Dimension Demonstration

Compute the dimension of the attractor of the Hénon map

Compute the Hénon attractor with the standard values

a=1.4;

b=0.3;

n=10000;

nTransient=100;

x=zeros(2,n+1);

for k=1:n+nTransient

x(:,k+1)=henon(x(:,k),a,b);

end

x=x(:,nTransient+1:end);

y=x(2,:);

x=x(1,:);

plot(x,y,'.','markersize',4)

axis equal tight off

Box-counting step

N=7;

epsilon=zeros(1,N);

Nboxes=zeros(1,N);

for k=1:N

K=2^(k+1);

[epsilon(k),Nboxes(k)]=countAndPlotBoxes(x,y,K);

end

Processing the box-counting step

clf; hold on;

p=polyfit(log(epsilon),log(Nboxes),1);

dim=abs(p(2));

loglog(epsilon,Nboxes,'o',epsilon,exp(polyval(p,log(epsilon))));

set(gca,'xscale','log','yscale','log')

xlabel('$\epsilon$')

ylabel('$N(\epsilon)$')

legend('data','least squares line');

fprintf('The calculated dimension is %0.2f.\n',abs(p(1)))

The calculated dimension is 1.27.

Definition of the Hénon map

The Hénon map, most commonly studied with a=1.4, b=.3

function X=henon(x,a,b)
y=x(2,:);x=x(1,:);
X=[1-a*x.^2+y
    b*x];
end

Function to count boxes for fixed

function [dx,Nboxes]=countAndPlotBoxes(x,y,K)

Scale and reposition the point set so that it fits in the box [0,1]x[0,1]

xmax=max(x); xmin=min(x); width=xmax-xmin;
ymax=max(y); ymin=min(y); height=ymax-ymin;
L=max(height,width);
y=(y-ymin)/L;
x=(x-xmin)/L;

Define the grid size

dx=1/K;

Define a square box with unit-length sides

xbox=[0 1 1 0 0];
ybox=[0 0 1 1 0];

Plot the point set

figure;clf; hold on;
plot(x,y,'.','markersize',4)

The main step:

Loop through all the sub-boxes of length and width dx. Set isFilled(row,col) to 1 if there is a point in the box, and to zero if not. If it is filled, plot a pink square.

Note that this makes use of two built-in MATLAB commands

inpolygon tests whether each point (x(row,col),y(row,col)) is in the box defined by xv and yv. This is vectorized and avoids the need for loops. (Not something you'd use very often.)
any returns 1 if any elements of a matrix are nonzero and returns 0 otherwise. The all command is similar. These are very useful utilities.

isFilled=zeros(K,K);
for row=1:K
    for col=1:K
        xv=(xbox+col-1)*dx;
        yv=(ybox+row-1)*dx;
        isFilled(row,col)=any(inpolygon(x,y,xv,yv));
        if isFilled(row,col)
            patch(xv,yv,zeros(1,5),'facecolor',[1 .7 .7],'edgecolor','k','facealpha',0.7)
        end
    end
end
Nboxes=sum(isFilled(:)); %count the number of filled boxes
axis equal tight off
title(sprintf('$\\epsilon=%0.3f, N_{\\rm boxes}=%i$',dx,Nboxes))
end