Markov chain green26 - Rhea

Markov Chains

I attempted to model a robot circulating about the origin with small noise in each step. You can directly plug in the following Matlab code. It generated the following images, showing the putative robot spiraling 'inward' or 'outward'. I assume this is a result of the butterfly effect; that is, noise early in the walk process has a greater effect on the final position of the robot than noise later in the walk process.

% set params
num_steps = 10000;
std_dev_rot = .01;
std_dev_noise_mag = .01;

% generate random initial state with complex magnitude 1
walk = zeros(1,num_steps);
walk(1) = exp(i*rand);
% generate next states (probabilistic modifications on current state)
for j=2:length(walk),
  walk(j) = walk(j-1)*normrnd(1,std_noise_mag)*(i^(-abs(normrnd(0,std_rot))));
end;
plot(walk); grid on; hold on;

% plot starting (green) and end (red) squares
plot(walk(1), 'gs','LineWidth', 2);
plot(walk(length(walk)), 'rs','LineWidth', 2);

% generate circle to compare with "robot" circulation
circ = zeros(1,1000);
circ(1) = i;
for j=2:length(circ),
  circ(j) = circ(j-1)*i^(2*pi/length(circ));
end;
plot(circ,':k','LineWidth', 2); hold off;