Verification of chaotic behavior in an experimental loudspeaker

Verification of chaotic behavior in an experimental loudspeaker
Verification of chaotic behavior in an experimental loudspeaker
Joshua D. Reiss
Department of Electronic Engineering, Queen Mary, University of London, Mile End Road, London E14NS,
United Kingdom
Ivan Djureka兲 and Antonio Petosicb兲
Department of Electroacoustics, Faculty of Electrical Engineering and Computing, University of Zagreb,
Unska 3, Zagreb, Croatia
Danijel Djurekc兲
Alessandro Volta Applied Ceramics, (AVAC) Laboratory for Nonlinear Dynamics, Kesten brijeg 5, Remete,
Zagreb, Croatia
共Received 24 January 2008; revised 23 April 2008; accepted 4 July 2008兲
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 a 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.
© 2008 Acoustical Society of America. 关DOI: 10.1121/1.2967843兴
PACS number共s兲: 43.38.Ja, 43.25.Ts, 43.25.Rq 关AJZ兴
I. INTRODUCTION
Loudspeakers are the most variable elements in any audio system and are responsible for marked audible differences between otherwise identical sound systems. The performance of loudspeakers 共i.e., their accuracy in reproducing
a signal without adding distortion兲 is significantly poorer
than that of other audio equipments. 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 共and many speaker designs fall further outside this range兲, whereas that of an amplifier may vary by
less than 0.1 dB. Thus an improved understanding of the
dynamics of a loudspeaker is of great importance since a
better loudspeaker design may yield a significantly better
performance in an audio playback system.
In a dynamic loudspeaker, sound is typically reproduced
by the movement of a current-carrying object in a magnetic
field due to the Lorentz force. An electrodynamic loudspeaker, as depicted in Fig. 1, is a type of a dynamic loudspeaker in which the magnetic field is produced by a permanent magnet. The audio signal from the amplifier, which is a
time-varying voltage, causes a current to flow through a
tightly wound coil of wire called the voice coil. The voice
coil is attached to a stiff but light cone and connected to a
fixed chassis by means of a flexible membrane. The spider, a
weblike membrane, suspends the voice coil in the center of
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
c兲
Electronic mail: [email protected]
b兲
J. Acoust. Soc. Am. 124 共4兲, October 2008
Pages: 2031–2041
the magnet and provides part of its restoring force. The surround connects the cone to the speaker chassis and keeps
nonaxial motion to a minimum so that the cone functions
approximately as a rigid piston. The interaction of the coil
and the driver’s magnetic system generates a mechanical
force that causes the coil, and thus the attached cone, to
move back and forth. This creates pressure waves in the air,
which reproduce the original audio signal coming from the
amplifier as sound.
A simplified model of an electrodynamic loudspeaker
operates as a driven damped harmonic oscillator.2 The dynamics of the displacement of the membrane, x, is given by
the second order time-dependent ordinary differential equation
M
dx
d 2x
+ kx = BlI0 cos共␻t兲,
2 + RM
dt
dt
共1兲
where M is the combined mass of the membrane and voice
coil, k is the elastic stiffness, B is the magnetic field, l is the
length of the voice coil wire, and I0 represents the drive
current with frequency f 0 = ␻0 / 共2␲兲. The friction term RM
consists of internal membrane friction and friction coming
from vibration in air. Note that this model is not used in the
analysis of the experiment but is given in order to show the
need for higher order nonlinear terms to accurately represent
the dynamics.
Both an inhomogeneous magnetic field and the
excursion-dependent stiffness of the suspension account for
the nonlinear behavior of the transducer. Thus, all coefficients in Eq. 共1兲 are dependent on displacement x. The mass
M was measured by weighing the detached vibrating parts
and was found to be 13.1 g. However, due to the elastic
0001-4966/2008/124共4兲/2031/11/$23.00
© 2008 Acoustical Society of America
2031
FIG. 2. Experimental setup with a loudspeaker and a laser distance meter.
FIG. 1. Diagram of an electrodynamic loudspeaker.
properties of the membrane, which spans between the voice
coil and the suspension on the perimeter, M should be replaced by an effective mass, M eff. M eff was evaluated from
the Q-factor of the loudspeaker3,4 and, as expected, was
found to be dependent on displacement. For higher driving
currents, the parameter Bl decreases with displacement
nearly symmetrically relative to a zero position.5 In these
experiments, Bl = 3.9 T m and reduces to ⬃3.6 T m for a
membrane displacement of ⫾6 mm, and corresponding nonlinearities can be neglected. In most commercially available
loudspeakers, the elastic stiffness has a quadratic dependence
on displacement. A systematic approach to describing such a
vibrating system, using the Duffing equation, was provided
by Woafo,6 where it was shown that related models of electromechanical transducers may exhibit chaos. However,
Woafo also noted that a linear version is used to describe
loudspeakers and that nonlinear terms, which are known to
cause subharmonics, are often neglected.6,7
The stiffness was recorded in static measurements by the
use of calibrated loads and corresponding measurements of
the membrane deflections and may be given as approximately k = m + nx + px2, where m = 480 N m−1, n = 31 N m−2,
and p = 7.5 N m−3. Due to the viscoelastic contribution of the
membrane polymer material, this disagrees substantially with
the stiffness calculated in the dynamic regime from the
simple resonance formula ␻ = k / M, even in the regime of
comparatively small drive currents 共I0 ⬍ 100 mA兲. Thus
higher order corrections coming from the dynamics of a vibrating viscoelastic body8,9 must be included.
When dealing with moderately low driving currents, a
significant agreement was found between a model based on
static measurements and the experimental data, especially for
an estimation of frequency dependence and electrical impedance under a variety of different operating regimes.10 Although simple models can be used to identify loudspeaker
2032
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
parameters, at high drive currents 共I0 ⬎ 1 A兲 they generally
do not account for the rich nonlinear dynamic behavior that
might be observed.11–13
Evidence of a possible chaotic behavior in a loudspeaker
was first observed by Wei et al. in 1986,14 where the appearance of subharmonics and broadband spectra at various drive
frequencies and voltages was noted. Recent work strengthened this conjecture,15–17 with the observation of hysteresis
and period doubling when the loudspeaker was driven at low
frequencies. This, in general, is not observed in the models.
Thus it is important to verify chaos and adjust the models
accordingly.
In this work, we analyze time series from an experimental electrodynamic loudspeaker system. We use a variety of
techniques from the chaotic time series analysis18 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.
II. EXPERIMENTAL SETUP
In these experiments, a low frequency loudspeaker was
used with a resonant frequency, recorded in air, of f
= 38 Hz, driving current of I0 = 10 mA, factor of Bl
= 3.9 T m, membrane diameter of 2R = 16 cm, rated rms
power of 60 W, and nominal impedance of 8 ⍀. According
to the manufacturer’s data, the voice coil inductance is L
= 0.9 mH, and the contribution of the inductive part ␻L to
the loudspeaker’s impedance can be neglected for driving
frequencies f ⬍ 100 Hz.
The experimental setup is depicted in Fig. 2. 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 an ultimate resolution of 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 the vibration amplitude with an accuracy of
2 ␮m and a sampling frequency of ⬃1 kHz 共a similar measurement apparatus was used to analyze nonlinear vibrations
of a loudspeaker in Wei et al.14兲. The analog/digital 共A/D兲
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
Reiss et al.: Loudspeaker chaos
FIG. 4. A bifurcation diagram of vibration amplitudes recorded in vacuo for
a fixed driving frequency of 53 Hz and a sweeping driving current of
20 mA/ s.
FIG. 3. Impedance measured in vacuo for various driving currents. The
inset shows the hysteresis of the cutoff frequency for positive and negative
frequency sweeps.
measurements, the impedance and vibration amplitude data
were recorded at 1 bar air pressure in a closed chamber and
compared to those evaluated in free laboratory atmosphere.
In a 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 a rated
power of 300 W via a series resistor 0.44 ⍀. 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 – 5 A. For
driving currents I0 ⬍ 100 mA, the back electromotive force is
comparable to the friction term in Eq. 共1兲, 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
共1 bar兲.19 The data recorded in vacuo are shown in Fig. 3. By
an increase in the driving current, the resonance curve became more and more distorted until an amplitude downturn
共cutoff兲 appeared near f = 43.5 Hz for I0 = 200 mA. This current indicated the starting value for an identification of the
chaotic regime. The inset in Fig. 3 depicts the hysteretic
property of the cutoff effect. That is, cutoff frequencies differ
for positive and negative frequency sweeps.
The onset of the amplitude cutoff 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 an evaluation of the appearance of subharmonics that precede the
chaotic state. An important feature of the unstable range was
the relatively small change in the impedance with increasing
driving current. In these experiments, loudspeaker impedance stayed at ⬃11 ⍀, 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.
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
A bifurcation diagram, as shown in Fig. 4, was produced
by fixing the frequency at f = 53 Hz, sweeping the driving
current at I0 = 1.5– 2.5 A at a rate of 20 mA/ s, 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 appeared20 and the system
vibrated with three amplitudes. The 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 are 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 a 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 minimized 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. 共1兲. Excerpts of the time series waveforms representing various driving currents are
given in Fig. 5. Figure 5共a兲 shows the time-dependent vibration amplitude at the starting driving current I0 = 2.4 A when
the recorded signal shows a nearly sinusoidal behavior. Figures 5共b兲–5共d兲 show new vibration amplitudes 共marked with
triangles兲, which appear with increasing driving current. The
corresponding averaged spectra over the whole time series
are given in Fig. 6. Figure 6共a兲 shows the expected behavior,
with a fundamental frequency corresponding to the drive frequency. The first subharmonic appeared at 28 Hz, as depicted in Fig. 6共b兲. A further increase in the driving current
resulted in a new subharmonic at 14 Hz 关see Fig. 6共c兲兴 and a
subsequent appearance of broadband behavior, 关Fig. 6共d兲兴.
The bifurcation diagram for measurements of vibration
amplitude in air is shown in Fig. 7 and was produced in the
same manner as Fig. 4. Vertical lines indicate values of driving currents for which the Feigenbaum ratio ␦1 / ␦2 = 4.669 is
fulfilled.21 A period 3 window cannot be seen, but this is not
a contra-indicator of chaos. Noise in the data acquisition sysReiss et al.: Loudspeaker chaos
2033
FIG. 5. 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 a broadband behavior.
tem may obscure the window, and the 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 Fig. 7.
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 a time series
analysis. A rather low frequency of 45 Hz was selected because the dynamics appeared less susceptible to parameter
drift and nonstationary behavior 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 were recorded with a 16 bit resolution, although
this is further limited by the 8 bit resolution of the laser
distance meter. The sample rate was 1024 Hz, so that the
data set is just over 4 min long and there are approximately
2034
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
FIG. 6. Average spectrum of vibration amplitude recorded in 1 bar air for
various driving currents corresponding to those in Fig. 5.
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.
III. DATA ANALYSIS AND RESULTS
To analyze a one dimensional experimental data set using chaotic time series analysis techniques, it is necessary to
FIG. 7. A bifurcation diagram of vibration amplitudes recorded in 1 bar air
for a driving current swept at 20 mA/ s and a fixed driving frequency of
56 Hz.
Reiss et al.: Loudspeaker chaos
transform the data using phase space reconstruction techniques. If only one variable from the system can be observed, X = 兵X共1兲 , X共2兲 , . . . , X共N兲 , . . . 其, then a D dimensional
time series of length N, Y = 兵Y共1兲 , Y共2兲 , . . . , Y共N兲其 is constructed from the original time series using a delay d as
follows:
Y共n兲 = 共X共n兲,X共n + d兲, . . . ,X共n + 共D − 1兲d兲兲,
共2兲
where we have assumed that X contains at least N + 共D
− 1兲d data points. If the time between samples represents one
period of data, then X represents a time series generated
from a map, or a Poincaré section of the system, in which
case the delay d used to generate Y is usually set to 1.
Assuming that the time series is stationary, that is, the
parameters that 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 behavior 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,
characterization of the chaotic behavior or lack thereof, and
identification of chaotic orbits.
In the following subsections, we construct delay coordinate embeddings from the scalar time series and then use this
technique to analyze the data and quantify the dynamical
system properties. We will use the notation introduced above
to describe the original scalar time series, X
= 兵X共1兲 , X共2兲 , . . . , X共N兲其, and a delay coordinate embedding
of the time series, Y = 兵Y共1兲 , Y共2兲 , . . . , Y共N兲其.
A. Nonstationarity and long term dynamics
A few simple tests that would identify strong drifts in
the data were first performed. Sliding windows of varying
lengths were applied to the data, and statistical quantities
such as mean, maximum, minimum, and standard deviations
were computed for each window. If the data were truly stationary, then these quantities would remain constant throughout the data. The results of the drift in the maximum and
minimum values of a one second window 共1024 data points兲
are depicted in Fig. 8.
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 behavior. However, this fluctuation is quite
small in relation to the full extent of the data 共for instance,
the variation in the maximum value is less than 2% of the
full scale of the data兲. Thus, although it may affect the results
of chaotic time series analysis methods, it is still small
enough such that the data are acceptable for analysis.
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
FIG. 8. Nonstationary behavior of the time series. Plotted are estimates of
the maximum and minimum values for overlapping windows of 1024
samples 共1 s兲 from the experimental data.
B. Poincaré sections
A common technique in a chaotic time series analysis is
to generate a Poincaré section, with one point per period,
from data sampled at a much higher frequency than the drive
frequency. Given that the system has a drive frequency of
45 Hz, 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 9 depicts
the Poincaré section plots using extracted peak amplitudes
and extracted times between zero crossings in the flow data.
Both techniques successfully capture the dynamics, although
the use of zero crossings appears slightly less noisy.
C. 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 function22,23
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
N−k
R共d兲 =
1
兺 关X共n兲 − ␮兴关X共n + d兲 − ␮兴,
共N − k兲␴2 n=1
共3兲
where ␮ is the mean and ␴2 is the variance of the data.
The mutual information of two random variables, a
苸 A and b 苸 B, is given by
I共A;B兲 =
p共a,b兲
兺 兺 p共a,b兲log p共a兲p共b兲 ,
a苸A b苸B
共4兲
where the convention 0 log 0 = 0 is 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共n兲 and b
denotes a range of values for X共n + d兲. These values must be
Reiss et al.: Loudspeaker chaos
2035
(a)
FIG. 11. A two dimensional plot of the experimental data with a delay of 7.
(b)
FIG. 9. Plot of the waveform for two methods of sectioning the data. The
top plot depicts peak amplitudes and the bottom plot depicts time intervals
between zero crossings.
chosen so as to provide a reasonable approximation to the
mutual information of the underlying dynamical systems
generating X共n兲 and X共n + k兲. Here, the mutual information
was calculated efficiently using a method described in Reiss
et al.,24 which partitions the range of values for X共n兲 and
X共n + k兲 recursively until there is no more hidden structure.
The mutual information often gives a better value because it takes nonlinear correlations into account. However,
FIG. 10. 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.
2036
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
for the loudspeaker data, the mutual information function
and the autocorrelation function were in strong agreement.
As shown in Fig. 10, the autocorrelation suggests a delay of
6 and the mutual information suggests a delay of 7. This was
in agreement with visual inspection since two and three dimensional plots revealed the most structure near these values
of delay 共see Fig. 11兲. 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 neighbor 共FNN兲
algorithm25 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 Fig. 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.
FIG. 12. Results of the FNN routine as applied to the flow data. 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.
Reiss et al.: Loudspeaker chaos
FIG. 13. Plot of the log correlation function vs log distance. With sufficient
low noise data, the slope of the plot may provide the correlation dimension.
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. However, the Grassberger–Proccacia algorithm also uses an approximation to the definition of correlation dimension.
Therefore, we attempted an alternative technique that allows
an estimation of the multiple definitions of the fractal dimension.
The delay coordinate embedded data can be gridded into
n dimensional boxes of equal length ␧, such that all vectors
lie within these N boxes. If a box is labeled i, then it has an
associated probability, Pi共␧兲, that a vector on the attractor
will reside within this box. The generalized entropies,
H0 , H1 , . . ., are defined in terms of the probabilities of vectors
occupying the boxes. For q = 0 , 1 , 2. . .,
Hq共␧兲 =
D. 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 give an
idea of the complexity of the underlying dynamics.
The correlation dimension is found by constructing a
function C共␧兲, which is the probability that two arbitrary
points from the delay coordinate embedding are closer together than a distance ␧,
N i−1
C共␧兲 =
2
兺 兺 H共␧ − 兩Y共i兲 − Y共j兲兩兲,
N共N − 1兲 i=1 j=1
共5兲
where H is the Heaviside unit step function. The correlation
dimension of an experimental time series is then given by
D = d log共C兲/d log共␧兲
共6兲
in the limit ␧ → 0 and N → ⬁. The correlation dimension may
be estimated by the slope of the curve log关C共␧兲兴 vs log共␧兲. A
noninteger result for the correlation dimension indicates that
the data are probably fractal. For too low or too high ␧ values, the results are 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,26,27 was used to
estimate fractal dimensions. The results of log2关C共␧兲兴 vs
log共␧兲 for the peak values from the original 45 Hz data are
depicted in Fig. 13. The correlation dimension cannot be
accurately estimated since there is not one 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. A
limited data set size reduces the region of the plateau and
increases uncertainty, and the presence of noise implies that
an accurate measurement can only be obtained for large ␧.
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
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
1
ln
1−q
N共␧兲
Pqi 共␧兲,
兺
i=1
q ⫽ 1,
N共␧兲
H1共␧兲 = −
Pi共␧兲ln Pi共␧兲.
兺
i=1
共7兲
In the following we keep the conventions common in dimension definitions and use the natural logarithm as opposed to
log base 2. The generalized dimension of order q is then
defined as
D共q兲 = − lim
␧→⬁
Hq共␧兲
.
ln ␧
共8兲
Under this definition, D共0兲, D共1兲, and D共2兲 are the box
counting dimension, the information dimension, and the correlation dimension, respectively. We also have the property
that if p ⬎ q, then D共p兲 艋 D共q兲.
Once the generalized entropies have been determined,
there are two ways to approximate the generalized dimensions for time series data. The first is if ␧ is sufficiently small
such that the limit is approximately correct and we have
enough data to get an accurate measurement for Hq共␧兲. In
that case we may use
D共q兲 = Dq共␧兲 = − Hq共␧兲/ln ␧.
共9兲
However, the preferred method is to simply look at the slope
of a plot of Hq共␧兲 vs ln共␧兲 since this has a quicker rate of
converge to the limit of ␧ → 0. We should mention that further information on theoretic properties can be determined
from the analysis of this sorting, such as the generalized
mutual
information
of
high
dimensional
data,
In共X1 , X2 , . . . , Xn兲, or the estimation of the metric entropy.28
For a large box size, the box counting dimension varies
widely from the others since the box counting dimension
D共0兲 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 midregion before diverging
slightly and then dropping to zero 共due to data set size兲. It is
this midregion that parallels the plateau region of the
Grassberger–Proccacia algorithm.26,27
Reiss et al.: Loudspeaker chaos
2037
E. 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 a
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,
␭i = lim
t→⬁
FIG. 14. Plot of the first four generalized dimensions from the original data.
Figures 14 and 15 present the results of our calculations
of the first four generalized dimensions performed on the
flow data embedded in five dimensions and the sectioned
data embedded in four dimensions, respectively. Displayed
are estimates of the first four generalized entropies for varying box sizes. Additional tests were also performed for the
embedding dimensions of 3–6 and for the next four generalized entropies. The results indicated that for p ⬎ q, D共p兲
艋 D共q兲, which agrees with the theory.
The estimates for fractal dimension were derived from
where the slope of of Hq共␧兲 vs ln共␧兲 showed the least deviation for successive values of ␧ 共␧ = 2−4 and 2−5兲. Estimates
ranged from 1.8 to 2.2 for all fractal dimensions calculated
on the sectioned data and from 2.6 to 3.0 for all fractal dimensions calculated on the flow data when the embedding
dimension was greater than or equal to 4. In both cases the
estimate agrees with our choice of the embedding dimension
and is also in rough agreement with the result from the
Grassberger–Proccacia 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
共approximately 10 000 points兲, which is known to cause a
slight overestimation of fractal dimension, and the relatively
high sample rate of the flow data, which skewed estimates
downward.29
2038
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
共10兲
Thus the spectrum of Lyapunov exponents, 兵␭1 , ␭2 , . . . , ␭D其
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 the identification of
Lyapunov exponents.
There should be an 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. The estimation of exponents from 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.30 Under the proper conditions, the
Lyapunov exponents should all, approximately, switch sign
when measured from a time reversal of the data.31
The Lyapunov dimension may be defined as
D␭ = L +
FIG. 15. Plot of the first four generalized dimensions from the sectioned
data.
1
pi共t兲
ln
.
t
pi共0兲
1
L
兺 ␭j ,
兩␭L+1兩 j=1
共11兲
where L is the maximum integer such that the sum of the L
largest exponents is still non-negative. That is, 兺Lj=1␭ j 艌 0
32
proposes
and 兺L+1
j=1 ␭ j ⬍ 0. The Kaplan–Yorke conjecture
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 the
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 共few orbits or a small number of points or
both兲 negatively affect all methods of analysis. Some criteria,
such as the confirmation of embedding parameter choices,
are a virtual necessity before any calculation is made. Others,
Reiss et al.: Loudspeaker chaos
TABLE I. Results of the estimation of the Lyapunov exponent共s兲 using three
different techniques. All calculations were performed with a four dimensional embedding of the peak value data.
Algorithm
␭1
␭2
␭3
␭4
Sum
Rosenstein et al.
Wolf et al.
Eckmann–Ruelle
0.403
0.391
0.380
0.054
−0.184
−0.484
−0.234
such as the agreement between the Lyapunov 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.33
Three methods of determining Lyapunov exponents
were implemented: the method of Eckmann and Ruelle34 for
determining the Lyapunov spectra and the methods of
Rosenstein et al.35 and Wolf et al.36 for determining the largest exponent. Since these methods are fundamentally different, one would not expect an agreement between the estimates to be simply due to them incorporating the same
mistakes. The sectioned data were used for all estimates
since these reduce dependence on the choice of delay time.
Abarbanel’s method was also applied,37 although this is
based on the same technique as Eckmann and Ruelle’s
method and the sectioned data provide very similar results.
Wolf et al.36 defined the exponents in terms of the rates
of divergence of volumes as they evolve around the attractor.
The method of Wolf et al. 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 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
method of Wolf et al.29 In previous work,36,38,39 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 method of Rosenstein et al. 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
an embedding dimension of 4, as suggested by the FNN
routine. The exponents are given in units of l/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
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
their final value when averaging local estimates of the dominant Lyapunov exponent over only 1000 points 共although the
entire sectioned data set was used to find near neighbors兲.
Several of our criteria are determined immediately upon
inspection. The dominant exponent results from all three
methods provide a rough agreement. One check on the algorithm of Wolf et al. was calculating the average angular displacement. This was typically less than 20%, well 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 共and for Abarbanel’s
technique37兲 was expected. For the algorithm of Wolf et al.,
the angular displacement errors are not likely to accumulate,
but each error may skew the largest positive exponent downward. These assumptions are confirmed by the slightly lower
estimate of the dominant exponent for the Eckmann–Ruelle
and Wolf algorithms as compared to the technique of Rosenstein et al. which is less susceptible to these errors.
However, it was not possible to confirm all criteria. The
measurement of the metric entropy is still ongoing. 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
the measurement of Lyapunov exponents from the original
flow data. Thus it was not possible to get an agreement between exponent estimates from the section and from the flow,
nor was it expected. For the 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 are reversed may not be a suitable criterion for
noisy data.
F. 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 共UPOs兲.
The identification of UPOs plays a critical role in many
chaos control algorithms.40 Most standard chaos control algorithms attempt to control the system onto a UPO while
operating within the chaotic regime. Small time-dependent
perturbations applied to an accessible parameter may then be
used to force the system onto the stable manifold and hence
enforce stability and a 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
Reiss et al.: Loudspeaker chaos
2039
FIG. 16. 共Color online兲 A delay coordinate embedding plot of the times
between zero crossings. Periods 1–3 points are identified, along with the
stable and unstable manifolds of the period 1 orbit.
occasional proportional feedback control schemes in tracking
and targeting of trajectories, and in the identification of symbolic dynamics.40,41 Thus we will also attempt to characterize
the UPOs exhibited during the chaotic state.
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 ␧,
储Y共n兲 − Y共n + p兲储 ⬍ ␧,
共12兲
in which case a periodic orbit exists in the vicinity of Y共n兲
and a least squares fit of all data in the region can be used to
estimate its exact location. The exact value of ␧ may be
varied depending on the size of the data set, the period p, and
the number of UPOs that one wishes to find. False positive
UPOs may be identified using the mean squared error of the
least squares fit.
Figure 16 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 s, corresponding to the
drive frequency of the system, 45 Hz. Its eigenvalues are
0.155 and −1.571, with corresponding eigenvectors 共1,0.155兲
and 共−0.637, 1兲. Similar results can be obtained for other
identified periodic orbits, and these results can be used to
implement a chaos control technique.
IV. FURTHER ANALYSIS AND CONCLUSIONS
An analysis was attempted on time series data from an
experimental electrodynamic loudspeaker in order to characterize the embedding dimension, the fractal dimension, the
Lyapunov exponents, and the UPOs. The obtained results
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.
2040
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
Various estimates of fractal dimension were performed,
including the measurement of correlation dimension,26,27 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.
However, the estimation of the dominant Lyapunov exponent, which is less reliant on a large data set size, provided
consistent results regardless of the method of estimation.10
We used several different techniques that provided a rough
agreement in their estimates of the dominant exponent, and
the results further agreed with the theory concerning the
Lyapunov spectrum and its properties. They reliably showed
evidence of a positive Lyapunov exponent, a strong indicator
of chaos.
Finally, we attempted to estimate the eigenvalues and
eigenvectors associated with detected UPOs. 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 well-known algorithms.
However, extraction of many empirical quantities, particularly fractal dimension, proved difficult due to a number
of issues. This may be accounted for partly by the nonstationarity and short data set size. Although the original data
set is over 200 000 points, it represents about 10 000 orbits.
This is insufficient for an 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. A 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 were
sampled at 1024 samples/ s, 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, 45 Hz, then this
would produce a natural Poincaré section. The analyzed data
had a 16 bit precision, but this was further limited by the
8 bit resolution of the A/D converter of the laser distance
meter. This meant that there was a significant uncertainty in
the location of nearby points. Since most analyses of chaotic
time series rely on an analysis of near neighbors in a locally
linear region, these resulted in inaccuracies in the estimation
of fractal dimension and Lyapunov exponents.
Current work is focused on improvements to the data
acquisition system. This would allow a more accurate analysis of the data, including measurements of how fractal and
Lyapunov dimensions 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 a
dynamical behavior from enhanced models of the loudspeaker. An accurate model of an electrodynamic loudReiss et al.: Loudspeaker chaos
speaker would represent a significant advance in the field,
particularly since model parameters could then be modified
to yield a loudspeaker design with optimal performance.
1
P. Klipsch, “Loudspeaker distortion,” Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP
’76, Philadelphia, PA, pp. 543–546.
2
D. Davis and E. Patronis, Sound System Engineering, 3rd ed. 共Elsevier,
New York, 2006兲.
3
E. S. Olsen and K. Thorbog, “Diaphragm area and mass nonlinearities of
cone loudspeakers,” Proceedings of the 99th AES Convention, New York,
1995, Convention Paper No. 4082.
4
M. Colloms, High Performance Loudspeakers, 6th ed 共Wiley, New York,
2005兲.
5
E. S. Olsen, “Nonlinear modeling of low frequency loudspeakers: A practical implementation,” Proceedings of the 102nd AES Convention, Munich, Germany, 1997, Convention Paper No. 4469.
6
P. Woafo, “Harmonic oscillations, stability and chaos control in a nonlinear electromechanical system,” J. Sound Vib. 259, 1253–1264 共2003兲.
7
P. Woafo, “Stability and chaos control in electrostatic transducers,” Phys.
Scr. 62, 255–260 共2000兲.
8
A. Steindl and H. Troger, “Chaotic oscillations of a fluid-conveying viscoelastic tube,” in Nonlinearity and Chaos in Engineering Dynamics, edited by J. M. T. Thompson and S. R. Bishop 共Wiley, Chichester, 1994兲, pp.
231–240.
9
J. Argyris, V. Belubekyan, N. Ovakimyan, and M. Minasyan, “Chaotic
vibrations of a nonlinear viscoelastic beam,” Chaos, Solitons Fractals 7,
151–163 共1996兲.
10
A. Petosic, I. Djurek, and D. Djurek, “Modeling of an electrodynamic
loudspeaker using Runge-Kutta ODE solver,” 122nd Convention of Audio
Engineering Society, Vienna, Austria, 2007.
11
T. Leishman and G. Dix, “Additional modeling details for the suspension
of a moving-coil loudspeaker,” J. Acoust. Soc. Am. 116共4兲, 2619 共2004兲.
12
R. A. Ribeiro, A. J. Serralheiro, and M. S. Piedade, “Application of Kalman and RLS adaptive algorithms to non-linear loudspeaker controller
parameter estimation: a case study,” IEEE International Conference on
Acoustics, Speech, and Signal Processing 共ICASSP兲, Philadelphia, 2005.
13
M. D. Fränken and J. Wassmuth, “Passive parametric modelling of dynamic loudspeakers,” IEEE Trans. Speech Audio Process. 9, 885–891
共2001兲.
14
R. J. Wei, Q. T. Tao, and W. S. Ni, “Bifurcation and chaos of direct
radiation loudspeaker,” Chin. Phys. Lett. 3, 469–472 共1986兲.
15
D. Djurek, I. Djurek, and A. Petosic, “Intrinsic membrane friction and
onset of chaos in an electrodynamic loudspeaker,” Proceedings of the
123rd Aes Convention, New York, NY, 2007.
16
I. Djurek, A. Petosic, and D. Djurek, “Chaotic state in an electrodynamic
loudspeaker controlled by gas pressure,” Proceedings of the 153rd ASA
Meeting, Salt Lake City, UT, 2007.
17
D. Djurek, I. Djurek, and A. Petosic, “Chaotic state in an electrodynamic
loudspeaker,” Proceedings of the 122nd Convention of the AES, Vienna,
Austria, 2007.
18
H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge
Nonlinear Science Series 共Cambridge University Press, Cambridge, England, 2004兲.
19
I. Djurek, D. Djurek, and A. Petosic, “Damping of an electrodynamic
loudspeaker by air viscosity and turbulence,” Proceedings of the 123rd
J. Acoust. Soc. Am., Vol. 124, No. 4, October 2008
Convention of Audio Engineering Society, New York, NY, 2007.
T. Y. Li and J. A. Yorke, “Period three implies chaos,” Am. Math. Monthly
82, 985–992 共1975兲.
21
M. J. Feigenbaum, “The universal metric properties of nonlinear transformations,” J. Stat. Phys. 21, 669–706 共1979兲.
22
A. M. Fraser, “Reconstructing attractors from scalar time series: A comparison of singular system and redundancy criteria,” Physica D 34, 391–
404 共1989兲.
23
A. M. Fraser and H. L. Swinney, “Independent coordinates for strange
attractors from mutual information,” Phys. Rev. A 33, 1134–1140 共1986兲.
24
J. D. Reiss, N. Mitianoudis, and M. B. Sandler, “Computation of generalized mutual information from multichannel audio data,” Proceedings of
the Audio Engineering Society 110th Convention, Amsterdam, The Netherlands, 2001.
25
M. B. Kennel, R. Brown, and H. D. I. Abarbanel, “Determining embedding dimension for phase-space reconstruction using a geometrical construction,” Phys. Rev. A 45, 3403–3411 共1992兲.
26
P. Grassberger and I. Procaccia, “Measuring the strangeness of strange
attractors.,” Physica D 9, 189–208 共1983兲.
27
P. Grassberger and I. Procaccia, “On the characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 共1983兲.
28
A. M. Fraser, “Information and entropy in strange attractors,” IEEE Trans.
Inf. Theory 35, 245–262 共1989兲.
29
J. D. Reiss, “The analysis of chaotic time series,” Ph.D. thesis, School of
Physics, Atlanta, Georgia Institute of Technology, 2001, pp. 250.
30
J. P. Crutchfield and N. H. Packard, “Symbolic dynamics of noisy chaos,”
Physica D 7, 201 共1983兲.
31
U. Parlitz, “Identification of true and spurious Lyapunov exponents from
time series,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 155–165 共1992兲.
32
P. Constantin and C. Foias, “Global Lyapunov exponents, Kaplan-Yorke
formulas and the dimension of attractors for 2D Navier-Stokes equations,”
Commun. Pure Appl. Math. 38, 1–27 共1985兲.
33
J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, “Testing for nonlinearity in time series: The method of surrogate data,” Physica
D 58, 77–94 共1992兲.
34
J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, and S. Ciliberto, “Liapunov
exponents from time series,” Phys. Rev. A 34, 4971–4979 共1986兲.
35
M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for
calculating largest Lyapunov exponents from small data sets,” Physica D
65, 117–134 共1993兲.
36
A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining
Lyapunov exponents from a time series,” Physica D 16, 285–317 共1985兲.
37
H. D. I. Abarbanel, “Tools for the analysis of chaotic data,” in Nonlinear
Dynamics and Time Series, Fields Institute Communications, edited by C.
D. Cutler and D. T. Kaplan 共American Math. Soc., Providence, RI, 1997兲,
pp. 1–16.
38
A. Wolf, “Quantifying chaos with Lyapunov exponents,” in Chaos, edited
by A. V. Holden 共Manchester University Press, Manchester, 1986兲.
39
A. Wolf and J. Vastano, “Intermediate length scale effects in Lyapunov
exponent estimation,” in Dimensions and Entropies in Chaotic Systems,
edited by G. Mayer-Kress 共Springer-Verlag, Berlin, 1986兲.
40
E. Schöll and H. G. Schuster, Handbook of Chaos Control, 2nd ed. 共Wiley,
New York, 2007兲, pp. 849.
41
K. Mischaikow, M. Mrozek, J. Reiss, and A. Szymczak, “Construction of
symbolic dynamics from experimental time series,” Phys. Rev. Lett. 82,
1144–1147 共1999兲.
20
Reiss et al.: Loudspeaker chaos
2041
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement