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the real time delay is ˄=ʔ t ). h is reconstruction of the phase space is called
time delay embedding . h e reconstruction of the phase space is not exactly the
same as the original phase space, but its topological properties are preserved
provided that the embedding dimension is sui ciently large. In practice,
the embedding dimension must be more than twice the dimension of the
attractor (i.e., m >2 d +1). h e reconstructed trajectory is then sui ciently
accurate for subsequent data analysis.
As an example we now explore the phase space portrait of a harmonic
oscillator such as an undamped pendulum. We i rst create the position
vector x1 and the velocity vector x2
clear
t = 0 : pi/10 : 3*pi;
x1 = sin(t);
x2 = cos(t);
h e phase space portrait
plot(x1,x2)
xlabel('x_1')
ylabel('x_2')
is a circle, suggesting an exact recurrence of each state at er one complete
cycle (Fig. 5.19). Using the time delay embedding we can reconstruct this
phase space portrait using only a single observation, e.g., the velocity vector,
and a time delay of i ve, which corresponds to a quarter of the period of our
pendulum.
tau = 5;
plot(x2(1:end-tau),x2(1+tau:end))
xlabel('x_1')
ylabel('x_2')
As we can see, the reconstructed phase space is almost the same as the
original phase space. Next, we compare this phase space portrait with one
for a typical nonlinear system, the Lorenz system (Lorenz 1963). Weather
patterns ot en to not change in a predictable manner. In 1963, Edward
Lorenz introduced a simple three-dimensional model to describe the
chaotic behavior exhibited by turbulence in the atmosphere. h e variables
dei ning the Lorenz system are the intensity of atmospheric convection, the
temperature dif erence between ascending and descending currents, and the
distortion of the vertical temperature proi les from linearity. Small variations
in the initial conditions can cause dramatically divergent weather patterns, a
behavior ot en referred to as the butterl y ef ect. h e dynamics of the Lorenz
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