IOP PUBLISHING
PLASMA PHYSICS AND CONTROLLED FUSION
Plasma Phys. Control. Fusion 55 (2013) 085015 (7pp)
doi:10.1088/0741-3335/55/8/085015
Permutation entropy analysis of
temperature fluctuations from a basic
electron heat transport experiment
J E Maggs and G J Morales
Department of Physics and Astronomy, University of California, Los Angeles, Los Angeles,
CA 90095, USA
Received 7 February 2013, in final form 19 May 2013
Published 10 June 2013
Online at stacks.iop.org/PPCF/55/085015
Abstract
The permutation entropy concept of Bandt and Pompe (2002 Phys. Rev. Lett. 88 174102) is
used to analyze the fluctuations in ion saturation current that spontaneously arise in a basic
experimental study (Pace et al 2008 Phys. Plasmas 15 122304) of electron heat transport in a
magnetized plasma. From the behavior of the Shannon entropy and the Jensen–Shannon
complexity it is found that the underlying dynamics are chaotic rather than stochastic. A
partitioning and scrambling technique is used to demonstrate that the exponential character of
the associated power spectrum arises from individual Lorentzian pulses observed in the
time series.
(Some figures may appear in colour only in the online journal)
present. The broadband frequency spectrum is exponential
in nature.
Two major questions motivated by these experimental
findings are: what is the reason for the characteristic
exponential frequency dependence of the fluctuation spectrum
during the anomalous transport period, and what is the
nature of the underlying dynamics that result in anomalous
heat transport? Two prominent theoretical approaches that
are candidates for answering these questions are based on
fundamentally different concepts: chaotic behavior [5, 6]
and stochastic processes [7–9]. Therefore, it is important
in answering these questions to utilize methods of signal
analysis that are capable of distinguishing between these
two dynamical systems. This investigation uses two such
methods: partitioning and temporal scrambling of time series,
and, permutation entropy analysis. The temporal scrambling
demonstrates that individual Lorentzian pulses are responsible
for the exponential character of the power spectrum, while
the permutation entropy analysis, based on the concepts of
Bandt and Pompe [10] and displayed in the entropy-complexity
plane [11], identifies that the dynamics in the temperature
filament are chaotic.
The paper is organized as follows. Section 2 presents
the signal scrambling analysis leading to the identification
of Lorentzian pulses as the origin of the exponential power
spectrum. Section 3 provides a brief introduction to the
permutation entropy method. Section 4 applies the entropy and
1. Introduction
The purpose of this paper is to conclusively identify the nature
of the underlying dynamics that cause anomalous transport in
a basic experiment of electron heat transport in a magnetized
plasma and to suggest that the techniques described here can
be usefully employed to examine signals from other plasma
experiments. The details of the experimental arrangement
and the major findings have been well documented [1–3], so
that only a cursory description is necessary for the present
investigation. The experiment typically uses a small (3 mm
diameter), single-crystal LaB6 cathode to inject a low-voltage
electron beam into a strongly magnetized, cold, afterglowplasma. The low-voltage beam acts as an ideal heat source and
produces a long (∼8 m), narrow (∼5 mm in radius) filament
of elevated electron temperature that is disconnected from
the walls of the device, and is surrounded by, an essentially
infinite, plasma of much lower temperature. The existence of
a transition from a period of classical transport, i.e. transport
due to Coulomb collisions, to one of anomalous transport has
been established through detailed measurements. During the
period of classical transport, drift-Alfvén waves grow linearly,
driven by the temperature gradient in the filament edge [4].
The transition in transport is characterized by a change in
the character of the frequency power spectrum of temperature
fluctuations, from a line spectrum in the classical transport
case, to a broadband spectrum when anomalous transport is
0741-3335/13/085015+07$33.00
1
© 2013 IOP Publishing Ltd
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Plasma Phys. Control. Fusion 55 (2013) 085015
J E Maggs and G J Morales
0.20
Log Power (arb. units)
1024 pieces
0.15
Lorentzian
pulse
256 pieces
64 pieces
0.10
0.05
s
16 pieces
0.00
Frequency (kHz)
2.4
2.5
2.6
Figure 2. A portion of one of the time signals from the temperature
filament experiment used in this analysis. A Lorentzian pulse (red
curve) with a width corresponding to the slope of the power spectra
is superposed on an isolated pulse appearing in the data and whose
peak is identified with the vertical black line. Scrambled signals are
constructed from 16 (red), 64 (orange), 256 (green) and 1024 (blue)
pieces. The temporal length of these pieces relative to the typical
Lorentzian pulse is indicated by the arrows of various colors.
complexity measures to experimental time series. Section 5
provides conclusions.
into 2n equal-length pieces and then randomly rearranging the
pieces. In the example presented here, each member of the
original ensemble of one hundred signals is broken into 16,
64, 256 and 1024 pieces and then randomly scrambled.
A portion of one representative time trace taken from
the original ensemble is shown in figure 2 together with a
superposed Lorentzian pulse (red solid curve), whose width,
τ = 5.0 µs, corresponds to the time constant deduced from
the power spectrum fit, exp(−2ωτ ), shown previously as the
dashed red line in figure 1. Also illustrated in figure 2, relative
to the representative Lorentzian pulse, are the sizes of the time
intervals used in the partitioning and scrambling process. The
temporal length of each segment in the partition with 64 pieces
(82 µs) is over 16 times the typical Lorentzian pulse width. In
contrast, the length of each segment in the 256-piece partition
(20 µs) is only about four times the typical pulse width, and
the interval with 1024 pieces is equal to the pulse width. It is
expected that scrambling the signal with partitions of 16 and
64 pieces will leave many of the pulses intact, while the 1024
piece partition should destroy most of the pulses.
The effect of the scrambling process on the ensemble
power spectra is shown in figure 3 in a log-linear format. For
ease of viewing, each power spectrum displayed is multiplied
by a factor of 102n (n = 0, 1, 2, 3, 4) to separate it from the
one below it. The power spectrum for the unscrambled signal
(bottom trace) is the same as displayed in figure 1. The curve
appears ‘smoother’ here because fewer points are used in the
displayed trace in order to reduce the file size of the figure.
A linear fit to the ‘exponential region’ and the corresponding
pulse width to which the slope corresponds are shown as dashed
lines (color coded) for each partition of the scrambled signal.
The partitions of 16 and 64 pieces do not affect the linear
(i.e. exponential) portion of the power spectrum very much.
The extent of the linear region decreases somewhat and the
2. Power spectrum
An ensemble of one hundred signals, each consisting of
5.25 ms of data, is used in the analysis of the power spectrum.
The temporal records are 8192 points in length collected with a
time increment of 0.64 µs. The individual signals correspond
to ion saturation current, Isat , collected by a small Langmuir
probe placed within the temperature filament. The signals
in this data set, collected when the probe is near the center
of the filament, display mainly negative pulses, presumably
representing decreased temperature. The ensemble-averaged
power spectrum of the signals is shown in figure 1 in a loglinear format. The power spectrum is exponential over the
range from 5–110 kHz with prominent lines at frequencies
between 5–50 kHz. The lowest frequency line, at about 8 kHz,
corresponds to a resonant thermal diffusion wave [12] while
the other peaks are drift-Alfvén waves [4]. The slope of the
exponential portion of the spectrum (represented by the red,
dashed, straight line) corresponds to a temporal Lorentzian
pulse whose width, τ , is 5 µs. Lorentzian pulses have the
functional form
A
,
1 + [(t − t0 )/τ ]2
2.3
Time (ms)
Figure 1. The ensemble-averaged fluctuation power spectrum of the
unscrambled signals. The peaks below 50 kHz arise from drift
waves and a thermal diffusion wave. The time constant associated
with the exponential fit to the power spectrum (i.e. exp(−2ωτ )) is
τ = 5.0 µs.
L(t) =
2.2
(1)
where t0 is the time of peak arrival, τ is the pulse width and A
is the peak amplitude. The power spectrum of the Lorentzian
pulse in equation (1) is proportional to exp(−2ωτ ).
The proximate cause of the exponential power spectra is
the occurrence, in the Isat time signals, of Lorentzian-shaped
pulses of the type shown in equation (1). This assertion is tested
by investigating the effects of partitioning and scrambling
the ensemble of signals [13]. The scrambled signals are
constructed from the ensemble by breaking the original signal
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Plasma Phys. Control. Fusion 55 (2013) 085015
J E Maggs and G J Morales
the total number of data points typically stored in an actual
experimental situation. Thus, in practical implementations of
the concept, the value of d is less than 7, and for the results
presented here the value, d = 5, is used. Once the Bandt–
Pompe probability distributions are computed for an ensemble
of signals, the statistical nature of the signal can be evaluated by
determining its location in the entropy-complexity plane [11].
There are a variety of entropy definitions in common usage
(Shannon, Tsallis and Renyi, for example) and even more
measures of statistical complexities, as presented by Martin
et al [14]. In the subsequent discussions, following Rosso
et al [11], Shannon’s formulation is used to evaluate entropy
and the Jensen–Shannon divergence serves as a measure of
statistical complexity.
The nature of the dynamics resulting in a particular
experimental time signal can be determined from its location in
the entropy-complexity plane, or, using the notation of Rosso
et al [11] , the so called CH-plane, when the normalized
Shannon entropy, H, and Jensen–Shannon complexity, CJS , are
used as measures. For a set of probabilities, P , of dimension N
where pj denotes the probability of occurrence for one of the
N possible states, pj 0; j = 1, 2, . . . , N and N
1 pj = 1,
the Shannon entropy, S and normalized Shannon entropy, H ,
are defined as [14]
N
S (P ) = −
pj ln pj ;
Figure 3. The ensemble averaged power spectra of the scrambled
signals. Dashed lines are the best fit to exponential dependence with
τ corresponding to the equivalent width of a Lorentzian pulse.
When the Lorentzian pulses are destroyed by the temporal
scrambling the exponential feature disappears.
slope decreases slightly for the 64-piece partition. However,
the 64-piece partition procedure does greatly reduce the ‘line’
features in the spectrum that arise from the presence of drift
waves. The 256-piece partition still exhibits a linear behavior,
albeit over a much reduced range of frequencies and the slope
is shallower, corresponding to a narrower pulse. However,
the effect of the 256-piece partition is very evident at low
frequencies where the power spectrum is now flat. Finally, the
1024-piece partition power spectrum is flat over the range from
5 to 100 kHz and is similar to ‘white’ noise. The systematic
change in the power spectra as the signals are partitioned into
smaller and smaller pieces until the pulses are destroyed clearly
demonstrates that the Lorentzian pulses are the cause of the
exponential part of the spectrum.
1
H (P ) = S (P )/max (S) = S (P )/ln (N ).
(2)
Note that the maximum Shannon entropy is obtained when
all states have equal probability, pj = 1/N ; j = 1, 2, , N.
This maximum entropy state is denoted as Pe . The Jensen–
Shannon complexity is defined as [14]
1
e
− 2 S (P ) − 21 S (Pe )
S P +P
S
2
CJ = −2 N +1
H (P ) .
ln (N + 1) − 2 ln (2 N ) + ln (N )
N
(3)
To illustrate the different regions where chaotic and stochastic
signals appear in the CH plane, figure 4 shows two well-known
examples of such dynamical behaviors, the chaotic logistic
map [15] and the stochastic fractional Brownian motion (fBm)
[16]. The green curve marked ‘fBm’ in figure 4 is the locus of
points of fractional Brownian motion, whose increments are
fractional Gaussian noise with Hurst exponent, He , ranging, in
steps of 0.05, from 0.025 to 0.975, (0.025 He 0.975). The
densely spaced collection of red dots represents the locations,
in the CH-plane, of the logistic map, xn+1 = rc xn (1.0 − xn )
for those values of rc in the range, 3.58 < rc < 4.0, for which
the logistic map exhibits chaotic behavior [15].
Also shown in figure 4 are two curves labeled ‘maximum
complexity’ and ‘minimum complexity’. The genesis of these
curves is discussed in detail by Martin et al [14], and only
a brief description is given here. For a chosen embedding
dimension, d, there are N = d! possible states in the Bandt–
Pompe probability space, i.e. the Bandt–Pompe probability
space has dimension N . Maximum Shannon entropy is
achieved when all states are equally populated, pj = 1/N for
j = 1, 2, 3, ..., N . Minimum complexity, for an entropy less
than maximum, corresponds to one state having probability,
3. Entropy and complexity measures
The statistical character of a time signal can be determined by
obtaining its permutation entropy and statistical complexity,
which can be computed from a probability distribution
introduced by Bandt and Pompe [10]. This quantity represents
the probability of occurrence of the various (Nt − d + 1)
realizations of the d! amplitude orderings of the d-tuples
that appear in a signal consisting of Nt discrete elements each
having an amplitude Aj with 0 j Nt −1. As a concrete
example for a case with d = 5, a signal of Nt = 20 000
elements has 19996 distinct 5-tuples (tj , tj +1 , tj +2 , tj +3 , tj +4 ).
Within each 5-tuplet, an ordering of the amplitude of the signal
recorded, e.g. (Aj , Aj +1 , Aj +2 , Aj +3 , Aj +4 ), is just one of a
possible set of 120 (i.e. 5!) permutations. The Bandt–Pompe
probability distribution of a particular time signal is determined
by computing the frequency of occurrence of each of the
possible permutations of the amplitude ordering observed in
the signal. The number of possible amplitude permutations
of a d-tuplet increases very rapidly with the value of d (e.g.
10! = 3 628 800), accordingly, this number can easily exceed
3
Plasma Phys. Control. Fusion 55 (2013) 085015
CJS
d=5
J E Maggs and G J Morales
t, is chosen small enough to capture the dynamics of interest
to the experimenter. The time step determines the Nyquist
frequency fN = 1/2t, and dynamical process that occur at
frequencies higher than fN are not resolved, but may affect
the power spectra through the phenomena of aliasing. It is
assumed here that care is taken by the experimenter to avoid
the effects of aliasing, so that only the effects of noise need
be considered. Noise in the time signals increases the entropy
when using the Bandt–Pompe probability. Permutations of
the basic amplitude ordering of a d-tuplet, [1, 2, 3, . . ., d]
that are not realized by a dynamic process of interest can,
however, be populated by noise. Maximum entropy occurs
when all possible amplitude permutations occur equally in a
time signal. Thus it is desirable to reduce the effects of noise
on experimental signals to better ascertain the true nature of the
dynamical process of interest. Of course, this is problematic
in some cases, because the dynamical process under study
may be stochastic and strongly resemble noise. A common
technique for reducing noise is to digitally filter the time
signals. Digital filtering alters the Fourier amplitude of the
signals, and, if the frequency range of the noise in the signals
can be identified, digital filtering can effectively eliminate
noise. Another technique for reducing the effects of noise
is wavelet de-noising [17] in which the wavelet coefficients,
instead of the Fourier amplitudes, of the signals are altered
in order to reduce the contributions of noise to the signals.
Both filtering and wavelet de-noising were explored, but these
techniques are found to greatly alter the locations in the CHplane of both chaotic and stochastic signals. A technique
for reducing the effects of noise that does not much alter
the location of stochastic processes in the CH-plane is subsampling, or the embedding delay [18].
The Bandt–Pompe probability computed for a specific
collection of d-tuplets garnered from a time signal involves
the concept of the d-tuplets representing structures in an
embedding space of dimension d [14, 18]. In its primal form,
the size, in time, of the structures investigated is dt, where,
t is the sampling time. As mentioned earlier, since the
number of possible structures grows as d!, it is not practical to
investigate larger structures in time by increasing the size of the
d-tuplet. Rather, larger structures in time can be investigated
by the technique of sub-sampling or embedding delay. In the
sub-sampled signal the interval between successive data points
is m t rather than t where m is taken to be a positive integer
in order to avoid interpolation. The sub-sampling technique
reduces the Nyquist frequency, and, of course, the number of
points in the signal, N , is reduced to N /m, but the total time
length of the signal is unaltered. Thus, sub-sampling limits
the upper frequency range of the dynamics under investigation
without changing the low frequency information. In contrast,
the lower range of frequencies available for investigation can
be limited by shortening the length of the time records used in
the analysis.
Often in the analysis of experimental data, certain features,
or frequency intervals, of the power spectra are of particular
interest. For example, in the temperature filament data
considered here, the range of frequencies over which the data
displays an exponential behavior is of particular interest, as
Logistic Map
MAXIMUM
COMPLEXITY
0.6
cB
Chaoti ehavior
0.4
Stochastic Beh
avi
or
0
fBm
MINIMUM
COMPLEXITY
0.2
0.2
0.4
H
0.6
0.8
1
Figure 4. Chaotic and stochastic dynamical processes occupy
different regions of the entropy-complexity plane. Chaotic
processes, as represented by the logistic map (densely spaced red
dots), reside in the lilac-shaded region, and stochastic processes,
represented by fractional Brownian motion (fBm) shown by the
green curve, reside in the un-shaded region. The embedding space
used to evaluate the Bandt–Pompe probability has dimension d = 5.
pk , with 1/N < pk 1, and all other states having equal
probabilities, pj = (1 − pk )/(N − 1); j = 1, 2, 3, . . .,
N , j = k. As a concrete example, for d = 5, and thus
N = 120, pk might be chosen to have 60 values between 1/120
and 1. The minimum complexity is then computed on a space
of dimension (60, 120). A realization of a time signal with
minimum complexity is a slowly changing signal with short
bursts of white noise.
In contrast, maximum complexity is computed from the
collection of N − 1 probability distributions associated with
probability spaces having dimensions ranging from 2 to N .
These N − 1 probability distributions have one state with
probability pk(i) , where 0 pk(i) 1/(N − i + 1) and
all other (N − i) states have equal probabilities, pj (i) =
(1 − pk(i) )/(N − i); j (i) = 1, . . ., N − i + 1, j (i) = k(i)
and where i = 1, ..., N − 1. As a concrete example of
one of the probability distributions in the N − 1 collection
of distributions used in determining the maximum complexity
curve, consider the case, i = N − 5. Then the probability
space dimension is 6 and pk(N −5) can range from 0 to 1/6. If
pk(N −5) = 0, the remaining 5 states have equal probabilities,
pj (N −5) = 1/5, if pk(N −5) = 1/12, pj (N −5) = 11/60, and
so forth. The maximum probability curve corresponds to the
locus of maximum complexities for the collection of entropycomplexity curves computed from this set of probability
distributions. If the various pk(i) vectors are chosen to have
dimension 60, as in the minimum complexity example, the
maximum probability is computed over a (119, 60, 120)
dimensional space.
4. Application to time signals
Experimental time signals gathered from plasma probes are
typically obtained from analog to digital converters in which a
voltage value is recorded at discrete time steps. The time step,
4
Plasma Phys. Control. Fusion 55 (2013) 085015
Hindmarsh
J E Maggs and G J Morales
(a)
(b)
Subsample
CJS
(c)
H
(d)
0.6
Gissinger
Log Power (arb. units)
d=5
Maximum
Complexity
G
L
s_D
D
s_L
s_H
0.4
(e)
(g)
D
H
L
G
0.2
s_fBm
fBm
(f )
Lorenz
Data
s_G
Minimum
Complexity
(h)
0.0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
0.2
0.4
H
0.6
0.8
1
1
Frequency (f/fN)
Figure 5. The effects of the sub-sampling operation are illustrated. (b),(d),(f ) and (h) are sub-sampled spectra extracted from time signals
whose spectra are shown in the adjacent left panels. Sub-sampling reduces the Nyquist frequency fN without changing the spectral shape.
The corresponding locations, in the CH plane, of the sub-sampled time series for the Hindemarsh (s H), Gissinger (s G), Lorenz (s L)
models and data (s D) all move toward the region of moderate entropy and maximum complexity, while sub-sampled fractional Brownian
motion (s fBm) is relatively invariant.
illustrated in figure 1. In employing the concept of Bandt–
Pompe probability, the sub-sampling technique can be used
to limit the range of frequencies in the power spectrum and,
thereby, limit the study of the dynamics to those time scales
of interest. This procedure is analogous to that employed in
analysis of stochastic time signals [19] in which the spectral
range of the power spectrum is limited to only those frequencies
over which the power spectrum exhibits a power law behavior.
Figure 5 illustrates the effects of sub-sampling on the
power spectra of chaotic signals obtained from numerical
solutions of some well-known nonlinear dynamics models:
Gissinger (G) [20], Hindemarsh (H) [21] and Lorenz (L) [22].
The effects on the data (D) used to generate figure 1 are
also indicated. All power spectra are shown as functions of
frequency, normalized to the Nyquist frequency, fN , whose
numerical value changes after sub-sampling. The power
spectra of the original signals recorded are shown in the leftside panels ((a), (c), (e), (g)). The corresponding power
spectra of the sub-sampled signals are shown in the adjacent,
left-side panels ((b), (d), (f ), (h)). Panel (h) is essentially
the same as figure 1. The sub-sampling interval is separately
chosen for each class of signal so that the Nyquist frequency
is reduced to where the spectral range of interest spans the
majority of the normalized frequency range. The signals
from nonlinear dynamics models (G, H and L) are calculated
using small time steps to ensure accuracy in the Runga–Kutta
integration, so that the Nyquist frequency of the calculated
signals is much higher than needed to capture the dynamics.
Thus, the sub-sampling intervals for these signals is large, 20
for the Gissinger model, 80 for the Hindemarsh model and
18 for the Lorenz model. In contrast, the data was only subsampled on an interval of 4 to limit the spectral range of interest.
The right side of figure 5 shows the locations, in the CHplane, of the original signals (G, H, L and D) together with
the sub-sampled signals (s G, s H, s L and s D). The subsampled signals move toward moderate entropy and higher
complexity. For comparison, the dashed curves display the
corresponding effect of sub-sampling, with an interval of 8,
on the stochastic signal associated with fractional Brownian
motion (fBm) shown earlier in figure 4. It is seen that the locus
of fractional Brownian motion (fBm) is not much changed by
the sub-sampling operation (s fBm). The power spectra of
the fractional Brownian motion signals are not shown because
they are power laws (by construction) over the entire frequency
range and remain so under the sub-sampling operation.
Figure 6 shows a comparison between data obtained
from the temperature filament experiment and the output of a
numerical model of chaotic advection [23, 24] that incorporates
the essential features of the experimental arrangement.
Figure 6(a) shows the location in the CH-plane of numerical
output from the model and data from the experiment. Spatial
contours of the temperature in the plane across the confinement
5
Plasma Phys. Control. Fusion 55 (2013) 085015
J E Maggs and G J Morales
Figure 6. (a) Locations in the CH plane of temperature filament data (crosses), and numerical output of chaotic advection model (triangles).
The shaded region, as in figure 4, delineates the region of chaotic behavior. The stochastic process of fractional Brownian motion (fBm) is
also shown for comparison. (b) Instantaneous spatial contours (x, y) of normalized electron temperature from the chaotic advection model
at a particular time. Time signals of temperature are obtained from the spatial locations indicated by ‘black’ dots to generate locations in CH
plane. (c) Same as in (b), but for experimental data.
by the authors [23] that the dynamics associated with transport
in the temperature filament is chaotic.
Figure 7 shows the changes in the location of the data
signals used to generate figure 3 under the partitioning and
scrambling operation. The signals are first sub-sampled, with
sub-sample interval m = 4, so that the spectral range is
limited to that shown in figure 1. A red triangle marked
(u) represents the ensemble averaged position of the subsampled, unscrambled signals. Other red triangles marked
with 16, 64, 256 and 1024 are ensemble-averaged positions of
the partitioned and scrambled, sub-sampled signals, with the
number of pieces in the partition indicated. Consistent with the
behavior of the power spectra shown in figure 3, the partitioned
and scrambled signals become less chaotic in nature and move
toward the white noise region of the CH plane as the number
of partitions grows.
magnetic field (x, y) obtained from the numerical solution of
the chaotic advection model are shown in (b) and measured
data (different than used in figure 1) from the temperature
filament experiment in (c). Time signals are taken from sixteen
locations along a circle with radius 0.6 cm as indicated by the
‘black’ dots in (b) and (c). Nyquist frequencies of both the
model and the experimental signals are adjusted by choosing
a sub-sampling interval that results in the exponential part of
the power spectra extending over at least half the frequency
range. The sub-sampled time signals from both the model and
experiment have 4000 points. The Bandt–Pompe probabilities
of time signals from the model and experiment are computed
for an embedding space with dimension, d = 5, so that
each sub-sampled time signal contains 3996 5-tuplets. For
a uniform distribution, the average number of realizations, per
tuplet, is roughly 33.
The locations of the 16 model points are shown as triangles
in figure 6(a) and the experimental data are shown as crosses.
Both the model output and experimental data are within the
shaded region that demarks ‘chaotic behavior’. The dynamics
of the model have been identified by orbit sampling techniques
to be chaotic, so it is reassuring that the sampled time series
falls in the ‘chaotic behavior’ region of the CH-plane. The
very close proximity of the experimental data to the chaotic
advection model data bolsters the assertion made previously
5. Conclusions
The method of ‘partitioning and scrambling’ has been shown
to destroy the exponential character of the power spectrum
of fluctuations in ion saturation current measured in a basic
experiment of electron heat transport. But importantly, the
destruction occurs only when the temporal scrambling interval
is chosen small enough to disrupt the individual Lorentzian
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Plasma Phys. Control. Fusion 55 (2013) 085015
J E Maggs and G J Morales
spectrum, ascertains that the underlying dynamic processes in
the temperature filament experiment are chaotic. Since it has
been well established [25–35] in a broad range of systems
that a signature of deterministic chaos is the generation of
exponential power spectra, the two independently established
findings of this study are entirely consistent with previous
knowledge common to the fluid turbulence and nonlinear
dynamics communities. The study contributes yet a further
example, from plasma physics, of this connection.
The relatively simple systems considered in this study
have illustrated the usefulness of combining the Pompe–Bandt
probability with a display in the entropy-complexity plane to
identify chaotic and stochastic behavior. This signal analysis
technique has the potential to uncover new features in a wide
range of fusion and basic plasma experiments.
CJS
Maximum
Complexity
d=5
0.6
u
16
64
0.4
fBm
256
Minimum
Complexity
0.2
1024
white noise
0
0.2
0.4
H
0.6
0.8
1.0
Figure 7. The locations, in the CH-plane, of the scrambled signals
used to generate the power spectra displayed in figure 3. As the
number of scrambled partitions is increased, the position in the CH
plane moves toward maximum entropy and minimum complexity,
approaching the location of ‘white noise’.
Acknowledgments
The work of JEM and GJM is performed under the auspices of
the BaPSF at UCLA, which is jointly supported by a DOE-NSF
cooperative agreement, and by DOE grant SC0004663.
pulses observed in the time signals. The exponential character
of the spectrum remains basically intact until the width of
the scrambling window is below 20 µs (the 256 and 1024
piece partitions), which is approximately the auto-correlation
time for a single pulse. In addition, the dynamical nature of
the signals, as indicated by their location in the C–H plane
does not change appreciably until the scrambling partition is
smaller than 20 µs in length. Thus, as long as the scrambling
process does not significantly affect the individual pulses,
both the statistical nature, as measured by the permutation
entropy, and the power spectrum remain relatively unchanged.
In previous work [3] it was demonstrated that the measured
value of the Lorentzian pulse width, τ , obtained from fitting
individual pulse events in the time series, is consistent with the
independent determination of τ from the slope of the power
spectrum in a log-linear plot. Combining this information with
the behavior of the spectrum when the scrambling technique
is applied, we conclude that the Lorentzian pulses observed in
the temperature filament data are responsible for the observed
exponential character of the power spectrum.
It has been illustrated how the Bandt–Pompe probability
can be used, in conjunction with the CH-plane, to clearly
separate chaotic from stochastic behavior in well-known
dynamical models that exhibit only one behavior or the other.
Furthermore, the effect of ‘time sub-sampling’ or ‘embedding
delay’ on the location of these systems in the CH-plane has
been established. The chaotic systems systematically move
into the region of moderate entropy and high complexity,
while the stochastic systems remain basically unchanged. This
implies that sub-sampling can extract chaotic signals that
are contaminated by extraneous noise that is not an intrinsic
property of the underlying dynamics. When the method is
applied to the data obtained from the heat transport experiment,
it is found that the location in the CH-plane moves from a
region on the edge of stochasticity to the middle of the region
occupied by chaotic systems. This behavior, independently of
the previous findings about the exponential origin of the power
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