# Pretty 2-Dimensional Chaotic Maps 14

Posted by **Loren Shure**,

Chaos theory has found uses across a broad set of scientific fields to explain some complicated observed behavior. In geophysics, my background, it can help explain the reversals of Earth's magnetic field, for example. Today I thought I'd share a chaotic system, called Gingerbreadman maps, whose equations make the system seem simple. That is, until you do some simulations.

### Contents

### Gingerbreadman Map Equations

The equations for the gingerbreadman map look simple enough. For any given point in space: , define the next point in the sequence by . I now show the same equations in this MATLAB code, accounting for some initial values `x0,y0` to seed the calculation.

dbtype 17:25 gingerbreadman

17 18 % main calculation 19 for i = 1:n 20 if i == 1 21 x(i) = 1 - y0 + abs(x0); 22 y(i) = x0; 23 else 24 x(i) = 1 - y(i-1) + abs(x(i-1)); 25 y(i) = x(i-1);

### Plot Results

I want to show several results from different starting values for `x0,y0`. First I'll set the random number generator seed for repeatable results.

rng(42) gingerbreadman

gingerbreadman

gingerbreadman

gingerbreadman

gingerbreadman

### Code Listing

Here's the full code listing for `gingerbreadman`. You can see it allows you to specify the initial conditions, and return points and initial conditions should you choose.
With no output arguments, `gingerbreadman` creates plots like you've been seeing.

`type gingerbreadman`

function [xout,yout, x0, y0] = gingerbreadman(x0,y0) % Gingerbreadman map producing a chaotic 2-D map. % Copyright 2011 The MathWorks, Inc. % if not enough inputs, assign random numbers if nargin < 2 x0 = randn(); y0 = randn(); end % iteration counter n = 10000; x = zeros(n,1); y = zeros(n,1); % main calculation for i = 1:n if i == 1 x(i) = 1 - y0 + abs(x0); y(i) = x0; else x(i) = 1 - y(i-1) + abs(x(i-1)); y(i) = x(i-1); end end % if output is requested, return gingerbread x,y values and % x0, y0 initial conditions % % otherwise plot results if nargout > 0 xout = x; yout = y; else scatter(x, y, '.'); end

### References

### Make the Plot Prettier!

Instead of using the plotting function `scatter`, post your thoughts (in code) here for a chance to win some MATLAB bling. I will look at entries up through ~~(was: Sunday, November 27, 2011)~~ Wednesday December 14, 2011 and announce the winner
(the one whose plot I find most interesting) shortly after that.

Get
the MATLAB code

Published with MATLAB® 7.13

## 14 CommentsOldest to Newest

One option to made the plot more meaningful:

(adjust ‘nbins’ to your liking)

nbins = 50;

xbin = linspace(min(x),max(x),nbins);

ybin = linspace(min(y),max(y),nbins);

xbin(end) = xbin(end)+eps(xbin(end));

ybin(end) = ybin(end)+eps(ybin(end));

c = hist3([x,y],’Edges’,{xbin,ybin});

c(c==0) = nan;

figure;

pcolor(xbin,ybin,c);

shading flat

colorbar

You can experiment with different initial conditions and color maps.

%% Housekeeping

close all

clear all

clc

%% Initialize

N=10000;

x=zeros(N,1);

y=zeros(N,1);

%% Initial Conditions

% x(1)=2*rand-1;

% y(1)=2*rand-1;

x(1)=-0.1;

y(1)=0.1;

%% Loop

for n=2:N

x(n)=1-y(n-1)+abs(x(n-1));

y(n)=x(n-1);

end

%% Plot

figure

plot(x,y,’.')

hold on

plot(x(1),y(1),’ro’)

axis equal

grid on

xlabel(‘x’)

ylabel(‘y’)

axis([-4 8 -4 8])

%% Plot 2

numPoints=128;

colorList=jet(numPoints);

figure

hAgedPlot=plot(x(1),y(1),’Marker’,’.’,'LineStyle’,'none’,'Color’,…

colorList(1,:));

hold on

hNewPlot=zeros(numPoints,1);

for n=1:numPoints

hNewPlot(n)=plot(x(1),y(1),’Marker’,’.’,'Color’,colorList(n,:),…

‘MarkerSize’,12);

end

set(gca,’Color’,'none’)

set(gcf,’Color’,'k’)

set(gcf,’Position’,[100 100 640 480])

axis([-6 10 -4 8])

set(gca,’Position’,[0 0 1 1])

axis off

for n=1:N

for m=1:numPoints

nn=n-m+1;

brokeEarly=false;

if nn0

set(hAgedPlot,’XData’,x(1:nn),’YData’,y(1:nn))

end

drawnow

end

As Steve Lord pointed out to me, the deadline is in the past! Change that to Dec. 14, 2011. Sorry for the mistake!

–loren

I used tags, but when I copy my code from the web back into MATLAB, the formatting is messed up and it doesn’t work.

(It’s not just “<" , it's also single quotes etc.)

Any advice?

I don’t have a good option for you. I copy and paste the code and then edit it myself. Sorry for the hassle.

–loren

Loren,

Here’s a suggestion for a winter themed version. The function is mostly the same. Only the plotting is different.

Enjoy.

~Jonathan

function [xout,yout, x0, y0] = gingerbreadman(x0,y0)

% Gingerbreadman map producing a chaotic 2-D map.

% Copyright 2011 The MathWorks, Inc.

% if not enough inputs, assign random numbers

if nargin 0

xout = x;

yout = y;

else

layers = 100;

numPerLayer = ceil(n/layers);

colors = winter(layers);

if numel(unique(x)) < numPerLayer

marker = '*-';

else

marker = '*';

end

hold off

for iLayer = 1:layers

if iLayer == layers

plot3(x((iLayer-1)*numPerLayer+1:end), y((iLayer-1)*numPerLayer+1:end), (iLayer-1)*numPerLayer+1:n, marker, 'MarkerEdgeColor', colors(iLayer,:), 'Color', colors(iLayer,:));

else

plot3(x((iLayer-1)*numPerLayer+1:iLayer*numPerLayer), y((iLayer-1)*numPerLayer+1:iLayer*numPerLayer), (iLayer-1)*numPerLayer+1:iLayer*numPerLayer, marker, 'MarkerEdgeColor', colors(iLayer,:), 'Color', colors(iLayer,:));

end

if iLayer == 1

hold on

end

end

if numel(unique(x)) < numPerLayer

view(0, 85)

else

view(0, 90)

end

end

I think part of my code listing was altered/removed in the post above.

Here it is again.

Yes, I had to use ‘& l t ;’ and ‘& g t ;’ (without spaces) for the less than and greater than symbols.

~Jonathan

function [xout,yout, x0, y0] = gingerbreadman(x0,y0)

% Gingerbreadman map producing a chaotic 2-D map.

% Copyright 2011 The MathWorks, Inc.

% if not enough inputs, assign random numbers

if nargin < 2

x0 = randn();

y0 = randn();

end

% iteration counter

n = 10000;

x = zeros(n,1);

y = zeros(n,1);

% main calculation

for i = 1:n

if i == 1

x(i) = 1 – y0 + abs(x0);

y(i) = x0;

else

x(i) = 1 – y(i-1) + abs(x(i-1));

y(i) = x(i-1);

end

end

% if output is requested, return gingerbread x,y values and

% x0, y0 initial conditions

%

% otherwise plot results

if nargout > 0

xout = x;

yout = y;

else

layers = 100;

numPerLayer = ceil(n/layers);

colors = winter(layers);

if numel(unique(x)) < numPerLayer

marker = '*-';

else

marker = '*';

end

hold off

for iLayer = 1:layers

if iLayer == layers

plot3(x((iLayer-1)*numPerLayer+1:end), y((iLayer-1)*numPerLayer+1:end), (iLayer-1)*numPerLayer+1:n, marker, 'MarkerEdgeColor', colors(iLayer,:), 'Color', colors(iLayer,:));

else

plot3(x((iLayer-1)*numPerLayer+1:iLayer*numPerLayer), y((iLayer-1)*numPerLayer+1:iLayer*numPerLayer), (iLayer-1)*numPerLayer+1:iLayer*numPerLayer, marker, 'MarkerEdgeColor', colors(iLayer,:), 'Color', colors(iLayer,:));

end

if iLayer == 1

hold on

end

end

if numel(unique(x)) < numPerLayer

view(0, 85)

else

view(0, 90)

end

end

I posted mine to File Exchange to avoid the reformatting problem.

http://www.mathworks.com/matlabcentral/fileexchange/34116

One thing it would be nice to extract automatically from the plot is some idea of what the boundary looks like. Getting the convex hull is easy, but the edge of a gingerbreadman is certainly non-convex! There are some interesting methods for finding such boundaries, so I’ve gone off the deep end and coded up the so-called ‘swinging arm’ method as a function. Simply pass it the vectors x and y containing the fractal points, a value for the length of the swinging arm, and a number of points required on the boundary. When the swinging arm length is set to infinity, the method returns the convex hull. You have to play with values for the arm length a bit to get a meaningful boundary shape: too small and it returns disconnected polygons; too large and we get the convex hull.

I’m not sure how pretty it is though!

function gingerboundary(xin,yin,l,bpoints)

% FUNCTION GINGERBOUNDARY(XIN,YIN,L,BPOINTS)

% function to implement swinging arm algorithm for finding a footprint of

% the gingerbreadman fractal set. XIN and YIN are vectors of the x and y

% positions of the points in the set. L sets the radius of nearest

% neighbours to scan, and BPOINTS is the number of boundary points.

% sort points by y-coordinate

p = [xin yin];

plot(p(:,1),p(:,2),’b.’);

n = size(p,1);

p = rot90(sortrows(fliplr(p)),2);

x = p(1,:); xold = x + [0 1];

b = x;

% scan loop

for i=1:bpoints

neighbours = p(sqrt(sum((p-repmat(x,n,1)).^2,2)) eps)

neighbours2 = [neighbours2; neighbours(j,:)];

end

end

% find ‘least clockwise’ point

s2 = size(neighbours2,1);

v = neighbours2-repmat(x,s2,1);

a = angle(v(:,1)+1i*v(:,2));

aprev = angle(xold(1)-x(1)+1i*(xold(2)-x(2)));

adiff = mod(a-aprev,2*pi);

% update

xold = x;

candidates = neighbours2(adiff==max(adiff),:);

x = candidates(1,:);

% plotting

%{

hold on

plot(p(:,1),p(:,2),’b.’);

plot(neighbours2(:,1),neighbours2(:,2),’r.’)

plot(b(:,1),b(:,2));

axis(‘square’);

hold off

drawnow

pause(1);

%}

% update boundary

b = [b; x];

end

% plot boundary

hold on;

plot(p(:,1),p(:,2),’b.’);

plot(b(:,1),b(:,2),’r');

axis(‘square’);

end

I haven’t checked all the other submissions, I hope there’s not yet one too close to mine…

With a minor change in the equation using that y(i) equals x(i-1), I simplified the calculation. However, I therefore produce one more value than can be plotted reasonably. Thus, I take the difference between consecutive values as color code.

n = 10000;

x = zeros(n+2,1);

x(1:2)=randn(1,2);

% main calculation

for cnt = 3:n+2

x(cnt) = 1 – x(cnt-2) + abs(x(cnt-1));

end

scatter(x(2:end),x(1:end-1),3,diff(x))

set(gca,’color’,'k’)

If you take x0 = x(2) and y0 = x(1), Loren’s equations come out with the same values (of course shifted by 2 indices).

This function uses a different approach.The function will map x and y data onto 2d array.This 2d array can be seen as an image.The image generated is grayscale.The darker the pixel more times the formula gets near to that value of x and y.This is same as visualizing a strange attractor.The function is a slow because it stores all the value of x and y.

function [x,y,x0,y0]=myGingerbreadManMap2(x0,y0)

%function takes about 3 seconds to complete

if nargin < 2

x0=randn();

y0=randn();

end

numIterate=1000000;

x=zeros(numIterate,1);

y=zeros(numIterate,1);

x(1)=x0;

y(1)=y0;

for n=2:numIterate

x(n)=1-y(n-1)+abs(x(n-1));

y(n)=x(n-1);

end

%find bounding box

maxX=max(x);

maxY=max(y);

minX=min(x);

minY=min(y);

%500×500 matrix as counter

img=zeros(500,500);

xdiff=(maxX-minX)/500;

ydiff=(maxY-minY)/500;

for n=1:numIterate

indx=floor((x(n)-minX)/xdiff);

indy=floor((y(n)-minY)/ydiff);

if (indx>0 && indy>0 && indx < 500 && indy < 500)

img(indx,indy)=img(indx,indy)+1;

end

end

%scale data

maxP=max(img(:));%maximum number of count

img=255*img/maxP;%scale counts in the range of 0 to 255

%diplay

img=uint8(img);

image(linspace(minX,maxX,500),linspace(maxY,minY,500),img);

axis equal

cmap=1-gray(256);

colormap(cmap);

%save image

%imwrite(img,cmap,’gingerbreadman.png’,’png’);

I was interested in which points were plotted last, so my code plots the first 3/4 of the points in white and then two other colors for the remaining 1/4. I also manually put a background around the plot for clarity.

The function takes a single input (no outputs) which determines the number of plots produced in a single figure, so an input of ’2′ produces 4 plots in a 2×2 grid.

function gingerbreadmansheet(N)

hold off

if nargin< 1

N = 1;

end

n = 10000;

for xtran = 0:(N-1)

for ytran = 0:(N-1)

x0 = randn();

y0 = randn();

x = zeros(1,n);

y = zeros(1,n);

for i = 1:n

if i == 1

x(i) = 1 – y0 + abs(x0);

y(i) = x0;

else

x(i) = 1 – y(i-1) + abs(x(i-1));

y(i) = x(i-1);

end

end

x = x+15*xtran;

y = y+15*ytran;

xv = [-2.5 -3.5 -2.5 -2.5 -3.5 -2 0 1 2 4 5.5 4.5 4.5 5.5 8.5 8.5 5.5 4.5 4.5 5.5 4 2 1 0 -2.5]+15*xtran;

yv = [-2.5 0 1 2 4 5.5 4.5 4.5 5.5 8.5 8.5 5.5 4.5 4.5 5.5 4 2 1 0 -2 -3.5 -2.5 -2.5 -3.5 -2.5]+15*ytran;

ss = get(0,’screensize’);

fill(xv,yv,[ 0.5451 0.2706 0.0745]);

hold on

plot(x(1:3*end/4), y(1:3*end/4),’.w’,'LineWidth’,3,’MarkerSize’,20/N^2);

plot(x((3*end/4+1):7*end/8), y((3*end/4+1):7*end/8),’.',’MarkerSize’,20/N^2,’MarkerEdgeColor’,[.95 0 0]);

plot(x(7*end/8+1:end), y(7*end/8+1:end),’.',’MarkerSize’,20/N^2,’MarkerEdgeColor’,[0 .5 0]);

set(gcf,’position’,[.05 .05 .85 .85]*ss(4))

set(gca,’xlim’,[-4 9+15*xtran],’ylim’,[-4 9+15*ytran],’color’,[.5 .5 .5])

end

end

Here is one more submission… I decided to make a pixelated gingerbread man with the output of the chaotic map, using a

lemniscate of Gerono as his green bow tie :)

function pixelatedGBM

rng(42)

[x,y] = gingerbreadman;

%scale and rotate our gingerbread man

r = [x y] * [cosd(135) -sind(135); sind(135) cosd(135)];

minR = min(r); maxR = max(r);

r = (r – repmat(minR,size(r,1),1))./repmat(maxR-minR,size(r,1),1);

r(:,2) = r(:,2).^1.5;

%create a pixel representation of it

N = 25;

b = linspace(0,1,N);

dif = b(2)-b(1);

[xb,yb] = meshgrid(b,b);

C = zeros(N);

x = r(:,1); y = r(:,2);

for i = 1:numel(xb)

C(i) = length(find(x >= xb(i) & x = yb(i) & y < yb(i)+dif));

end

C = reshape(C,size(xb));

%smooth a little for better results

smooth = @(A,L) ((eye(size(A,1)) + L ^ 2 * diff(eye(size(A,1)),2)' * diff(eye(size(A,1)),2) + 2 * L * diff(eye(size(A,1)))' * diff(eye(size(A,1)))) \ A);

D = smooth(smooth(C,0.1)',0.1)';

%let's draw it :)

set(figure,'Position',[0 0 300 400],'Color',[1 1 1])

movegui(gcf,'center')

set(surf(xb,yb,zeros(size(D)),D),'ZData',xb.*0-0.01);

view(2); shading flat; grid off; axis off equal;

colormap([1 1 1; pink(19)])

hold on

t = linspace(0,2*pi,50);

set(fill((sin(t))/9+0.5,(sin(2*t))/18+0.65,[0 .8 0]),'EdgeAlpha',0)

set(fill((sin(t))/30+0.5,(cos(t))/30+0.65,[0 .9 0]),'EdgeAlpha',0)

t = linspace(0,2*pi,5);

set(fill((sin(t))/20+0.5,(cos(t))/20+0.5,[.8 0 0]),'EdgeAlpha',0)

set(fill((sin(t))/20+0.5,(cos(t))/20+0.3,[.8 0 0]),'EdgeAlpha',0)

plot3(r(1:10:end,1),r(1:10:end,2),r(1:10:end,1)*0-0.005,'.','MarkerSize',3,'MarkerEdgeColor',[1 1 1]);

end

Nice, the < were also messed up into tags in my submission…

here is the original code in pastebin:

http://pastebin.com/hGjWYpji