bifurcation diagram of fractional discret henon map

how can i plot the bifurcation diagram of the fractional discrete henon map ( matlab code)?

Answer 1

[Shishir Reddy]

Hi @sabrina

As per my understanding, you would like to plot the bifurcation diagram of the fractional discrete Hénon map. Kindly refer the following steps to understand about the same.

1. Define the standard Map:

The classical Hénon map is:

x(n+1) = 1 - a*x(n)^2 + y(n);
y(n+1) = b*x(n);

2. Choose a Fractional Difference Method:

One common approach is using Caputo-like fractional differences for discrete systems. The fractional weights can be computed as follows -

alpha = 0.9;   % fractional order
M = 100;       % memory length
w = zeros(1, M);
for k = 0:M-1
    w(k+1) = (-1)^k * gamma(alpha+1) / (gamma(k+1) * gamma(alpha - k + 1));
end

3. Sweep the Bifurcation Parameter

Sweep the parameter ‘a’ over a desired range. For each value, initialize the system and iterate the fractional Hénon map using the precomputed weights. After discarding initial transients, store the remaining x values and plot them against ‘a’ to build the bifurcation diagram.

4. Plot the Bifurcation Diagram

Once you’ve collected the long-term x values for each a, you can plot them like this:

plot(a_values, x_values, '.', 'MarkerSize', 1);
xlabel('a'); ylabel('x'); title('Bifurcation Diagram');

I hope this helps.

[sabrina]

Thank you for your help, but there are still some errors, and the program is not working.

Answer 2

[sabrina]

I’m still struggling to understand how to properly draw the bifurcation diagram, especially in the context of fractional models. I would appreciate any guidance or resources you could share.