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Statement of problem:

Taken's theorem shows that we can project a version of the stable attractor for the Lorenz system by looking at a time series form. (The theory is not so important in this case, I'm more concerned with the algorithm I'm implementing on matlab and making it work.)

The Lorenz system of course is given by $\begin{equation}\begin{cases}\frac{dx}{dt}=\sigma y-\sigma x\\ \frac{dy}{dt}=x\rho-y-xz\\ \frac{dz}{dt}=xy-\beta z\end{cases}\end{equation}$.

I'm only considering the case for parameters given by $\sigma=10,\,\rho=28,\,\beta=8/3$.

So if $(x(t),y(t),z(t))$ is a solution, then the mapping has the following simplified form $$\phi_{\tau}(x(t),y(t),z(t))=\langle x(t),x(t-\tau),x(t-2\tau),...,x(t-K\tau)\rangle.$$

Here $\tau>0$. So I'm trying to implement the time delay mapping on matlab for values $K=1$ and $K=2$ and subsequently find the value $\tau$ that will give me the right version of the attractor. By the way, I used euler's method to solve the Lorenz system in this case.

function lorenz
sigma=10;
rho=28;
beta=8/3;
h=0.1;
n=10;
tau=1; 
x=[0;-4;-1];%initial conditions% 
for i=1:n
    z=x(:,i)+h*euler(x(:,i),sigma,rho,beta); 
    x=[x z];
    if i>=3
        x(2,:)=x(1,i-tau);
        x(3,:)=x(1,i-2*tau);
    end
end 
figure(1)
plot3(x(1,:),x(2,:),x(3,:));
figure(2)
plot(x(1,:));

function x=euler(x_old,sigma,rho,beta)
 x=[0;0;0];
 x(1,1)= sigmax_old(2,1)-sigmax_old(1,1);
 x(2,1)= rhox_old(1,1)-x_old(2,1)-x_old(1,1)x_old(3,1);
 x(3,1)= x_old(1,1)x_old(2,1)-betax_old(3,1);

So am I doing this right? I'm getting it to graph but it's not the same as the Lorenz attractor

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user5896534
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  • Euler is not suitable for chaotic dynamics, if not divergent you get a chaotic dynamic that is rather unrelated to the original system. The step size $0.1$ is too large here even for higher order methods. Good results are obtained for RK4 and step size $0.05$ or smaller. The direct plot suffers from the similarity of the samples, I got good results plotting the corresponding iterated differences, see https://math.stackexchange.com/questions/3587778/lorenz-equations-embedding-and-takens-theorem. – Lutz Lehmann Oct 17 '21 at 07:02
  • I'm having trouble coding the time delay map, is there a trick to doing it? It's saying there's an invalid use of operator. – user5896534 Oct 19 '21 at 00:54

0 Answers0