MATLAB Implementation of 3D Baker Map with Visualization

Below is a MATLAB code example for implementing the 3D Baker map:

function [x,y,z] = bakerMap3D(x0,y0,z0,a,b,c)

% Define number of iterations for the chaotic system N = 10000;

% Initialize arrays to store trajectory coordinates x = zeros(N+1,1); y = zeros(N+1,1); z = zeros(N+1,1);

% Set initial conditions for the dynamical system x(1) = x0; y(1) = y0; z(1) = z0;

% Implement the 3D Baker map algorithm with conditional transformations % The algorithm divides the 3D space into 8 regions based on parameter boundaries (a,b,c) % Each region applies specific linear transformations to stretch and fold the space for i=1:N if x(i)>=0 && x(i)<=a && y(i)>=0 && y(i)<=b && z(i)>=0 && z(i)<=c x(i+1) = 2*x(i)/a; y(i+1) = 2*y(i)/b; z(i+1) = 2*z(i)/c; elseif x(i)>=a && x(i)<=2*a && y(i)>=0 && y(i)<=b && z(i)>=0 && z(i)<=c x(i+1) = 2*(x(i)-a)/a + 1; y(i+1) = 2*y(i)/b; z(i+1) = 2*z(i)/c; elseif x(i)>=0 && x(i)<=a && y(i)>=b && y(i)<=2*b && z(i)>=0 && z(i)<=c x(i+1) = 2*x(i)/a; y(i+1) = 2*(y(i)-b)/b + 1; z(i+1) = 2*z(i)/c; elseif x(i)>=a && x(i)<=2*a && y(i)>=b && y(i)<=2*b && z(i)>=0 && z(i)<=c x(i+1) = 2*(x(i)-a)/a + 1; y(i+1) = 2*(y(i)-b)/b + 1; z(i+1) = 2*z(i)/c; elseif x(i)>=0 && x(i)<=a && y(i)>=0 && y(i)<=b && z(i)>=c && z(i)<=2*c x(i+1) = 2*x(i)/a; y(i+1) = 2*y(i)/b; z(i+1) = 2*(z(i)-c)/c + 1; elseif x(i)>=a && x(i)<=2*a && y(i)>=0 && y(i)<=b && z(i)>=c && z(i)<=2*c x(i+1) = 2*(x(i)-a)/a + 1; y(i+1) = 2*y(i)/b; z(i+1) = 2*(z(i)-c)/c + 1; elseif x(i)>=0 && x(i)<=a && y(i)>=b && y(i)<=2*b && z(i)>=c && z(i)<=2*c x(i+1) = 2*x(i)/a; y(i+1) = 2*(y(i)-b)/b + 1; z(i+1) = 2*(z(i)-c)/c + 1; elseif x(i)>=a && x(i)<=2*a && y(i)>=b && y(i)<=2*b && z(i)>=c && z(i)<=2*c x(i+1) = 2*(x(i)-a)/a + 1; y(i+1) = 2*(y(i)-b)/b + 1; z(i+1) = 2*(z(i)-c)/c + 1; end end

% Create 3D visualization of the Baker map trajectory plot3(x,y,z)

% Add axis labels for clarity xlabel('x') ylabel('y') zlabel('z')

% Set plot title title('3D Visualization of Baker Map Dynamics')

% Configure plot appearance properties grid on axis equal view(30,30)

% End function end

This example provides MATLAB code for implementing the 3D Baker map. You can use this code to simulate the Baker map dynamics and generate 3D visualizations. The Baker map is a chaotic mapping system commonly employed in information encryption/decryption, data compression, and image processing applications. The algorithm works by iteratively applying piecewise linear transformations that stretch and fold the phase space, creating complex chaotic behavior. If you're interested in chaotic systems, this code serves as a foundation for exploring the properties and applications of the Baker map, including sensitivity to initial conditions and mixing properties that make it suitable for cryptographic applications.