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% Math 3520: Differential Equations and Dynamical Systems
%  Complex dynamics, Julia and Fatou sets, Mandelbrot set. 
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%% Complex squaring map p(z)=z^2+a
a=[-1,0];           % Example a=-1+0i, so p(z)=z^2-1
% Implement complex squaring in R^2: (z(i)+i*z(2))^2+a
csqr = @(z) [z(1).*z(1)-z(2).*z(2)+a(1), 2*z(1).*z(2)+a(2)];

N = 10000; 

% Generate N pairs (xn,yn) for plotting as points
X=[0,0];   % Initial values [x0,y0] used by Mandelbrot
for(i=2:N)
 X(i,:) = csqr(X(i-1,:)); 
end
figure; hold on;   % New graph, accumulate points
plot(X(:,1), X(:,2), ".");    % plot trajectory
title("Complex Squaring Map Iteration");
xlabel("Real Part"); ylabel("Imaginary Part");