Confirmation of chaos in a loudspeaker system using time series analysis

Confirmation of chaos in a loudspeaker system using time series analysis
Audio Engineering Society
Convention Paper
Presented at the 125th Convention
2008 October 2–5
San Francisco, CA, USA
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Confirmation of chaos in a loudspeaker
system using time series analysis
1
Joshua D. Reiss1, Ivan Djurek2, Antonio Petosic2 and Danijel Djurek3
Centre for Digital Music, Queen Mary, University of London, Mile End Road, London, E14NS, U. K.
[email protected]
2
University of Zagreb, Faculty of Electrical Engineering and Computing, Dept. of Electroacoustics
Unska 3, Zagreb, Croatia
[email protected]; [email protected]
3
AVAC – Alessandro Volta Applied Ceramics, Laboratory for Nonlinear Dynamics
Kesten brijeg 5, Remete, Zagreb, Croatia
[email protected]
ABSTRACT
The dynamics of an experimental electrodynamic loudspeaker is studied by using the tools of chaos theory and time
series analysis. Delay time, embedding dimension, fractal dimension and other empirical quantities are determined
from experimental data. Particular attention is paid to issues of stationarity in the system in order to identify sources
of uncertainty. Lyapunov exponents and fractal dimension are measured using several independent techniques.
Results are compared in order to establish independent confirmation of low dimensional dynamics and a positive
dominant Lyapunov exponent. We thus show that the loudspeaker may function as a chaotic system suitable for low
dimensional modeling and the application of chaos control techniques.
1.
INTRODUCTION
Loudspeakers are the most variable elements in any
audio system, and are responsible for marked audible
differences between otherwise identical sound systems.
Loudspeaker performance (i.e., their accuracy in
reproducing a signal without adding distortion) is
significantly poorer than that of other audio equipment.
For example, harmonic distortion in a typical
loudspeaker can be 100 to 1000 times greater than that
of amplifiers[1]. The frequency response of a
loudspeaker is often referenced as being within ±3 dB
of perfect linearity (and many speaker designs fall
further outside this range), whereas an amplifier may
vary less than 0.1 dB.
An electrodynamic loudspeaker consists of a membrane
suspended to a fixed rim and put in motion by the
Reiss, Djurek, Petosic and Djurek
Chaos in a loudspeaker
Lorentz force exerted on the voice coil positioned in the
field of a permanent magnet. The Lorentz force
oscillates in the same phase and frequency as the current
generated by the sound radiation, and it is commonly
accepted that the membrane vibrates with amplitude
being linearly dependent on the amplitude of the input
AC signal, while natural frequency is expected to be
independent of vibration amplitude, i.e. the system
operates in a linear regime. However, this is true only
for small driving AC currents (<10 mA). For higher
currents the vibration amplitude deviates from the linear
dependence and natural frequency changes with
changing amplitude of the input signal.
It is of great importance to study such a nonlinear
system in terms of the laws and rules of widely quoted
nonlinear phenomena, including stability theories. Their
development makes possible the use of new physical
tools in understanding the loudspeaker vibration
properties. Thus an improved understanding of the
dynamics of a loudspeaker is of great importance, since
when taking into account the nonlinear phenomena, a
loudspeaker might be better designed and yield
significantly better performance in an audio playback
system.
Evidence of possible chaotic behavior in a loudspeaker
was first observed by Wei, et al in 1986[2], where the
appearance of subharmonics and broadband spectra at
various drive frequencies and voltages was noted. Tong,
et al[3] also identified chaos through the measurement
of Lyapunov exponents and fractal dimension, although
their results were not held up to scrutiny and may be
considered unreliable (for instance, they do not report
how many data points were used and the results are not
repeatable). Recent work strengthened the conjecture of
chaotic behavior [4-6] with the observation of
hysteresis, also reported in [7], and period doubling
when the loudspeaker is driven at low frequencies. This
in general is not observed in the models and thus it is
important to verify chaos and adjust the models
accordingly.
The question arises whether study of the chaotic state
may be useful in the construction of the loudspeaker. It
was found[6] that chaos appears in loudspeakers with
comparatively high intrinsic friction (Ri ~ 0.6-0.7 kg/sec
measured at driving current I0 = 10-100 mA), as
evaluated from the resonance line width. This should be
compared to a non-chaotic loudspeaker with Ri ~ 0.170.20 kg/sec, and high intrinsic friction would normally
be an indication of the poor quality of the loudspeaker.
However, resonance line width by itself does not
necessarily imply a reduction of the harmonic nature of
the loudspeaker, when considered as a forced oscillator.
Of primary importance is the dependence of the intrinsic
friction on the vibration amplitude. This dependence
contributes to the temperature rise on the membrane
surface. This is low in high quality loudspeakers, due to
the high heat capacity of the membrane material. The
temperature rise in turn reduces the thermo-elastic noise
in the membrane and contributes to the quality of sound
reproduction.
A dynamical system is represented by a set of nonlinear
equations[8, 9] given in the form
dψ i
= Fi (ψ 1 ⋅⋅⋅ψ 2 )
i = 1,.., N
(1)
dt
which might be applied to a nonlinear forced oscillator
described by the equation of motion
d2x
dx
a 2 + b + k ⋅ x = f1 = f 0 ⋅ cos φ
dt
dt
(2)
dx
φ = ψ 2 = ωt , ψ 1 =
dt
In the linear regime when the electrodynamic
loudspeaker (EDL) is driven by a small current (I0~10
mA), the coefficients in equation (2) are nearly constant,
and stiffness k is expressed by the natural frequency
ω0 and mass M in the form k=ω02M. The solution to
equation (2) can be expressed as x=A·cosωt.
In this form, displacement x in equation (2) properly
describes only the motion of the voice coil, while the
points on the membrane, because of its flexibility, suffer
combined displacement x+z, z being the solution of the
Bessel equation which is given in the radial coordinate r
⎛ d 2 z 1 dz ⎞
d2z
d2z
ρ ⋅ 2 + E ⋅ Δz = ρ 2 − E ⎜ 2 +
(3)
⎟ = f2
dt
dt
r dr ⎠
⎝ dr
E and ρ are the Young modulus and density of the
membrane polymeric material, respectively. Equation
(3) expresses force density coming from the inertial and
elastic shear term. After integration over the membrane
volume both terms can be added to the left side of
equation (2).
However, the Bessel modes derived from equation (3)
do not contribute to the intrinsic friction of the
membrane, but such a friction results from the
viscoelastic losses and these losses are manifested by
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Chaos in a loudspeaker
fluctuations in the membrane surface visible as tilts[10].
Such tilts are stochastic surface fluctuations and are
associated to vibration modes having frequencies
extending up two orders of magnitude higher than
loudspeaker’s natural frequency.
In these experiments, stiffness k was evaluated from the
static measurements by the use of calibrated loads and
the
evaluation
of
corresponding
membrane
displacements. It was found that stiffness has a form
(m = 930 Νm-1, n = −114·103 Νmk=m+n·x+p·x2
3
-3
2
, p = 32·10 Nm ), and k obtains a minimum value at
x~1.8 mm. A short analysis of equation (2) shows that it
would be exceedingly difficult to explain amplitude
bifurcations and the existence of a chaotic state in the
loudspeaker by this nonlinearity, even if coefficients m,
n, and p are varied over a broad range of values far
from the commonly accepted values indicated by the
technical performance of the loudspeaker. Instead, the
only effect observed by the use of simulations based
upon this quadratic nonlinearity was the well known
amplitude cut-off[11], as shown in the inset of Figure 1.
Another experiment showed that the membrane's
properties play an important role in the dynamics of the
system. The same membrane was reinforced by
phenolic resin[12], leaving the elastic suspension intact.
Static measurement of the stiffness after the
reinforcement revealed that k increased to about 1230
Nm-1 at the origin, and this is to be contrasted with the
concept of the EDL membrane as being a point mass
suspended on an elastic spring. Furthermore, an increase
of the stiffness by such reinforcement suggests a
substantial contribution of membrane elasticity to the
vibration properties of the EDL.
frequency calculated from the formula k=ω02·M, k being
measured statically by the use of the calibrated loads.
An initial decrease of the calculated resonance
frequency from static measurements correlates to a
decrease of the resonant frequency obtained from
dynamic measurements, but a strong increase of the
latter at A>5 mm might not be explained bythe simple
quadratic nonlinearity in equation (2). Properties of the
membrane material also play an important role in the
restoring force.
In this respect, investigation of the dynamics of the
chaotic state can provide very useful information
concerning the composition and elastic properties of the
membrane. This, in turn, could enable important
improvements to the membrane design.
Figure 1. Frequency dependence of the impedance of the
loudspeaker. The inset shows the frequency dependence of
the vibration amplitude for driving current I0=4 A.
In addition to the elastic properties of the membrane
governed by the Young modulus, viscoelasticity of the
composite polymer material also plays an important
role. Viscoelastic losses in the membrane are expressed
by intrinsic membrane friction Ri entering the second
term in equation (2), and these losses are brought about
as a hysteresis in the stress-strain diagram of the
membrane material. Systems with such hysteresis obey
memory properties which manifest in the loudspeaker as
a time dependent stiffness[13], and in the literature
dealing with stochastic processes are commonly referred
as an after effect.
Furthermore, one has to consider that a resonance
frequency dependent on vibration amplitude (Figure 2)
deviates from the dependence of the resonance
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Chaos in a loudspeaker
Figure 3. Experimental set up with loudspeaker and laser
distance meter.
Figure 2. Amplitude dependent resonant frequency; (○)
calculated from the statically measured stiffness, (●)
measured in dynamic regime. Inset shows current
dependent intrinsic membrane friction Ri.
In this work, we analyze time series from an
experimental electrodynamic loudspeaker system. We
use a variety of techniques from chaotic time series
analysis[14] to show that the system is indeed chaotic
and exhibits low dimensional dynamics suitable for
further analysis and the implementation of chaos control
techniques. By quantifying the nonlinear behavior, we
also provide empirical observations which may be used
to refine the modeling and design of loudspeakers.
2.
EXPERIMENTAL SET-UP
In these experiments, a low frequency loudspeaker was
used with a resonant frequency, recorded in air, of f =
38 Hz, driving current I0 = 10 mA, factor B·l = 3.9 Tm,
membrane diameter 2R = 16 cm, rated RMS power of
60 W, and nominal impedance of 8 ohms. According to
the manufacturer’s data, the voice coil inductance is L =
0.9 mH, and the contribution of the inductive part ω·L to
the loudspeaker's impedance can be neglected for
driving frequencies f < 100 Hz.
The experimental set-up is depicted in Figure 3. The
loudspeaker was placed in a stainless steel chamber. Air
pressure within the chamber was measured by the use of
an absolute capacitive gauge with ultimate resolution
0.01 mbar. A glass window on the top of the chamber
ensured the transparency to the light beam from the
laser distance meter which measured vibration
amplitude with an accuracy of 2 μm, and sampling
frequency of ~ 1 kHz (a similar measurement apparatus
was used to analyse nonlinear vibrations of a
loudspeaker in Wei, et al[2]). The A/D converter
resolution of the laser distance meter was 8 bits. This
allowed the acquisition of 128 amplitude levels in both
up and down vibration directions. In order to check the
possible influence of the chamber wall friction on the
course of measurements, the impedance and vibration
amplitude data were recorded at 1 bar air pressure in
closed chamber and compared to those evaluated in free
laboratory atmosphere. In the frequency range near 50
Hz recorded data, notably impedance, showed no
significant difference.
For impedance and amplitude measurements, the
loudspeaker was connected to an audio amplifier with
rated power of 300W via a series resistor 0.44 Ω. A
rather small resistor value was used because of the
possibility of driving higher currents. However, small
resistance gave rise to increased influence of the back
electromotive force. A satisfactory compromise was
found with the total voltage swing across the
loudspeaker, clipping not included, of +/- 50 V, which
in turn provided driving currents I0 = 4 to 5A. For
driving currents I0 < 100 mA, the back electromotive
force is comparable to the friction term in Eq. (2), while
for higher currents intrinsic friction increased and
became the dominant contribution to the impedance.
The loudspeaker vibration amplitude dependent on
frequency was measured for various driving currents in
an evacuated space and in normal atmospheric pressure
(1 bar)[15]. The data recorded in vacuo are shown in
Figure 4. By an increase of the driving current the
resonance curve became more and more distorted until
an amplitude downturn (cut-off) appeared near f = 43.5
Hz for I0 = 200 mA. This current indicated the starting
value for identification of the chaotic regime. The inset
in Figure 4 depicts the hysteretic property of the cut-off
effect. That is, cut-off frequencies differ for positive and
negative frequency sweep.
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Chaos in a loudspeaker
Onset of the amplitude cut-off was followed by a
subsequent frequency sweep which gave rise to the
erratic vibration amplitude. This unstable range
extended up to 54 Hz. This was chosen as the fixed
frequency for evaluation of the appearance of
subharmonics which precede the chaotic state. An
important feature of the unstable range was the
relatively small change of the impedance with
increasing driving current. In these experiments,
loudspeaker impedance stayed at ~ 11 Ω, irrespective of
whether the system was operated in air or in an
evacuated chamber. This in turn meant that the driving
system could be considered as a current source, even in
the case when the amplifier was used as a voltage
source.
Figure 4. Impedance measured in vacuo for various
driving currents. The inset shows the hysteresis of the cutoff frequency for positive and negative frequency sweeps.
Figure 5. A bifurcation diagram of vibration amplitudes
recorded in vacuo for fixed driving frequency 53 Hz and
20 mA/sec sweeping driving current.
A bifurcation diagram, as shown in Figure 5, was
produced by fixing the frequency at f = 53 Hz, sweeping
the driving current from I0 = 1.5 A to 2.5A at a rate of
20 mA/sec, and recording the amplitude of the
displacement of the loudspeaker. At I0 ~ 1.81, a first
bifurcation pitchfork appeared, which was followed by
multiple period doublings, until at I0 = 2.15 A the
characteristic period-3 window appeared[16] and the
system vibrated with 3 amplitudes. Existence of period
doubling and a period 3 window was a strong indicator
of chaos.
However, the period 3 window was observed only in an
evacuated space, and data from a loudspeaker operating
in an evacuated chamber is unsuitable for time series
analysis techniques. This is primarily because the voice
coil bonding agent evaporates when in a vacuum, which
in turn changes the resonant frequency during the course
of the measurements. In addition, for long time and
heavy duty loudspeaker operation it is important to
remove heat from the loaded voice coil, and this is more
easily accomplished in an air atmosphere.
Measurements in 1 bar air were performed in a closed
chamber, since this minimised parameter drift due to
free air convection in the laboratory. The driving
frequency was fixed at 56 Hz, and the driving current
was increased up to values when higher harmonics
appeared as a result of the nonlinear restoring term in
Eq. (2). Excerpts of the time series waveforms
representing various driving currents are given in Figure
6. Figure 6a shows the time dependent vibration
amplitude at the starting driving current I0 = 2.4 A,
when the recorded signal shows nearly sinusoidal
behavior. Figure 6b, c and d show new vibration
amplitudes (marked with triangles) which appear with
increasing driving current. The corresponding averaged
spectra over the whole time series given in Figure 7.
Figure 7a shows the expected behavior, with a
fundamental frequency corresponding to the drive
frequency. The first subharmonic appeared at 28 Hz, as
depicted in Figure 7b. Further increase of the driving
current resulted in a new subharmonic at 14 Hz (see
Figure 7c) and the subsequent appearance of broadband
behavior, Figure 7d.
The bifurcation diagram for measurements of vibration
amplitude in air is shown in Figure 8, and was produced
in the same manner as Figure 5. Vertical lines indicate
values of driving currents for which the Feigenbaum
ratio δ1/δ2 = 4.669 is fulfilled[17]. A period 3 window
cannot be seen, but this is not a contra-indicator of
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Chaos in a loudspeaker
chaos. Noise in the data acquisition system may obscure
the window, and existence of such a window is not
considered a necessary condition. Whereas heating the
voice coil makes long term measurements difficult for
the evacuated loudspeaker experiment, reverberation
and air circulation added noise to the short term
measurements used in generating the bifurcation
diagram of Figure 8.
Leaving the loudspeaker to operate at a driving
frequency of 45 Hz, the driving current was selected at a
value slightly below 2.8 A, at which point the first
subharmonic became attenuated. This indicated the
starting point for recording the vibration amplitude
included in time series analysis. A rather low frequency
of 45 Hz was selected because the dynamics appeared
less susceptible to parameter drift and nonstationary
behaviour in this range.
The time series analysis presented in the following
sections is derived from a 247,392 point experimental
flow data set. The data was recorded with 16bit
resolution, though this is further limited by the 8 bit
resolution of the laser distance meter. The sample rate
was 1024Hz, so that the data set is just over four
minutes long and there are approximately 22.76 samples
per drive period. In the results that follow, units are not
typically given on the measurements since they have
been scaled and transformed by the data acquisition
system.
Figure 6. Short time series plots of the vibration amplitude
recorded in 1 bar air for a fixed driving frequency of 56
Hz, and various driving currents. The triangles indicate
the appearance of period doubling, leading to aperiodic
behavior observed at 3.6A.
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Chaos in a loudspeaker
Figure 8. A bifurcation diagram of vibration amplitudes
recorded in 1 bar air for a driving current swept at 20
mA/sec and fixed driving frequency of 56 Hz.
3.
DATA ANALYSIS AND RESULTS
To analyse a 1 dimensional experimental data set using
chaotic time series analysis techniques, it is necessary to
transform the data using phase space reconstruction
techniques. If only one variable from the system can be
observed, X={X(1), X(2), ...X(N),...}, then a Ddimensional time series of length N, Y={Y(1), Y(2), ...
Y(N)} is constructed from the original time series using
a delay d as follows.
Y (n) = ( X (n), X (n + d ),... X ( n + ( D − 1) d ) )
(4)
where we have assumed that X contains at least N+(D1)d data points. If the time between samples represents
one period of data, then X represents time series
generated from a map, or Poincare section of the
system. In which case the delay d used to generate Y is
usually set to 1.
Figure 7. Average spectrum of vibration amplitude
recorded in 1 bar air for various driving currents
corresponding to Figure 6.
Assuming that the time series is stationary, that is, the
parameters which govern the dynamics are not
significantly changing over time, then with sufficient
data and the appropriate choice of the delay parameter d
and the embedding dimension D, Y will accurately
represent the dynamical behaviour of the system. Once
Y has been constructed, then further analysis of this
multidimensional time series may be used to estimate
various quantities related to the structure of the phase
space, such as the dimensionality of the attractor,
characterisation of the chaotic behavior or lack thereof,
and identification of chaotic orbits.
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Chaos in a loudspeaker
Figure 9. Nonstationary behavior of the time series.
Plotted are estimates of the maximum and minimum
values for overlapping windows of 1024 samples (one
second) from the experimental data.
In the following subsections, we construct delay
coordinate embeddings from the scalar time series, and
then use this technique to analyse the data and quantify
its dynamical system properties. We will use the
notation introduced above to describe the original scalar
time series, X={X(1), X(2), ...X(N)} and a delay
coordinate embedding of the time series, Y={Y(1), Y(2),
... Y(N)}.
3.1. Nonstationarity and long-term dynamics
A few simple tests were first performed that would
identify strong drifts in the data. Sliding windows of
varying length were applied to the data and statistical
quantities such as mean, maximum, minimum and
standard deviation were computed for each window. If
the data was truly stationary, then these quantities
would remain constant throughout the data. Results of
the drift in the maximum and minimum values of a one
second window (1024 data points) are depicted in
Figure 9.
Figure 10. Plot of the waveform for two methods of
sectioning the data. The top plot, part (a), depicts peak
amplitudes and the bottom plot, part (b), depicts time
intervals between zero crossings.
It can be seen that the dynamics of the system are not
entirely stationary. For instance, the maximum value
undergoes an upward trend, particularly near the
beginning of the time series. This nonstationarity was
also confirmed by the measurement of other statistical
quantities such as the skewness and kurtosis for
windowed data.
When the dynamics change over time in an
experimental system it is often difficult to determine the
cause. The behavior may be caused by long term
dynamics which are inherent to the system or by a
simple transient before settling into some behaviour.
However, this fluctuation is quite small in relation to the
full extent of the data (for instance, variation in the
maximum value is less than 2% the full scale of the
data) and thus, though it may affect the results of
chaotic time series analysis methods, it is still small
enough that the data is acceptable for analysis.
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Chaos in a loudspeaker
Figure 11. Two techniques for estimating an embedding
delay from the experimental data. The first method uses
the first minimum of the mutual information function and
the second uses the first zero crossing of the
autocorrelation function. The methods suggest a delay
between 6 and 7.
3.2. Poincaré sections
Figure 12. Results of the FNN routine as applied to the
original flow data and the two Poincare sections. An
appropriate embedding dimension is found when the
percentage of false near neighbors drops to a constant
value. This indicates that the embedding dimension should
be at least 4.
R(d ) =
A common technique in chaotic time series analysis is
to generate a Poincare section, with one point per
period, from data sampled at much higher than the drive
frequency. Given that the system has a drive frequency
of 45Hz, a natural Poincaré section would be to sample
the system at the drive frequency. Since this was not
possible due to the limited sampling frequencies of the
data acquisition system, we considered several
techniques for extracting a Poincaré section. These
included the peak amplitude values, their second
derivatives, times between peaks and times at which the
amplitudes cross a fixed value. Figure 10 depicts the
Poincare section plots using extracted peak amplitudes
and extracted times between zero crossings in the flow
data. Both techniques successfully capture the
dynamics, though the use of zero crossings appears
slightly less noisy.
3.3. Embedding parameters
A reasonable value for the delay may be suggested
either by the first zero crossing of the autocorrelation
function or by the first minimum of the mutual
information function[18, 19], as either value is plotted
as a function of delay. For the time series data, given a
delay d, the autocorrelation is found simply from
1
( N − k )σ 2
N −k
∑ [ X (n) − μ ][ X (n + d ) − μ ]
(5)
n =1
where μ is the mean and σ2 is the variance of the data.
The mutual information of two random variables,
a ∈ A and b ∈ B , is given by
I ( A; B ) =
p ( a, b)
∑∑ p(a, b) log p(a) p(b)
(6)
a∈ A b∈B
where the convention 0log0=0 was used. In the case of
the mutual information between a time series and a
delayed version of itself, a represents a range of values
for X(n) and b a range of values for X(n+d). These
values must be chosen so as to provide a reasonable
approximation to the mutual information of the
underlying dynamical systems generating X(n) and
X(n+k). Here, the mutual information was calculated
efficiently using a method described in Reiss, et al.[20],
which partitions the range of values for X(n) and X(n+k)
recursively until there is no more hidden structure.
The mutual information often gives a better value
because it takes nonlinear correlations into account.
However, for the loudspeaker data, the mutual
information function and the autocorrelation function
were in strong agreement. As shown in Figure 11, the
autocorrelation suggests a delay of 6 and the mutual
information a delay of 7. This was in agreement with
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Chaos in a loudspeaker
visual inspection since 2 and 3 dimensional plots
revealed the most structure near these values of delay
(see Figure 13). Unfortunately, they also reveal a
complexity or noise dependence that makes the fine
scale structure very difficult to detect.
A modified form of the false nearest neighbors
algorithm[21] (FNN) was chosen as the primary
technique for determining the embedding dimension.
The modification is intended to take into account
stochastic phenomena which result in FNNs occurring
regardless of the embedding dimension. An appropriate
embedding dimension is found when the percentage of
FNNs drops to a constant value. As shown in Figure
12, the percentage of false neighbors approaches a
constant value with an embedding dimension of 5 for
the flow data, and an embedding dimension of 4 for
either sectioned data set. This is in agreement with the
observation that a Poincaré section should have one less
dimension than the original data.
3.4. Fractal dimension
The dimensionality of a chaotic attractor is typically a
noninteger value. That is, the attracting region of the
phase space will not completely fill out a region of that
space. In this section we use several different methods
to estimate the dimensionality of the loudspeaker data,
which further gives an idea of the complexity of the
underlying dynamics.
inaccurate, so the slope must be measured in the
midrange of the curve. A good value should be in the
region where measurements of the dimension are most
stable.
The Grassberger-Proccacia algorithm[22, 23] was used
to estimate fractal dimension. Results of log2(C(ε))
versus log(ε) for the peak values from the original 45Hz
data are depicted in Figure 14. The correlation
dimension cannot be accurately estimated since there is
not a significant region where the slope remains
constant. This is because estimates of correlation
dimension using the Grassberger-Proccacia algorithm
are highly susceptible to noise and data set size. Limited
data set size reduces the region of the plateau and
increases uncertainty, and the presence of noise implies
that accurate measurement can only be obtained for
large ε. For higher dimensional data these problems are
aggravated since minimal noise and exponentially more
data are required to identify the plateau region. In
addition, data with a high sample rate may exhibit
strong correlations that skew the estimates. The
approximations due to noise, data set size,
nonstationarity and so on are inherent in the data set.
But the Grassberger-Proccacia algorithm also uses an
approximation to the definition of correlation
dimension. Therefore, we attempted an alternative
technique that allows estimation of multiple definitions
of the fractal dimension.
The correlation dimension is found by constructing a
function C(ε) that is the probability that two arbitrary
points from the delay coordinate embedding are closer
together than a distance ε.
C (ε ) =
2
N ( N − 1)
N
i −1
∑∑ H (ε − | Y (i) − Y ( j) |)
(7)
i =1 j =1
where H is the Heaviside unit step function. The
correlation dimension of an experimental time series is
then given by
D = d log(C ) / d log(ε )
(8)
in the limit ε → 0 , and N → ∞ . The correlation
dimension may be estimated by the slope of the curve
log(C(ε)) versus log(ε). A noninteger result for the
correlation dimension indicates that the data is probably
fractal. For too low or too high ε values, the results are
Figure 13. A two dimensional plot of the experimental data
with a delay of 7.
The delay coordinate embedded data can be gridded into
n-dimensional boxes of equal length ε, such that all
vectors lie within these N boxes. If a box is labelled i,
then it has an associated probability, Pi(ε), that a vector
on the attractor will reside within this box. The
generalized entropies, H0, H1, … are defined in terms of
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Chaos in a loudspeaker
the probabilities of vectors occupying boxes. For
q=0,1,2…,
N (ε )
1
H q (ε ) =
ln ∑ Pi q (ε )… q ≠ 1
1 − q i =1
(9)
N (ε )
H1 (ε ) = − ∑ Pi (ε ) ln Pi (ε )
i =1
In the following we keep the conventions common in
dimension definitions and use the natural logarithm as
opposed to the log base 2. The generalized dimension of
order q is then defined as
H q (ε )
D (q ) = − lim
(10)
ε →∞ ln ε
Under this definition, D(0), D(1), and D(2) are the box
counting dimension, the information dimension and the
correlation dimension, respectively. We also have the
property that if p>q, then D(p)≤D(q).
Once the generalized entropies have been determined,
there are two ways to approximate the generalized
dimensions for time series data. The first is if ε is
sufficiently small such that the limit is approximately
correct, and we have enough data to get an accurate
measurement for H q (ε ) . In which case we may use
D(q) ≈ Dq (ε ) = − H q (ε ) / ln ε
(11)
However, the preferred method is to simply look at the
slope of a plot of Hq(ε) vs ln(ε), since this has a quicker
rate of converge to the limit of ε−>0. We should
mention that further information theoretic properties can
be determined from the analysis of this sorting, such as
the generalized mutual information of high dimensional
data, In(X1,X2,…Xn), or estimation of the metric
entropy[24].
For large box size, the box counting dimension varies
widely from the others, since the box counting
dimension D(0) is more susceptible to errors. It is also a
poor quantity to use since it says nothing about the
density of the attractor, only about its shape. However,
the box counting dimension and all the others converge
in the mid-region, before diverging slightly and then
dropping to zero (due to data set size). It is this mid
region that parallels the plateau region of the
Grassberger-Proccacia algorithm[22, 23].
dimensions and the sectioned data embedded in four
dimensions, respectively. Displayed are estimates of
the first four generalized entropies for varying box size.
Additional tests were also performed for the embedding
dimensions 3-6, and for the next four generalized
entropies.
The
results
indicated
that,
for
p > q, D( p ) ≤ D(q ) , which agrees with theory.
The estimates for fractal dimension were derived from
where the slope of of Hq(ε) vs ln(ε) showed the least
deviation for successive values of ε (ε =2-4 and 2-5).
Estimates ranged from 1.8 to 2.2, for all fractal
dimensions calculated on the sectioned data, and 2.6 to
3.0 for all fractal dimensions calculated on the flow
data, when embedding dimension was greater than or
equal to 4. In both cases the estimate This agrees with
our choice of the embedding dimension, and is also in
rough agreement with the result from the GrassbergerProccacia algorithm.
However, in general, the
dimensionality of the sectioned data is less than 1 plus
the dimensionality of the flow data. This may be
accounted for primarily by the small data set size for the
sectioned data (approximately 10,000 points), which is
known to cause a slight overestimation of fractal
dimension, and the relatively high sample rate of the
flow data, which skewed estimates downwards[25].
3.5. Lyapunov exponents
The Lyapunov exponents characterize how chaotic a
system is. For a D-dimensional dynamical system,
consider the infinitesimally small D-sphere centered
around a point on the attractor. As this evolves, the
sphere will become an D-ellipsoid due to the deforming
nature of the attractor flow. Then the ith Lyapunov
exponent is defined in terms of the exponential growth
rate of the ith principal axis of the ellipsoid.
1 p (t )
λ i = lim ln i
(12)
t →∞ t
pi (0)
Figure 15 and Figure 16 presents the results of our
calculations of the first four generalised dimensions
performed on the flow data embedded in five
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Chaos in a loudspeaker
Dλ = L +
L
1
λL +1
∑λ
(13)
j
j =1
where L is the maximum integer such that the sum of
the L largest exponents is still non-negative. That is,
L +1
L
∑
j =1
Figure 14. Plot of the log correlation function versus log
distance. With sufficient, low noise data, the slope of the
plot may provide the correlation dimension.
Thus the spectrum of Lyapunov exponents,
{λ1, λ2, ..., λD} describes the rate of growth of the
distance between nearby trajectories in phase space.
Chaos is often defined by the existence of a positive
dominant Lyapunov exponent, which indicates that
nearby trajectories will, over time, diverge
exponentially away from each other.
The determination of Lyapunov exponents from noisy,
experimental data is a difficult task. Although many
methods have been presented, there are also numerous
examples of these methods breaking down when tested
against real data, as well as questions concerning the
validity of the methods. Thus the results of exponent
determination were held to scrutiny. Criteria were
established for identification of Lyapunov exponents.
There should be agreement between exponents as
measured from different algorithms, and some
measurement of error should be provided. Embedding
parameters used in estimating exponents must be
confirmed independently by other methods. The results
should remain consistent under various parameter
settings for exponent estimation. For flow data, a zero
exponent should be clearly found. Estimation of
exponents from each both flow and sectioned data
should be in agreement. The sum of all the exponents
must be negative, and the sum of the positive exponents
should be less than or equal to the metric entropy. In
fact, for many cases they should be equal.[26] Under the
proper conditions, the Lyapunov exponents should all,
approximately, switch sign when measured from a time
reversal of the data.[27]
The Lyapunov dimension may be defined as
λj ≥0,
and
∑λ
j
<0.
The
Kaplan-Yorke
j =1
conjecture[28] proposes that this is equal to the
information dimension. Within error bounds, this seems
to be true. Therefore, a final criterion is that the
Lyapunov dimension estimates should agree with
information dimension estimates.
It is doubtful that all criteria can be satisfied unless one
is dealing with a long, noise-free time series of low
dimensional simulated data. Noise, high dimensionality
and short time series length (few orbits or small number
of points or both) negatively affect all methods of
analysis. Some criteria, such as confirmation of
embedding parameter choices are a virtual necessity
before any calculation is made. Others, such as
agreement between the Lyapunov dimension and
information dimension, are very strong indicators that
Lyapunov exponents have been reasonably determined.
Still other criteria require calculations of quantities that
are particularly difficult to compute from time series,
such as metric entropy. Previous authors chose to reject
the use of estimated Lyapunov exponents as
discriminating statistics[29].
Three methods of determining Lyapunov exponents
were implemented; the method of Eckmann and
Ruelle[30] for determining the Lyapunov spectra, and
the Rosenstein[31] and Wolf, et al.,[32] methods for
determining the largest exponent. Since these methods
are fundamentally different, one would not expect
agreement between the estimates to be simply due to
them incorporating the same mistakes. The sectioned
data were used for all estimates since this reduces
dependence on the choice of delay time. Abarbanel’s
method was also applied[33], though this is based on the
same technique as Eckmann and Ruelle’s method and
for sectioned data provides very similar results.
Wolf[32] defined the exponents in terms of the rates of
divergence of volumes as they evolve around the
attractor. The Wolf method involves following a
trajectory in its path around the attractor. The rate of
growth between points on this trajectory and a nearby
trajectory is used to estimate the largest Lyapunov
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Chaos in a loudspeaker
exponent. When the distance between these trajectories
becomes too large, then a new nearby trajectory is found
that is within a reasonable angular distance from the
previous nearby trajectory. In order to minimize
parameter dependence, we used a variation on the Wolf
method[25]. In previous work,[32, 34, 35] both
maximum allowable displacement and maximum
angular displacement were left as free parameters. We
use one parameter: the number of near neighbors to
consider after a given number of iteration steps.
The Rosenstein method involves looking at average
divergence rates of nearest neighbors. It also finds the
dominant exponent. The Eckmann and Ruelle method
involves using a small neighborhood of points and
iterating them forward to estimate the local Jacobian,
and then determining the Lyapunov spectrum from the
eigenvalues of the Jacobians around the attractor.
In Table I, results of exponent calculations are provided.
All calculations were performed with a time delay of 1,
and embedding dimension of 4, as suggested by the
False Nearest Neighbors routine. The exponents are
given in units of 1/time, where the time scale is defined
so that the time between samples is 1. Many more
calculations were performed until a reasonable and
stable parameter regime was found for all methods. In
general, exponent calculations converged to within 5%
of their final value when averaging local estimates of
the dominant Lyapunov exponent over only 1,000
points (although the entire sectioned data set was used
to find near neighbors).
Table I. Results of estimation of the lyapunov exponent(s)
using three different techniques. All calculations were
performed with a four dimensional embedding of the peak
values data.
Algorithm
λ1
Rosenstein
0.403
Wolf
0.391
EckmannRuelle
0.380
λ4
Sum
λ2
λ3
0.054
-0.184 -0.484 -0.234
within reasonable bounds. For the Lyapunov spectrum,
the second exponent is very close to zero and the sum of
the exponents is negative.
The folding of the attractor brings diverging orbits back
together. So any effects of nonlinearities will most
likely serve to move all exponents closer to zero. Also
increasing the number of near neighbors used may
underestimate the value because this allows a larger
distance between neighbors. Hence a slight
underestimate of the positive exponents for the
Eckmann-Ruelle algorithm (and for Abarbanel’s
technique[33]) was expected. For the Wolf algorithm,
the angular displacement errors are not likely to
accumulate, but each error may skew the largest
positive exponent downwards. These assumptions are
confirmed by the slightly lower estimate of dominant
exponent for the Eckmann-Ruelle and Wolf algorithms
as compared to Rosenstein’s technique, which is less
susceptible to these errors.
However, it was not possible to confirm all criteria.
Measurement of the metric entropy is still ongoing
work. Sectioning the data introduced additional noise.
More importantly, the uncertainty in sample values
tended to dominate over the divergence on a small time
scale, thus introducing errors into measurement of
Lyapunov exponents from the original flow data. Thus
it was not possible to get agreement between exponent
estimates from the section and from the flow, nor was it
expected. For measurement of the Lyapunov spectrum,
uncertainty in the values of other exponents meant that
it was not possible to get a reliable estimate of the
Lyapunov dimension. Time reversal results were also
inconclusive at best. However, simulated data with the
addition of noise would not usually switch the signs of
the exponents under time reversal either. So the sign
change of exponents when the data is reversed may not
be a suitable criterion for noisy data.
Several of our criteria are determined immediately upon
inspection. The dominant exponent results from all
three methods provide rough agreement. One check on
the Wolf algorithm was calculating the average angular
displacement. This was typically less than 20%, well
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Chaos in a loudspeaker
Figure 15. Plot of the first four generalized dimensions
from the original data .
Figure 16. Plot of the first four generalized dimensions
from the sectioned data.
3.6. Unstable periodic orbits
In addition to the occurrence of a positive Lyapunov
exponent, a chaotic system may also be characterized by
having an infinite number of unstable periodic orbits
(UPOs). The identification of unstable periodic orbits
plays a critical role in many chaos control
algorithms[36]. Most standard chaos control algorithms
attempt to control the system onto a UPO while
operating within the chaotic regime. Small timedependent perturbations applied to an accessible
parameter may then be used to force the system onto the
stable manifold and hence enforce stability and periodic
behavior. The drive frequency is the most preferable
candidate to use as the varied parameter in a control
scheme. This is because it is easily adjustable and a
small change in drive frequency often yields appropriate
changes in the dynamics. it is useful in occasional
proportional feedback control schemes, tracking and
targeting of trajectories, and in the identification of
symbolic dynamics[36, 37]. Thus we will also attempt
to characterize the UPOs exhibited during the chaotic
state.
be used to estimate its exact location. The exact value of
ε may be varied depending on the size of the data set,
the period p and the number of UPOs that one wishes to
find. Identification of false positive UPOs may be
determined using the mean squared error of the least
squares fit.
Figure 17 shows the delay coordinate embedding of the
Poincaré section using times between zero crossings. In
this figure, we identified period 1, period 2 and period 3
orbits. From the least squares fit estimate of the local
dynamics,
the
eigenvalues
and
eigenvectors
corresponding to the stable and unstable manifolds can
be found. For the period 1 orbit, we have a fixed point
located at 0.0222 seconds, corresponding to the drive
frequency of the system, 45 Hz. Its eigenvalues are
0.155 and -1.571, with corresponding eigenvectors
(1,0.155) and (-0.637,1). Similar results can be obtained
for other identified periodic orbits, and these results can
be used to implement a chaos control technique.
4.
CONCLUSIONS AND FURTHER WORK
Analysis was attempted on time series data from an
experimental electrodynamic loudspeaker in order to
characterize the embedding dimension, fractal
dimension, the Lyapunov exponents, and the unstable
periodic orbits. Results were obtained which indicate
that the system is governed by low dimensional chaotic
dynamics, and thus is highly amenable to control,
tracking, synchronization, noise reduction and so forth.
Particular care was made in verifying the presence of a
positive Lyapunov exponent.
By looking for when the dynamics approach the same
region after a given number of iterates, periodic orbits
can be found. UPOs of period p are found simply by
establishing a threshold ε,
|| Y (n) − Y (n + p ) ||< ε
(14)
In which case, a periodic orbit exists in the vicinity of
Y(n) and a least squares fit of all data in the region can
Figure 17. A delay coordinate embedding plot of the times
between zero crossings. Period 1, 2 and 3 points are
identified, along with the stable and unstable manifolds of
the period 1 orbit.
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Chaos in a loudspeaker
Various estimates of fractal dimension were performed,
including measurement of correlation dimension[22,
23], information dimension, and box counting
dimension for different embedding dimensions.
Although there was qualitative confirmation of low
dimensional behavior, consistent results for quantitative
values for the fractal dimension were not achieved.
limited by the 8 bit resolution of the A/D converter of
the laser distance meter. This meant that there was
significant uncertainty in the location of nearby points.
Since most analysis of chaotic time series relies on
analysis of near neighbors in a locally linear region, this
resulted in inaccuracies in estimation of fractal
dimension and Lyapunov exponents.
However, estimation of the dominant Lyapunov
exponent, which is less reliant on large data set size,
provided consistent results regardless of the method of
estimation[26]. Several different techniques were used
which provided rough agreement in their estimates of
the dominant exponent, and the results further agreed
with theory concerning the Lyapunov spectrum its
properties. They reliably showed evidence of a positive
Lyapunov exponent, a strong indicator of chaos.
Current work is focused on improvements to the data
acquisition system. This would allow more accurate
analysis of the data, including measurements of how
fractal dimension and Lyapunov dimension change with
parameter settings, and use of chaotic time series
prediction methods on the data. This could also be used
for further direct comparison with dynamical behavior
from enhanced models of the loudspeaker. An accurate
model of an electrodynamic loudspeaker would
represent a significant advance in the field, particularly
since model parameters could then be modified to yield
a loudspeaker design with optimal performance.
Finally, we attempted to estimate the eigenvalues and
eigenvectors associated with detected unstable periodic
orbits. These may be easily identified. Control may be
applied to allow the loudspeaker to operate as desired
within the chaotic regime. Tracking and maintenance
should also be possible, since the appropriate dynamics
have been found for the application of several wellknown algorithms.
However, extraction of many empirical quantities,
particularly fractal dimension, proved difficult due to a
number of issues. This may be accounted for partly due
to nonstationarity and short data set size. Though the
original data set is over 200,000 points, it represents
about 10,000 orbits. This is insufficient for effective
calculation of fractal dimension in the presence of noise
and long term dynamics. Furthermore, although the
dynamics are somewhat stable, there is still a gradual
change in various statistical quantities when examined
using a sliding window through the data. This tends to
distort various measurements. Longer data set size could
capture the long term dynamics.
However, we believe that the difficulty in estimating
some measures of nonlinear behavior and
dimensionality is primarily due to low sample rates and
low resolution due to limitations in the data acquisition
system. First, the data was sampled at 1024 samples per
second, or approximately 23 samples per period. A
higher sampling rate would yield more accurate
Poincaré sections. Alternatively, if the sampling data
could be sampled at exactly the drive frequency, 45Hz,
then this would produce a natural Poincaré section. The
analysed data had 16 bit precision, but this was further
The investigators are also devising experimental
conditions for the control of chaos[36] in the
loudspeaker. In addition to more practical applications,
such as robust operation of the loudspeaker in the
chaotic regime, this work can be useful for composing
fractal music[38]. Unlike other chaotic dynamic
systems, loudspeaker operate in two-dimensions by
changing two input parameters, frequency and driving
current. So, the necessary two-dimensional patterns
provided for successful composing can be created by
two variables. Similar to other composing instruments,
the loudspeaker may be used in both roles
simultaneously, as a composing and music reproduction
device. This is an advantage over the usual theory of
fractal composing which assumes complicated
interpolating electronic stages based upon synthesizers.
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Chaos in a loudspeaker
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