xn +1 = l xn (1 xn) |
l = 4r.
xn n.
) ( 1);
() 2;
() 4.
%logistic_guld
clear all
r=[0.1 0.6 0.8 0.9];
for i=1:4
clear x
x(1)=0.75;
for n=2:25%150
l(i)=4*r(i);
x(n)=4*r(i)*x(n-1)*(1-x(n-1));
end
plot(x)
ss=num2str(r(i));
ssl=num2str(l(i));
title(['r = ' ss ' lambda = ',ssl])
xlabel('n');
ylabel('x');
grid on
figure
end
. 8.1,. r = 0.1 r = 0.6.
. 8.1,6. r = 0.8 r = 0.9.
1978. .
, f () , , , r = 0.9, . , . .
8.1.
. (8.6) r = 0.2 r = 0.24 x 0. , = 0 . , r = 0 x 0. , , .
. (8.6) r = 0.26, 0.5, 0.7, 0.72, 0.74 0.748. ( r = 0.748 1000 .) = 0? , 0 .
, = 0 ? , , .. , 1. r?
x 0, 1, , xn , . , r .
. (8.6) r = 0.752, 0.76, 0.8 0.862. ( r = 0.752 1000 .) , 0.75, , .. , 1, , 2.
|
|
r, * , 1* x 2* r = 3/4. ( 1* 2*) 2.
. , . .
. (8.6) r = 0.863 0.88? ?
. r = = 0.89, 0.891 0.8922?
8.2.
. r > rc = 0.892486417967... , . x 0 = 0.500 0.501. , 10%?
. , . r = 0.91 x 0 = 0.5 200 . , = x /10 = 10* . , . . r < rc?
. r = 0.958. ?
. 8.2. r. . .
(8.6) r (. (8.2). . 8.2 , .
8.3.
. , , r. n . 0.8 < r < 0.9. ?
. , 4 32. 4 ?
. .
8.3.
, , , , , !
|
|
. 8.3. xn +1 = 4 rxn (l xn) r = 0.6 x 0 = 0.05. = 0 , = 0.58333 .
r f (x). . 8.3 f (x) r = 0.6. , = , = f (x) * = 0 * = 0.58333, .. f () * = 0 * = 0.58333 . 0 , . { = 0, = 0} = f (x) { x 0, 0 = f (x 0)}. { x 0, y 0} { y 0, y 0}. , 1 = 0. x 2. { x 1, y 0} = f (x). = 1 = f (x 1) ; 2. , :
1) = f (x);
2) = x;
3) 1 2 .
. 8.3 0 = 0.05 r = 0.6. , x 0 ( ≠ 0 ≠ 1), x * = 7/12 ≈ 0.58333. ( . 8.3.) ( 1). x 0 = 0 . .
x * = 0 * = 0.58333 r = 0.6? = f (x) f. ( 45) x . , , 450, .. | df (x)/ dx | x = x * < 1, * , , , | df (x)/ dx | x = x * > 1, * . f (x), . 8.3, , x = 0 , f (x) x = 0 . f (x) = 0.58333 .
, 0 < r < 3/4 .
, r 3/4 < r < 1? , r f () 2. , ..
xi = f (f (xi)), i = 1, 2 | (8.7) |
f (x) g (x) = f (f (x)). , r? , g (x) . f 4 h (x) = g (g (x)) = f (f (f (f (x)))). , .. 1 → 2 → 4 → 8 → 16 → 32 →.... 8.4.
|
|
8.4.
. xn r , r < 3/4 f (x) . f (x) = 1/2, f (x) . g (x) = f (f (x)? g (x) x = 1/2? * g (x)? g (x) ? r 1 r, f (x) . , r 1 = 0.75.
. f (x) = 0.785. f (x)? ? g (x) 1* x 2* . ( 0 = 0.1 x 0 = 0.3.) g (x) ? 1* x 2* f (x)? , g (x) 1* 2* .
. g (x). r g (x) g (x) . r = r( 1) g 1/2? ? r . r . , r = r 2 = 0.8623 g (x) 1 .
. , r, r 2, r ≈ 0.87, h (x) = g (g (x)) . r = r (2), 1/2? r = r (2). r = r 3 h ?