Help! Bifurcation diagram for difference equation

Hello everyone! I need help figuring out how to plot a bifurcation diagram when dealing with difference equations. I tried following the indications on the page http://www.math.le.ac.uk/people/rld8/ma1251/lab3.html, but I cannot figure out what I'm doing wrong. This is what I wrote:

clear all;
close all;

Npre=100; 
Nplot=200;
x = zeros(Nplot,1);
for lambda = 0:0.01:5
  x(1) = 2.95;
  x(2)=x(1)+0.2;
  x(3)=x(1)-0.2;
  for i = 3:Npre
%   x(i+3) = x(i+2)+0.19*100*(0.5*0.3*(3-x(i+2))-0.5*0.5*(x(i+2)-3))./(1+exp(-lambda*(2.1*((1/3)-0.01)+15*((x(i+1)-x(i))./x(i)))));
    x(3)=x(3)+0.19*100*(0.5*0.3*(3-x(3))-0.5*0.5*(x(3)-3))./(1+exp(-lambda*(2.1*((1/3)-0.01)+15*((x(2)-x(1))./x(1)))));
  end
  for i = 1:Nplot-3
  x(i+3) = x(i+2)+0.19*100*(0.5*0.3*(3-x(i+2))-0.5*0.5*(x(i+2)-3))./(1+exp(-lambda*(2.1*((1/3)-0.01)+15*((x(i+1)-x(i))./x(i)))));
  end
  plot(lambda*ones(Nplot,1), x, '.', 'markersize', 2);
  hold on;
end
title('Bifurcation diagram');
xlabel('lambda');  ylabel('x_i');
set(gca, 'xlim', [0 5], 'ylim', [1.5 4.5]);
hold off;

[darova]

I don't understand the question. I run your code and got this:

Can you exaplain more? Maybe a simple sketch or simething?

[Carolina Biliotti]

Thank you darova for your help! I am getting the same figure as you, but my aim is actually to replicate the following plot:

It is clear that there are some mistakes in the code... It is my first time trying to plot a bifurcation diagram on my own, any advice is well taken!! Thank you again.

[darova]

Can you attach your original eqaations?

I have somer problem with the link you gave

[Carolina Biliotti]

http://www.math.le.ac.uk/people/rld8/ma1251/lab3.html

This is an example I used to build my bifurcation, I found it on the following matlab answer:https://www.mathworks.com/matlabcentral/answers/319767-how-to-plot-a-bifurcation-diagram-for-differential-equation-is-there-any-formula-to-plot-the-bifurc

The original equation is the following:

function[x]=time_series(alpha, beta, lambda, gamma, Num, gf, gc, w, F, r, D, N, x0)
alpha=2.1;
beta=15;
D=1;
F=3;
r=0.01;
Num=100;
gf=0.5;
gc=0.3;
lambda=1;
gamma=0.19;
w=0.5;

x=zeros(N,1);
x(1)=x0;
x(2)=x0+0.2;
x(3)=x0-0.2;
for i=1:N-3
   x(i+3)=x(i+2)+gamma*Num*((w*gf*(F-x(i+2))+(1-w)*gc*(x(i+2)-F))./(1+exp(-lambda*(alpha*((D/F)-r)+beta*((x(i+1)-x(i))./x(i))))));
 
end

[Carolina Biliotti]

I forgot, in the above code N=1100, where the first 1000 observations are just to lose the effect of the initial condition, which is x0=2.95

[darova]

There is too much text! I understand this: you have diff equation and want to draw something

Where is the diff equation? Can you show? What needs to be done to draw bifurcation diagram?

[Carolina Biliotti]

This is the difference equation:

x(i+3)=x(i+2)+gamma*Num*((w*gf*(F-x(i+2))+(1-w)*gc*(x(i+2)-F))./(1+exp(-lambda*(alpha*((D/F)-r)+beta*((x(i+1)-x(i))./x(i))))))

As I understood correctly, to draw a bifurcation diagram we need to create a loop to show the values approached asymptotically of some system of equations or map as a function of a bifurcation-changing parameter (lambda, in my case). The solid line in the picture I attached are the stable values of the system (so the system is stable until lambda approaches lambda_T). After lambda_t the system becomes unstable, first through period-doubling bifurcations, then reaching chaotic behavior.

Thank you again for your help.

[darova]

Try to change lambda value

Don't forget about limits

set(gca, 'xlim', [0 5], 'ylim', [1.5 4.5]);