[資訊] Logistic map 的 matlab 程式

[ANUBISANKH (Pourquoi？)]

幾年前寫的，會跑出兩個圖。第一個是把理論計算和實驗方法取得的 Lyapunov 指數隨
參數變化 plot 出來；第二個是把 fixed points 找出來，可看出隨參數變化有 period
doubling。


clc;
close all;
clear all;

I=100;    % Iteration
n=5;      % separation between pairs
A=1;      % f(x)=ux(1-x),
B=4;      %"u" from A to B
ai=0.001;  % resolution of parameter "u"

f=round((B-A)/ai);
for u=1:f
    U=u*ai;
    x(u,1)=0.501;

    % Iteration and calculating theoretical Lyapunov exponent
    sum=0;
    for i=1:I
        x(u,i+1)=(U+A)*x(u,i)*(1-x(u,i)); % logistic map
        sum=sum+log(abs((U+A)*(1-2*(1-x(u,i)))));
    end
    Lt(u)=sum/I;

    % find closest points then calculate largest Lyapunov exponent
    counter=0;
    Lsum=0;
    for i=2:I+1
        if abs(x(u,1)-x(u,i))<0.05
        if 0<abs(x(u,1)-x(u,i))
        if i+n<I
        if 0<abs(x(u,1+n)-x(u,i+n))
            counter=counter+1;
            L(counter)=(1/n)*log(abs(x(u,1+n)-x(u,i+n))/abs(x(u,1)-x(u,i)));
            Lsum=Lsum+L(counter);
        end
        end
        end
        end
    end
    Lave(u)=Lsum/counter;

end

% plot theoretical and experimental data of L.E. together
figure(1)
plot([A+ai:ai:B],Lave,[A+ai:ai:B],Lt)

figure(2)
u=1:f;
i=round(0.7*I):I;
fixed=plot((u*ai+A),x(u,i),'.')
set(fixed,'MarkerSize',4.5);

--
"Bureaucrats!" Mallory scoffed cheerily. "They might known this would happen,
if they'd properly studied Catastrophist theory. It is a concatenation of
synergistic interactions; the whole system is on the PERIOD-DOUBLING ROUTE TO
CHAOS!"
                                                     ----《Difference Engine》

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