function lorenzgui
%LORENZGUI Plot the orbit around the Lorenz chaotic attractor.
% This function animates the integration of the three coupled
% nonlinear differential equations that define the Lorenz Attractor,
% a chaotic system first described by Edward Lorenz of MIT.
% As the integration proceeds you will see a point moving in
% an orbit in 3-D space known as a strange attractor.
% The orbit ranges around two different critical points, or attractors.
% The orbit is bounded, but may not be periodic and or convergent.
%
% The mouse and arrow keys change the 3-D viewpoint. Uicontrols
% provide "pause", "resume", "stop", "restart", "clear", and "close".
%
% A listbox provides a choice among five values of the parameter rho.
% The first value, 28, is the most common and produces the chaotic
% behavior. The other four values values produce periodic behaviors
% of different complexities. A change in rho becomes effective only
% after a "stop" and "restart".
%
% Reference: Colin Sparrow, "The Lorenz Equations: Bifurcations,
% Chaos, and Strange Attractors", Springer-Verlag, 1982.
if ~isequal(get(gcf,'name'),'Lorenz Gui')
% This is first entry, just initialize the figure window.
rhos = [28 99.65 100.5 160 350];
shg
clf reset
p = get(gcf,'pos');
set(gcf,'color','black','doublebuff','on','name','Lorenz Gui', ...
'menu','none','numbertitle','off', ...
'pos',[p(1) p(2)-(p(3)-p(4))/2 p(3) p(3)])
% Callback to erase comet by jiggling figure position
klear = ['set(gcf,''pos'',get(gcf,''pos'')+[0 0 0 1]), drawnow,' ...
'set(gcf,''pos'',get(gcf,''pos'')-[0 0 0 1]), drawnow'];
% Uicontrols
paws = uicontrol('style','toggle','string','start', ...
'units','norm','pos',[.02 .02 .10 .04],'value',0, ...
'callback','lorenzgui');
stop = uicontrol('style','toggle','string','close', ...
'units','norm','pos',[.14 .02 .10 .04],'value',0, ...
'callback','cameratoolbar(''close''), close(gcf)');
clear = uicontrol('style','push','string','clear', ...
'units','norm','pos',[.26 .02 .10 .04], ...
'callback',klear);
rhostr = sprintf('%6.2f|',rhos);
rhopick = uicontrol('style','listbox','tag','rhopick', ...
'units','norm','pos',[.82 .02 .14 .14], ...
'string',rhostr(1:end-1),'userdata',rhos,'value',1);
else
% The differential equation is ydot = A(y)*y
% With this value of eta, A is singular.
% The eta's in A will be replaced by y(2) during the integration.
rhopick = findobj('tag','rhopick');
rhos = get(rhopick,'userdata');
rho = rhos(get(rhopick,'value'));
sigma = 10;
beta = 8/3;
eta = sqrt(beta*(rho-1));
A = [ -beta 0 eta
0 -sigma sigma
-eta rho -1 ];
% The critical points are the null vectors of A.
% The initial value of y(t) is near one of the critical points.
yc = [rho-1; eta; eta];
y0 = yc + [0; 0; 3];
% Integrate forever, or until the stop button is toggled.
tspan = [0 Inf];
opts = odeset('reltol',1.e-6,'outputfcn',@lorenzplot,'refine',4);
ode45(@lorenzeqn, tspan, y0, opts, A);
end
% ------------------------------
function ydot = lorenzeqn(t,y,A)
%LORENZEQN Equation of the Lorenz chaotic attractor.
% ydot = lorenzeqn(t,y,A).
% The differential equation is written in almost linear form.
% ydot = A*y
% where
% A = [ -beta 0 y(2)
% 0 -sigma sigma
% -y(2) rho -1 ];
A(1,3) = y(2);
A(3,1) = -y(2);
ydot = A*y;
% ------------------------------
function fin = lorenzplot(t,y,job,A)
%LORENZPLOT Plot the orbit of the Lorenz chaotic attractor.
persistent Y
if isequal(job,'init')
% Initialize axis and comet, R = axis settings, L = length of comet.
rho = A(3,2);
switch rho
case 28, R = [ 5 45 -20 20 -25 25]; L = 100;
case 99.65, R = [ 50 150 -35 35 -60 60]; L = 240;
case 100.5, R = [ 50 150 -35 35 -60 60]; L = 120;
case 160, R = [100 220 -40 40 -75 75]; L = 165;
case 350, R = [285 435 -55 55 -105 105]; L = 80;
otherwise, R = [100 250 -50 50 -100 100]; L = 150;
end
set(gcf,'pos',get(gcf,'pos')+[0 0 0 1])
drawnow
set(gcf,'pos',get(gcf,'pos')-[0 0 0 1])
drawnow
if get(gca,'userdata') ~= rho, delete(gca), end
set(gca,'color','black','pos',[.03 .05 .93 .95],'userdata',rho)
axis(R);
axis off
comet(1) = line(y(1),y(2),y(3),'linestyle','none','marker','.', ...
'erasemode','xor','markersize',25);
comet(2) = line(NaN,NaN,NaN,'color','y','erasemode','none');
comet(3) = line(NaN,NaN,NaN,'color','y','erasemode','none');
Y = y(:,ones(L,1));
uics = flipud(get(gcf,'children'));
paws = uics(1);
stop = uics(2);
set(paws,'string','pause','callback','','value',0);
set(stop,'string','stop','callback','','value',0);
beta = -A(1,1);
eta = sqrt(beta*(rho-1));
yc = [rho-1; eta; eta];
line(yc(1),yc(2),yc(3),'linestyle','none','marker','o','color','g')
line(yc(1),-yc(2),-yc(3),'linestyle','none','marker','o','color','g')
ax = [R(2) R(1) R(1) R(1) R(1)];
ay = [R(3) R(3) R(4) R(3) R(3)];
az = [R(5) R(5) R(5) R(5) R(6)];
p = .9;
q = 1-p;
grey = [.4 .4 .4];
line(ax,ay,az,'color',grey);
text(p*R(1)+q*R(2),R(3),p*R(5),sprintf('%3.0f',R(1)),'color',grey)
text(q*R(1)+p*R(2),R(3),p*R(5),sprintf('%3.0f',R(2)),'color',grey)
text(R(1),p*R(3)+q*R(4),p*R(5),sprintf('%3.0f',R(3)),'color',grey)
text(R(1),q*R(3)+p*R(4),p*R(5),sprintf('%3.0f',R(4)),'color',grey)
text(R(1),R(3),p*R(5)+q*R(6),sprintf('%3.0f',R(5)),'color',grey)
text(R(1),R(3),q*R(5)+p*R(6),sprintf('%3.0f',R(6)),'color',grey)
fin = 0;
cameratoolbar('setmode','orbit')
uicontrol('style','text','units','norm','pos',[.38 .02 .34 .04], ...
'foreground','white','background','black','fontangle','italic', ...
'string','Click on axis to rotate view')
elseif isequal(job,'done')
fin = 1;
else
% Update comet
L = size(y,2);
Y(:,end+1:end+L) = y;
comet = flipud(get(gca,'children'));
set(comet(1),'xdata',Y(1,end),'ydata',Y(2,end),'zdata',Y(3,end));
set(comet(2),'xdata',Y(1,2:end),'ydata',Y(2,2:end),'zdata',Y(3,2:end))
set(comet(3),'xdata',Y(1,1:2),'ydata',Y(2,1:2),'zdata',Y(3,1:2))
Y(:,1:L) = [];
drawnow;
% Pause and restart
uics = flipud(get(gcf,'children'));
paws = uics(1);
stop = uics(2);
rhopick = uics(4);
rho = A(3,2);
while get(paws,'value')==1 & get(stop,'value')==0
set(paws,'string','resume');
drawnow;
end
set(paws,'string','pause')
fin = get(stop,'value') | get(rhopick,'value')==rho;
if fin
set(paws,'value',0,'string','restart','callback','lorenzgui')
set(stop,'value',0,'string','close', ...
'callback','cameratoolbar(''close''), close(gcf)')
end
end