Attractor reconstruction for the Lorenz equations

Simulate the Lorenz equations, outputing the results with small uniform spacing in time.

sigma=10; b=8/3; r=28;

f=@(t,y)lorenzODE(t,y,sigma,r,b);

fixedPoint=[sqrt(b*(r-1)),sqrt(b*(r-1)),r-1];

perturbation=[1 1 1];

y0=fixedPoint+perturbation;

tFinal=100;

tTransient=20;

dt=0.01;

tspan=0:dt:(tTransient+tFinal); %Note that by specifying tspan with more than two elements, this gives the output array t

[t,y]=ode45(f,tspan,y0);

p=find(t>=tTransient,1,'first');

t=t(p:end);y=y(p:end,:);

plot3(y(:,1),y(:,2),y(:,3))

xlabel('$x$'); ylabel('$y$');ylabel('$y$')

Reconstruct using time-delay embedding

You can move the slider back and forth to change the lag in the time delay

x=y(:,1);

dim=3;

lag=7;

ps=[];

for i=1:dim

ps(:,i)=x(1+(i-1)*lag:end-(dim-(i-1))*lag);

end

plot3(ps(:,1),ps(:,2),ps(:,3))

xlabel('$x(t)$'); ylabel('$x(t+\tau)$');zlabel('$x(t+2\tau)$')

title(sprintf('$\\tau=%0.2f$',lag*dt))

Function definitions

The ordinary differential equation

function dydt= lorenzODE(~,y,sigma,r,b)
dydt=zeros(3,1);
dydt(1) = sigma*(y(2)-y(1));
dydt(2) = r*y(1)-y(2) - y(1)*y(3);
dydt(3)= y(1)*y(2)-b*y(3);
end