graphics_toolkit("gnuplot")
grid on
nx=100000; a=3.9; x(1,1:4)=[0.1 0.33 0.54 0.88];
j=1:4;
for i=2:nx
x(i,j)=a*x(i-1,j).*(1-x(i-1,j));
end
x=x(1:nx,j);
dx=x(2:nx,j)-x(1:nx-1,j);
figure, plot(x(end-49:end,1),"or-","markerfacecolor","r","linewidth",2)
hold on
plot(x(end-49:end,2),"ob-","markerfacecolor","b","linewidth",2)
title(["X   (Nx = " num2str(nx) ",  a = " num2str(a) ")"])
hold off
xmed(j)=mean(x(:,j))
xvar(j)=var(x(:,j))
x0=x;
x0(:,j)=x(:,j)-xmed(j);
nr=39,r=zeros(nr+1,length(j));
for i=0:nr 
r(i+1,j)=sum(x0(1:nx-i,j).*x0(i+1:nx,j))./xvar(j)/(nx-i-1); 
end
figure, plot([0:nr],r(:,1),"or-","markerfacecolor","r","linewidth",2)
hold on
plot([0:nr],r(:,2),"ob-","markerfacecolor","b","linewidth",2)
plot([0:nr],r(:,3),"og-","markerfacecolor","g","linewidth",2)
title(["Func. autocorr. r   (Nx = " num2str(nx) ",  a = " num2str(a) ")"])
hold off
figure
semilogy([0:nr],abs(r(:,1)),"or-","markerfacecolor","r","linewidth",2)
hold on
semilogy([0:nr],abs(r(:,2)),"ob-","markerfacecolor","b","linewidth",2)
semilogy([0:nr],abs(r(:,3)),"og-","markerfacecolor","g","linewidth",2)
title(["Func. autocorr. r   (Nx = " num2str(nx) ",  a = " num2str(a) ")"])
hold off
dxmed(j)=mean(dx(:,j))
dxvar(j)=var(dx(:,j))
dx0=dx;
dx0(:,j)=dx(:,j)-dxmed(j);
nr=39,r=zeros(nr+1,length(j));
for i=0:nr 
dr(i+1,j)=sum(dx0(1:nx-1-i,j).*dx0(i+1:nx-1,j))./dxvar(j)/(nx-1-i-1); 
end
figure, plot([0:nr],dr(:,1),"or-","markerfacecolor","r","linewidth",2)
hold on
plot([0:nr],dr(:,2),"ob-","markerfacecolor","b","linewidth",2)
plot([0:nr],dr(:,3),"og-","markerfacecolor","g","linewidth",2)
title(["Func. autocorr. de dx   (Nx = " num2str(nx) ",  a = " num2str(a) ")"])
hold off
figure
semilogy([0:nr],abs(dr(:,1)),"or-","markerfacecolor","r","linewidth",2)
hold on
semilogy([0:nr],abs(dr(:,2)),"ob-","markerfacecolor","b","linewidth",2)
semilogy([0:nr],abs(dr(:,3)),"og-","markerfacecolor","g","linewidth",2)
title(["Func. autocorr. de dx   (Nx = " num2str(nx) ",  a = " num2str(a) ")"])
hold off