So far we have discussed four measurements that allow us to visualize and identify the attractor coming from a nonlinear process:
To get a time series we hook up the output signal from the
nonlinear process to an input channel of the oscilloscope and use
the time base of the scope to generate the temporal dimension of the
plot. To obtain a power spectrum,
we use a spectrum analyzer or digitize the data and use an FFT.
An experimental phase space portrait can be plotted on an oscilloscope
either by recording two system variables directly, such as (x, y) or 
, or from the delayed variable 
, where 
is the
delay time. Lastly, in a forced system, the Poincaré section is
obtained from the phase space trajectory by strobing it once each
forcing period using the ``z'' blanking on the back of the oscilloscope.
Now to identify an attractor, we monitor these four diagnostic tools as we vary a system parameter. A bifurcation point is easy to identify by using these tools, and the existence of a particular bifurcation sequence, say a sequence of period doubling bifurcations, is a strong indicator for the possible existence of chaotic motion.
The use of these diagnostic tools is illustrated schematically in Figure 3.27 for the period doubling route to chaos .
Figure 3.27: Period doubling route to chaos:
(a) period one, (b) period two, (c) period four, and (d) chaotic.
(Adapted from Tredicce and Abraham [1].)
For the parameter values 
, 
, and
, an examination of any one of these diagnostics is sufficient
to identify the existence of a periodic motion, as well as its period.
For 
, these four diagnostics, as well
as the fact that the strange Poincaré section arose from
a sequence of bifurcations from a periodic state, support the claim that the
motion is chaotic.
In particular, the Poincaré section is useful for distinguishing low-dimensional chaos from noise. The chaotic Poincaré section illustrated in Figure 3.27 resembles the familiar one-humped map studied in Chapter 2. In contrast, the Poincaré map for a noisy signal from a stochastic process fills the whole oscilloscope screen with a random collection of dots (see Fig. 3.28).
Figure 3.28: Poincaré maps from periodic, quasiperiodic, chaotic, and
noisy (random) processes.
Thus motion on a strange attractor has a strong spatial correlation not present in a purely random signal. In the next section we will quantify this observation by introducing the correlation dimension, a measure that allows us to distinguish whether the signal is coming from a low-dimensional strange attractor or from noise.
Figure 3.29: Quasiperiodic route to chaos: (a) equilibrium, (b) periodic, (c)
quasiperiodic, and (d) chaotic.
(Adapted from Tredicce and Abraham [1].)
The quasiperiodic route to chaos is illustrated in Figure 3.29. In this case, the phase portrait for a quasiperiodic motion would resemble a Lissajous pattern, which may be hard to distinguish from a slowly evolving chaotic orbit. The Poincaré map, on the other hand, is useful in distinguishing between these two cases. The sequence of dots forming the Poincaré map in the quasiperiodic regime lie on a closed curve that is easy to distinguish from the spread of points in the Poincaré map for a strange attractor.