Introduction to Acoustics

Fig. 8.26 The point set of a strange attractor of an experimental,
driven pendulum. (Courtesy of M. Kaufmann)
points of the set M inside the ball of radius r around
P(Fig.8.24). Examples for a line and a surface give
the right integer dimensions of one and two (Fig. 8.25),
because for a line we have N(r) ∼ r 1 for r → 0 and thus
D = 1, and for a surface we have N(r) ∼ r 2 for r → 0
and thus D = 2.
Chaotic or strange attractors usually give a fractal
(noninteger) dimension. The pointwise dimension may
not be the same in this case for each point P. The distribution
of dimensions then gives a measure of the
inhomogeneity of the point set. Figure 8.26 gives an
example of a fractal point set. It has been obtained by
sampling the angle and the angular velocity of a pendulum
at a given phase of the periodic driving for
parameters in a chaotic regime.
To investigate the dynamic properties of a (strange
or chaotic) set the succession of the points must be
retained. As chaotic dynamics is brought about by
a stretching and folding mechanism and shows sensitive
dependence on initial conditions (see, e.g., [8.32])
the behavior of two nearby points is of interest: do
they separate or approach under the dynamics? This
L(t 0)
t 0
L'(t 1)
L(t 1)
L'(t 2)
L(t 2)
t 1 t 2
L'(t 3)
t 3
Fiducial trajectory
Fig. 8.27 Notations for the definition of the largest Lyapunov
exponent
Surrogate
data
• Dimensions
• Lyapunov
exponents
• Statistical
analysis
Nonlinear Acoustics in Fluids 8.12 Acoustic Chaos 291
System
Measurement
Time series
Embedding
Reconstructed
state space
Characterization
Diagnosis
Linear
time series
analysis
Linear filter
Nonlinear
noise reduction
• Modelling
• Prediction
• Controlling
Fig. 8.28 Operations in nonlinear time-series analysis.
(Courtesy of U. Parlitz)
behavior is quantified by the notion of the Lyapunov exponent.
In Fig. 8.27 the calculation scheme [8.134] is
depicted. It gives the maximum Lyapunov exponent via
the formula
λmax = 1
tm − t0
m�
L ′
(tk)
log2 L(tk−1)
k=1
� bits
s
�
. (8.224)
As space does not permit us to dig deeper into the
operations possible with embedded data, only a graph
depicting the possible operations in nonlinear timeseries
analysis is given in Fig. 8.28. Starting from the
time series obtained by some measurement, the usual approach
is to do a linear time-series analysis, for instance
by calculating the Fourier transform or some correlation.
The new way for chaotic systems proceeds via
embedding to a reconstructed state space. There may be
some filtering involved in between, but this is a dangerous
operation as the results are often difficult to
predict. For the surrogate data operation and nonlinear
noise reduction the reader is referred to [8.131–133].
From the characterization operations we have mentioned
here the dimension estimation and the largest
Lyapunov exponent. Various statistical analyses can be
done and the data can also be used for modeling, prediction
and controlling the system. Overall, nonlinear time
series analysis is a new diagnostic tool for describing
(nonlinear) systems.
Part B 8.12