%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Math 3520: Differential Equations and Dynamical Systems
%  Fixed-point iteration and discrete dynamical systems 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Iterate to find the fixed point for f(x)=cos(x)
formal long;   % show 15 digits to observe convergence
x=0.1; % x_0, initial value anywhere in R
while(1)       % Use Ctrl-C to terminate
 x=cos(x)      % x_{n+1} from x_{n}
 pause(0.1);   % enough time to see the result
end

%%% Cobweb graph coordinates
function [X,Y] = cobweb(F,X0,steps)
% Input
%  F  = function handle, computes X(n+1) from X(n)
%  X0 = initial value X(1)
%  steps = number of visits to X=Y line (e.g., 10)
% Output
%  X,Y = coordinates to plot a cobweb starting at X0
  X(1) = X0;  % Initial value for iteration
  Y(1) = 0;   % Initial cobweb point is on X-axis
  for j=2:2:2*steps  % roundtrip to X=Y takes 2 steps
    X(j) = X(j-1);   % up/down to the graph of F
    Y(j) = F(X(j));  % 
    X(j+1) = Y(j);   % left/right to X=Y line
    Y(j+1) = Y(j);   % 
  end
end

%% Cobweb graph for cos(x) from x0=0.2;
F = @(x) cos(x); 
[X,Y]=cobweb(F, 0.2, 10);
figure;                  % New plot
t=0:0.01:1; plot(t,t);   % line (x,y)=(t,t), 0<t<1
hold on;  plot(t,F(t));  % iteration function
plot(X, Y);
title("Cobweb for Iteration of F");
xlabel("X(n)"); ylabel("X(n+1)"); 

%% Cobweb graph for 3x-x^3 from x0=1.8;
F = @(x) 3*x-x.*x.*x; 
[X,Y]=cobweb(F, 1.8, 10);
figure;                  % New plot
t=min(X):0.01:max(X); plot(t,t);   % line (x,y)=(t,t)
hold on;  plot(t,F(t));  % iteration function
plot(X, Y);
title("Cobweb for Iteration of F");
xlabel("X(n)"); ylabel("X(n+1)"); 

%% Logistic map cobweb
r=3.5;   % use 0<r<4
F = @(x) r*x.*(1-x);  % use componentwise ".*"
[X,Y]=cobweb(F, 0.2, 10);
figure;                  % New plot
t=0:0.01:1; plot(t,t);   % line (x,y)=(t,t), 0<t<1
hold on;  plot(t,F(t));  % iteration function
plot(X, Y);
title("Cobweb for Iteration of F");
xlabel("X(n)"); ylabel("X(n+1)");