In the string system there is no difficulty in specifying and measuring the major system variables. Usually, though, we are not so lucky. Imagine an experimental dynamical system as a black box that generates a time series, x(t). In practice, we may know little about the process inside the black box. Therefore, the experimental construction of a phase space trajectory, or a Poincaré map, seems very problematic.
One approach to this problem--the experimental reconstruction of a phase
space trajectory --is
as follows. We start out by assuming that the
time series is produced by a deterministic dynamical system that can
be modeled by some nth-order ordinary differential equation.
For this particular example we assume that the system is
modeled by a third-order differential system, as is the case for planar
string vibrations. Then a given trajectory of the
system is uniquely specified by the value of the time series and its
first and second derivatives at time 
:
This suggests that in reconstructing the phase space we can begin with our measured time series, and then use x(t) to calculate two new phase space variables y(x(t)) and z(x(t)) defined by
In estimating the pointwise derivatives of x(t) in an experiment we can
proceed in at least two ways: first, we can process the
original signal through a differentiator, and then record (digitize) the
original signal along with the differentiated signals; or second,
we could digitize the signal, and then compute the derivatives
numerically. While both techniques are feasible, each is
fraught with experimental difficulties because
differentiation is an inherently noisy process. This is
because approximating a derivative often involves taking the
difference of two numbers that are close in value. To see this, consider
the numerical derivative of a digitized time series 
defined by
For instance, let 
, 
,
and 
.
Then 
; that is, the value of the first derivative is
already buried in the noise, and the problem just gets worse when
taking higher-order derivatives.
However, let's look at equation (3.59) again. If the sampling time
is evenly spaced, then 
is constant, so
That is, almost all the information about the derivative is
contained in a variable constructed by taking the difference of two points
in the original time series . This idea can be generalized
as follows. Instead of defining the new variables for the reconstructed
phase space in terms of the derivatives, we can recover almost all of the same
information about an orbit from the embedded variables defined by [26]:
where r and s are integers. Each new embedded variable is defined by taking a time delay of the original time series. Clearly, any number of embedded variables can be created in this way, and this method can be used to reconstruct a phase space of dimension far greater than three.
There are many technical issues associated with the construction of an embedded phase space. An in-depth discussion of these issues can be found in reference [1]. The first concern is determining a good choice for the delay times , r and s. One rule of thumb is to take r to be small, say 3 or 4. In fact, we could define the second variable as
and think of it as a velocity variable. The second embedding time should be much larger than r, but not too large. To be more specific, consider the planar oscillations of a string again. In this example a natural cycle time is given by the period of the forcing term. A good choice for s is some sizable fraction of the cycle time. For instance, let's say our digitizer is set to sample the signal 64 times each period. Then a sensible choice for r might be 4, and for s might be 16, or one-quarter of the forcing period. Figure 3.24 shows a plot of a chaotic trajectory in the Duffing oscillator in both the original phase space and the phase space reconstructed from the embedding variables.
Figure 3.24: Trajectory of the Duffing oscillator in (a) phase space and (b)
the embedded phase space with delay time 
.
The similarity of the two representations lends support to the claim that a trajectory in the embedded phase space provides a faithful representation of the dynamics.
Constructing a real-time two-dimensional embedded phase space is straightforward. An embedded signal is obtained by sending the original signal through a delay line. The current signal and the delayed signal are sent to the oscilloscope, thereby giving us a real-time representation of the phase space dynamics from our black box.
