File:Diffeomorphism of a square.svg

% Compute a diffeomorphism from a square to a square which leave
% the boundary fixed.

function main()

   N = 20; % num of grid points
   epsilon = 0.1; % displacement for each small diffeomorphism
   num_comp = 10; % number of times the diffeomorphism is composed with itself
   
   S = linspace(-1, 1, N);

   [X, Y] = meshgrid(S);

   Z = X; W = Y;

   % take num_comp compositions of the same small diffeomorphism
   for iter = 1:num_comp

      for i=1:N
         for j=1:N
            
            [Z(i, j), W(i, j)] = small_diffeo(Z(i, j), W(i, j), epsilon);
            
         end
      end

   end
   
   % graphing settings
   lw = 2;
   mycolor = [1, 0, 0.1];
   small = 0.1;
   
   figure(1); clf; hold on;
   for i=1:N
      plot(X(:, i), Y(:, i), 'linewidth', lw, 'color', mycolor);
      plot(X(i, :), Y(i, :), 'linewidth', lw, 'color', mycolor);
   end
   axis([-1-small, 1+small, -1-small, 1+small]);
   axis equal; axis off;
   
   figure(2); clf; hold on;
   for i=1:N
      plot(Z(:, i), W(:, i), 'linewidth', lw, 'color', mycolor);
      plot(Z(i, :), W(i, :), 'linewidth', lw, 'color', mycolor);
   end
   axis([-1-small, 1+small, -1-small, 1+small]);
   axis equal; axis off;
   
function [z, w] = small_diffeo(x, y, epsilon);

   A1=epsilon*(cos(pi*x)+1)*(cos(pi*y)+1)/4.0;
   A2=epsilon*cos(pi*x/2)*cos(pi*y/2);

   A = (A1+A2)/2;
   
   z = x +(-y)*A;
   w = y +( x)*A;