xk),
c . ,
k=0
(xk)
xk 1
c
x00
x2(x)dx 0 (0, 1]. xn = f(xn1) xn Crn (n). , = 1. xn = f(xn1) (x) = x f(x).
xn xn1 = (xn1)., xn n (x), - [1,+),
(n) (n 1) = ((n 1)). (n) (n1) (n1), (x) (x)((x)). ,
(1)(n)
dt
(t) n,
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.. xn = (n) 1(n),
(y) =
(1)y
dt
(t).
(xn) (xn1) , , , . , (xk) (xk1) == 1 + o(1) k = 1, . . . , n, (xn) = n + o(n). , xn == 1(n+o(n)). (y) (y), . xn = xn1 x2n1.
(y) =
x1y
dt
t2 1y= (y).
(xn) (xn1)
(xn) (xn1) = 1xn 1xn1
=xn1 xnxnxn1
1
n . , (xn) = 1xn n. , xn 1n n. , .
2 [5]. xn+1 = f(xn), n = 0, 1, . . . , f(x) = x(x)+ (x)(0 < x X). (x) > 0, (x) 0. :
limx0
(x) = 0, limx0
(x) = 0, limx0
(x)
(x)= 0.
(t) (0 t .
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xn (n) (n).
. (x) ,
xk1xk
d
() xk1 xk
(xk1= 1 (xk1)
(xk1).
,
t(xk) k k1i=0
(xi)
(xi)+
Xx0
d
().
limi
(xi)
(xi)= 0,
limk
t(xk)
k 1.
Pk =(xk1)(xk)
, Qk =(xk1)(xk)
.
Pk = 1 +(k)(xk1 xk)
(xk,
xk < k < xk1. ,
Pk = 1 + (k)
(1 (xk1)
(xk1)
)Pk.
limk
Pk = 1,
Pk (k) 0 k . ,
limk
Qk = limk
Pk(xk1)(xk1)
= 0.
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83
xk1xk
d
() xk1 xk
(xk)= Pk Qk,
t(xk) k +k1i=0
(Pi 1Qi) +X
x0
d
()
, ,
limk
t(xk)
k 1. ,
limk
t(xk)
k= 1.
,
Xxk
d
() k k .
.
(t) =t(t)(t)
(t > 0)
, .. (t) > c > (t) < 0, (t) = ((t)) < 0. (0) = X,
(t) = X exp
( t0
()
d
).
(t+ h)
(t)= exp
( t+ht
()
d
) h > 0 (
t
t+ h
)c 2 +5. , r, I = [0, 1] f(x, r), I. f(x, r). r > 2 +
5 - |f (x, r)| > 1 I1 = [0, q0] [q1, 1], f(q0) = f(q1) = 1, r. , r f(x, r). r (4, 2+
5) - . I1 |f (x, r)| 1. , r - f(x, r). -
.
1. f : R R . f(x). - :
(I) C > 0 > 1, |(f (n))(x)| Cn x n 1.(II) N 1, |(f (n))(x)| > 1 x n N .(III) n0 1, |(f (n0))(x)| > 1 x .(IV) x nx 1, x, |(f (nx))(x)| > 1.
. , (IV)(III). - x nx, | ddxf (nx)(x)| > 1 ddxf (nx)(x) . - Ux x x > 1, y Ux | ddxf (nx)(y)| > x. {Ux|x } . , - {Ui}ki=1. {i}ki=1 1
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{ni}ki=1 , | ddxf (ni)(y)| > i y Ui. = max{n1, . . . , nk}, 0 = min{1, . . . , k}, m = min
x{|f (x)|}.
, |f (x)| . , m>0. k , - k0m
> 1, n0 = k + . , n0 x | ddxf (n0)(x)| > 1. - x n0, x . 1 , x U1. = n1 - 2 , f
()(x) U2. + n2 > k, . , 3.
{1, . . . , j}. =ji=1
ni. j+1 , f()(x) Uj+1. + nj+1 > k. j . ,
k 1. , j k 0 > 1. , x , | ddxf (n0)(x)| > 1. (III).(III)(II). n0 = 1 (III), . , n0 > 1.
= min{ ddx
f (n0)(x)}, m = min
x{|f (x)|},
> 1, m > 0. n0 > 1, m 1. k , - mn01k > 1. N = n0k + (n0 1). n > N , n = n0(k + ) + i, > 0 0 i n0 1. x
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ddx
f (n)(x) = d
dxf (n0(k+)(f (i)(x))
ddx
f (i)(x) k+mi
kmn01 > > 1.(II)(I). N = 1 (II), . N > 1.
m1 = minx
{ ddx
f (N)(x)}, m = min
x{|f (x)|}.
, m1 > 1. , m 1, N > 1.
= m1/N1 , C = (m/)
N1.
, > 1 C > 0. n > 0 n = kN + i, k 0 0 i N 1. x d
dxf (n)(x)
= ddx
f (kN)(f (i)(x)) d
dxf (i)(x)
mk1mi = kNmi == kNi(m/)i kN+i(m/)(N1) = Cn.(I)(IV). n ,
Cn > 1. ddx
f (n)(x) Cn > 1. nx = n x .
, -
, (IV).
,
(I).
r - f(x, r) r > 4. , ,.. . -
, r f(x, r). r > 0 f(x, r) p1 = 11/r f (p1, r) = 2r. |f (p1, r)| > 1, r > 3. p0 = 1/r. p0 p1 1/2.
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. 1.
, f(p0, r) = p1 ( r > 1). ,
f([p0, q0], r) = f([q1, p1], r) = [p1, 1].
, x, J = (p0, q0) (q1, p1), J f(x, r), .. f(x, r) / J . .
2. r > 4 c x J , n 2, f (n)(x, r) [p0, p1).. x J . f(x, r) (p1, 1) f (2)(x, r) (0, p1)., n 1 f (2+n)(x, r) [p0, p1). - f (2)(x, r) [p0, p1), . f (2)(x, r) (0, p0). , n 1 f (2+n)(x, r) [p0, p1). . f(z, r) > z z (0, p0), f (n+2)(z, r) p0. - f (n+2)(z, r) n z0 p0. , z0 f(x, r). 0 < z0 < p1. , p1 f(x, r).
3. r > 4, q0 p0 < p0. , (p0, q0) (q1, p1) , (0, p0) (p1, 1).
. , r > 4 - 2p0 > q0. , p0 = 1/r
q0 =1
21
4 1r.
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, 0 < 1 4/r < 1. 1 4/r > 1 4/r. 1/2
1
4 1r>
1
2 2r,
21
r>
1
21
4 1r.
, -
r.
. r > 4, r f(x.r).
. xr. , x>1/2. x 1. r 1. x p1, n = 1, f (x, r) f (x, p1) =
= r + 2 < 1. x = q1, f (n)(q1, r) = 0 n 2
| ddx
f (n)(q1, r)| = |f (q1, r)| |f (1, r)| |f (0, r)|n2 =
= rn1r2 4r = rn
1 (4/r), , 1 n. x, q1 p1. 2 - , n, f (n)(x, r) [p0, p1). In,j In, x (, In -, , [0, 1] f (n)(x, r)). In,j [q1, p1), . , In,j [q1, p1). f (n)(x, r) In,j [0, 1] (. 2.4), In,j -:
In,j = Ln,j Kn,j Rn,j, f (n)(Ln,j, r) = [0, p0], f
(n)(Kn,j, r) = (p0, p1), f(n)(Rn,j, r) = [p1, 1].
Ln,j In,j [q1, p1) Rn,j In,j [q1, p1), 3 , |f (n)(Ln,j, r)| > |Ln,j| |f (n)(Rn,j, r)| > |Rn,j| (, |A| A). , f (n)(x, r) - In,j. f
(n)(x, r), -, y Ln,j z Rn,j, - | ddxf (n)(y, r)| > 1 | ddxf (n)(z, r)| > 1. f (n)(x, r) [p0, p1),
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x Kn,j. , y x < z. f (n)(x, r) [y, z], ( 1.9) -, | ddxf (n)(x, r)| > 1. , In,j [q1, p1). In,j :
In,j = Ln,j Kn,j Rn,j,
f (n)(Ln,j, r) = [0, p0], f(n)(Kn,j, r) = (p0, p1), f
(n)(Rn,j, r) = [p1, 1]. -, x Kn,j, f
(n)(x, r) [p0, p1). x (q1, p1), Ln,j Rn,j [q1, p1). [q1, p1) , In,j . , Ln,j [q1, p1), Rn,j -. . In,j [q1, 1] In,j [q1, p1) 6= , p1 In,j. , |f (n)(Ln,j, r)| > |Ln,j|. y Ln,j, | ddxf (n)(y, r)| > 1. , | ddxf (n)(p1, r)| > 1, p1 - . , x [y, p1] f (n)(x, r) [y, p1]. , | ddxf (n)(x, r)| > 1.
.
. r > 4, r .
, r > 2+5.
4. -
[0, 1] [0, 1],
f(x) =
3x, 0 x 1/3,1, 1/3 x < 1/2,1/2, x = 1/2,0, 1/2 < x 2/3,3x 2, 2/3 x 1.
. 2.
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. 2. f(x)
x , f(x) . x = .0abc . . . , f(x) = .abc . . . . x = .1abc . . . ,
f(x) =
1, , 1, 0,1/2, c 1,0, , 1, 2.
x = .2abc . . . , f(x) = .abc . . . . , x 1, f(x) f(x) (x = 0, 1/2, 1). , - 1 ( 000 . . . , 222 . . . ), . ,
, 0 2. .
1/3, 2/3, 1/9, 2/9, . . . , , , -
, 0 2. - x0 f
(n)(x0) , , - , -
. , x0 = .200220220 . . . , x1 = .00220220 . . . , x2 = .022022 . . . ,x3 = .22022022 . . . , x4 = .2022022 . . . , x5 = x2, x6 = x3 .. ,
, .
a, b, c, . . . x0 , - x0, x1 = f(x0), x2 = f
(2)(x0), . . . (0, 1/3) a , (2/3, 1) b , (0, 1/3) - c .. , a, b, c, = 1, 1, 2, 2, 3, 3, . . . , x0 = .020022000222 . . . ( ).
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3 95
, ,
, , -
, -
. ,
.
.
5. 3
J f : J J . , 3 f(x) , - S J ( ), :
1) p, q S (p 6= q)
limn sup |f
(n)(p) f (n)(q)| > 0,
limn inf |f
(n)(p) f (n)(q)| = 0,2) p S q
limn sup |f
(n)(p) f (n)(q)| > 0. f(x) ( - ) - J , S J c . , , 3, . , 2)
( ) 1).
6.
, 0, 1, 2. - , ..
3/2? -, .
1-, 2-, (3/2)-.
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96
. - 0 ,
. -
- ( .
.. ).
,
. -
. -
. , -
, . . -
.
X . N() , , -X. , X d = dimX, 0 d 1, 0 N() C/d, C - , d- X, ..
C = lim0
dN().
d
d = dimX = lim0
lnN()
ln 1, (1)
lim0
lnN()
ln 1= lim
0lnCd
ln 1= lim
0d ln 1 + lnC
ln 1= d.
(1) , d-. d X . X , N() = 1. - 0. X L, N() = L/. d = 1. C. = 1/3 N(), C, N(1/3) = 2. = 1/9, , N(1/9) = 4 = 1/3m N(1/3m) = 2m. - C
d = limm
ln 2m
ln 3m=
ln 2
ln 3= 0.630929 . . . .
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.. C . 2
. r1 = r2 = r, ,
d =ln 2
ln r1.
, -
d (0, 1). , r1 6= r2
rd1 + rd2 = 1,
(0, 1). , . , N(/2k) = 2k.
d = limk
ln 2k
ln 2k
= 1.
. X - [0, 1]. |l| l. - X - li, .
L(X, ) = infi
|li|, (2)
- X. (2) , -
, L(X, ) , , . ( )
L(X) = lim0
L(X, )
.
, X X , L(X) L(X ). Xn. - {lni},
i
|lni| < L(Xn, ) + 2n L(Xn) +
2n.
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- nXn, in
|lni| . {li} - X,
i
|li| L(X, ) + 1 L(X) + 1 = K , , 0, L(X) = 0. L(X) , . , 0, L(X) = < 0 L(X) = 0 > 0. L(X) 0 , . 0 - X. dimX. ,
dimX = sup{ : L(X) =} = inf{ : L(X) = 0}.
:
1) L(X) > 0 dimX ; 2) dimX L(X) =;
3) L(X)
-
99
-
, ,
.
. X(p) , 1 p, .. x = 0, 12 . . . X(p),
limn
1
n
nk=1
k = p.
, X(1/2) 1. , p 6= 1/2, X(p) 0. (. [2]),
dimX(p) = 1ln 2
[p ln p+ (1 p) ln(1 p)].
r, xn() n- -
r. , x =n=1
xn()/rn.
Ni(, n) , i x1(), . . . , xn(). X(p0, p1, . . . , pr1) x,
limn
Ni(, n)
n= pi, i = 0, 1, . . . , r 1.
,
dimX(p0, . . . , pr1) = 1ln r
r1i=0
pi ln pi.
, -
ln 2/ ln 3, .. .
7.
-
.
xn+1 = f(xn).
f(x) , - |df/dx|. x0 x0 +
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100
O(x0) O(x0 + ) ,
f(x0 + ) f(x0) = f (x0) + o(). , (n+ 1)- ,
f (n)(x0 + ) f (n)(x0) = ddx
f (n)(x)
x=x0
+ o() = n1i=0
f (f (i))(x0) + o().
.
= limn
1
nln ddx
f (n)(x0) = lim
n ln
(n1i=0
|f (f (i))(x0)|)1/n
=
= limn
1
n
n1k=0
ln |f (xk)|,
xk = f(k)(x0), . , .
xn+1 = axn.
xn = an , = ln |a|. ln |a|>0, . ln |a| < 0, . - (p > 0)
g(x) =
{px, x < 1/2,p px, x 1/2. g(x) = p ,
= ln p.
, > 0 p > 1 . p < 1, < 0. . , (0, 1) x = 0. p = 2 ,
f(x) = 4x(1x). ,
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f(x) g(x) h g(n) = f (n) h, h(x) , - . ,
.
f(x) = 4x(1x) = ln 2.
xn = sin2(2n arcsin
x0),
xn+1 = 4xn(1 xn). , (. [24]), -
,
f(x, r) = rx(1 x), 0 < r 4, , .
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1. , .. /
.. . .: , 1977. 368 .
2. , . / . .
.: , 1969. 239 .
3. , .. / .. . .: , 1978.
142 .
4. , .. / .. //
. : , 1974.
. 109 138.
5. , .. /
.. . : , 1983. 296 .
6. , .. -
/ .. , .. , .. : 92.33 /
, 1992. 55 .
7. / .. [ .].
.: , 1992. 431 .
8. , .. , / .. -
, .. . : , 1992. 208 .
9. , .. .
3 . .2. / .. . .: , 1969. 800 .
10. , .. / .. -
, .. , .. . : , 1986.
280 .
11. / . [ .]. :
, 1989. 216 .
12. , . . / . . .: ,
1988. 240 .
13. Arterborn, D.R. A Cantor set of nonconvergence / D.R. Arterborn, W.D.
Stone // Amer. Math. Mon. 1989. V. 96, 7. P. 604 608.
14. Coppel, W.A. An interesting Cantor set / W.A. Coppel // Amer. Math. Mon.
1983. V. 90, 7. P. 456 460.
15. Devaney, R.L. An introduction to chaotic dynamical systems / R.L. Devaney.
Menlo Park, California, Reading, 1986. 321 p.
102
Copyright & A K-C
-
103
16. Guckenheimer, J. On the bifurcation of maps on the interval / J. Guckenheimer
// Invent. Math. 1977. V. 39, 2. P. 165 178.
17. Guckenheimer, J. The dynamics of density dependent population models / J.
Guckenheimer, G.R. Oster, A. Ipaktchi // J. Math. Biology. 1977. V. 4, 2.
P. 101 147.
18. Ho, C.W. A graph theoretic proof of Sharkovsky's theorem on the periodic
points of continuous functions / C.W. Ho, C. Morris // Pacic J. Math. 1981.
V. 96, 2. P. 361 370.
19. Kraft, R.L. Chaos, Cantor sets, and hyperbolicity for the logistic maps /
R.L. Kraft // Amer. Math. Mon. 1999. V. 106, 5. P. 400 408.
20. Li, T.Y. Period three implies chaos / T.Y. Li, J.A. Yorke // Amer. Math.
Mon. 1975. V. 82, 10. P. 985 992.
21. Maeda, Y. Discretization of non-lipschitz continuous o.d.e. and chaos / Y. Maeda,
M. Yamaguti // Proc. Japan Acad., Ser. A. 1996. V. 72, 2. P. 43 45.
22. May, R.M. Biological populations obeying dierence equations: stable points,
stable cicles and chaos / R.M. May // J. Theor. Biology. 1975. V. 51, 3.
P. 511 526.
23. May, R.M. Bifurcation and dynamic complexity in simple ecological models
/ R.M. May, G.F. Oster // Amer. Natur. 1976. V. 110, 974. P. 573 599.
24. de Melo, W. One-dimensional dynamics. Ergebnisse der Mathematik und
ihrer Grenzgebiete (3) / W. de Melo, S. van Strien. Berlin: Springer-Verlag, 1993.
605 p.
25. Phillipson, P.G. Analytics of period doubling / P.G. Phillipson // Commun.
Math. Phys. 1987. V. 111, 1. P. 137 149.
26. Rogers, T.D. Chaos in the cubic mapping / T.D. Rogers, D.C. Whitley //
Math. Modelling. 1983. V. 4. P. 9 25.
27. Saito, N. Exact solutions of simple nonlinear dierence equation systems that
show chaotic behavior / N. Saito, A. Szabo, T. Tsuchiya // Z. Naturforsch. 1983.
V. 38a, 9. P. 1035 1039.
28. Singer, D. Stable orbits and bifurcation of maps of the interval / D. Singer//
SIAM J. Appl. Math. 1978. V. 35, 2. P. 260 267.
29. Smale, S. The qualitative analysis of a dierence equation of population
growth / S. Smale, R.F. Williams// J. Math. Biology. 1976. V. 3, 1. P. 1 4.
30. Stran, F.D.Jr. Periodic points of continuous functions / F.D.Jr. Stran //
Math. Mag. 1978. V. 51, 2. P. 99 105.
31. Whitley, D.C. Discrete dynamical systems in dimensions one and two /
D.C. Whitley // Bull. London Math. Soc. 1983. V. 15, 3. P. 177 217.
32. Whittaker, J.V. An analytical description of some simple cases of chaotic
behavior / J. V. Whittaker // Amer. Math. Mon. 1991. V. 98, 6. P. 489 504.
Copyright & A K-C
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, ..
, ..
2.10.06. 6084/16. Data Copy.. . . 6,1. .-. . 5,9. 100 .
- -
.
.. , . , 94, . 37.
. (0852) 73-35-03
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