Geoscience Reference
In-Depth Information
a
b
Fig. 5.21 a
h e phase space portrait for the Lorenz system. In contrast to the simple periodic
system, the trajectories of the Lorenz system obviously do not follow precisely the previous
course, but recur very close to it.
b
h e reconstruction of the phase space portrait using only
the i rst state and a time delay of 6 reveals a topologically similar phase portrait to a, with the
two typical ears.
tau = 6;
plot3(x(1:end-2*tau,1),x(1+tau:end-tau,1),x(1+2*tau:end,1))
grid, view([100 60])
xlabel('x_1'), ylabel('x_2'), zlabel('x_3')
reveals a similar phase portrait with the two typical ears (Fig. 5.21). h e
characteristic properties of chaotic systems can also be observed in this
reconstruction.
h e time delay and embedding dimension need to be chosen from a
previous analysis of the data. h e delay can be estimated with the help of the
autocovariance or autocorrelation function. For our example of a periodic
oscillation,
Movie
5.3
clear
t = 0 : pi/10 : 3*pi;
x = sin(t);
we compute and plot the autocorrelation function
for i = 1 : length(x) - 2
r = corrcoef(x(1:end-i),x(1+i:end));
C(i) = r(1,2);
end
plot(C)
