# Experimental Investigation of the Transition to Spatiotemporal Chaos with a System-Size Control Parameter

Authors: Daniel R. Spiegel; Elliot R. Johnson
Journal: Research Letters in Physics
 (2008)

---

## Abstract

Using a localized laser-heating method to allow the use of system size as a control parameter, we experimentally investigate, using liquid-crystal electroconvection with soft boundary conditions, the onset of spatial temporal chaos (STC) with increasing system size. We find that temporal periodicity is significantly quenched as the system size increases. The increase of the fourth moment (kurtosis) of the temporal Fourier transform provides a very useful quantitative measure of the loss of temporal periodicity (hence the onset of STC) as the pattern size increases, and also provides a simple means for determining a natural chaotic length scale. This length scale is comparable to the length of vertical rows observed in the original pattern. Our experiments, thus, imply that there are well-defined building blocks, which in our case are easily visualized, that control the dynamics in STC liquid crystal convection. The results of our experiments appear to be consistent with the conclusions of recent STC computer simulations carried out by Fishman and Egolf.


---

## Body

## 1. Introduction

The understanding of
chaotic  dynamics in a spatially ex-tended system
with many degrees of freedom that is driven far from equilibrium, referred to
as spatiotemporal chaos (STC), is still far from complete. Indeed, a rigorous
predictive statistical STC model does not exist [1], in
spite of the large amount of recent attention dedicated to STC including a
number of important computer simulations. Fishman and Egolf
[1] have
found that deviations from extensivity, revealed in high-resolution simulations
of the one-dimensional complex Ginzburg-Landau (CGL) equation, did indeed occur
for small system sizes, and concluded that CGL STC must be composed of
definitive weakly interacting building blocks, each containing about two
degrees of freedom. Certainly, one may then hope to formulate anonequilibrium
statistical mechanical model of STC based on the building block interactions
[2].
According to Fishman and Egolf, microextensive behavior
[3, 4]
is observed in large systems because building blocks
separated by a large distance should be essentially uncorrelated.Although the use of the system size as a control
parameter has been a powerful tool in STC computer simulations, it is very
difficult to vary this parameterexperimentally
[5–16] in a well-controlled manner, and the usual experimental
hard boundary conditions (either Neumann or Dirichlet) can significantly
placate the chaotic behavior one is attempting to study in smaller STC systems
[17]. In the case of planar electroconvection in
nematic liquid crystals, however, we have shown recently
[18, 19]
that modest laser heating can be used to produce spatially localized patterns,
since the electroconvection threshold voltage decreases with increasing
temperature [19]. By varying the size of the
laser-cross-section incident on the sample, we therefore have a systematic
manner of controlling the size of a convection pattern without introducing hard
boundary conditions.In planar eletroconvection, the voltage control parameter
is defined asϵ=(V/V0)2−1,
where V0 is the threshold electroconvection voltage at
the applied AC frequency. In our experiments, the laser-generated temperature
profile results in a threshold voltage that depends on position. The first
rolls appear where the temperature is largest, at a threshold voltage of V0,min.
The voltage-control parameter, therefore, takes its maximum value ϵmax=(V/V0,min)2−1 at this position. We set ϵmax=0.75 at an applied frequency of 377 Hz (which is
well above the voltage required to generate STC [16, 20]) throughout this work by measuring V0,min independently for each individual
laser-controlled pattern investigated. Our approach is very different than
previous STC electroconvection experiments [21–23].Our basic experimental procedure can be understood
with reference to Figure1, in which x (the director axis) and y are taken in the horizontal and vertical
directions, respectively. Following our usual
procedure [18],
we employ 40μm-thick samples of MBBA doped at 0.2 wt% with
a blue-absorbing dye (methyl red). On the left-hand side of the figure, we show
two different 488 nm laser-beam cross-sections. The length in the y direction of the laser-line cross-section
incident on the sample is set by cylindrical lenses and an adjustable
knife-edge slit that is placed about 2 cm above the sample. The pattern aspect
ratio is defined by the fact that the mean roll spacing is about 1/3 the sample thickness. The convection patterns
generated using these cross-sections, each driven using a voltage-control
parameter of ϵmax=0.75,
are shown on the right-hand side of the figure. In Figure 1, the pattern size is
larger in the y-direction for the laser-cross-section that is
longer in the y-direction. In this manner, we may use the
length of the laser cross-section as a system-size control parameter while
holding ϵmax fixed. The laser cross-section is not
spatially uniform due to knife-edge and single-slit diffraction; however, this
nonuniformity is not reflected in convection patterns. We believe this is due
to strong advection within the pattern creating a uniform temperature profile
after the patterns forms, as we haveargued previously
[18].Figure 1
Two different laser cross-sections, shown on the left,
are used to generate two spatiotemporal chaotic (STC) patterns of different
sizes.x and y are horizontal (along the director) and
vertical, respectively. Each pattern is driven at a fixed voltage-control
parameter of ϵmax=0.75.
The horizontal white bar in the lower-right corner represents a length of 100 μm,
and the vertical white bar on the right, which we will argue, represents the
natural chaotic length scale for our experiments,
and corresponds to a length of 171μm.
The laser light transmitted through the sample is blocked from the detector
using an optical filter; the actual positions of the laser cross-sections within
the STC patterns are indicated by the white arrows.Spacetime contour plots for pattern dynamics obtained
at two different system sizes are shown in Figure2. To construct the spacetime
portraits, we study the variation of the pattern intensity over a long strip,
parallel to the director (x-direction), located vertically near the
center of the pattern. The length of the laser-beam cross-section employed,
which controls the size of the pattern in the y direction, is noted at the top of each figure.
Although both profiles show significant disorder, there is an important
difference: the smaller-sample-size profile on the left displays some regions,
where temporal periodicity is clearly evident (e.g., near x=600⁢μm on the left-hand spacetime portrait). To
better understand this difference, we have in Figure 3 Fourier transformed the
vertical time axis of Figure 2, showing the Fourier-frequency power spectra at
different positions along the x-axis. The power-spectrum intensities have
been multiplied by a factor of the Fourier frequency f so that 1/f-noise would not dominate the spectra. In
these “space-frequency” diagrams, it is clear that the
smaller-system-size (left-hand) temporal Fourier spectra at different spatial
positions is considerably more peaked in frequency than the larger-system-size
spectra on the right.Spacetime
contour diagram for patterns of different sizes, with the “laser” number
at the top representing the length of the laser-beam cross-section, which sets
the width of the pattern perpendicular to the director.
(a)(b)“Space-frequency diagrams”: contour
plots in which the time axis of Figure2 has been Fourier transformed to obtain
the power spectra, multiplied by the frequency f to reduce the effects of 1/f noise.
(a)(b)In order to quantify the increasing spread in Fourier
power spectra with increasing system size, one needs to calculate moments of
the power spectra. We have found that the second
moment (variance) does not couple to the outer
wings of the power spectra; thus, we have instead calculated a fourth moment (a
kurtosis-type functionK): K=∑i(Pifi)(fi−f¯)4/∑i(Pifi),
where the sum is over temporal frequencies fi,
and where Pi is the component of the Fourier power spectrum
at temporal frequency fi,
spatially averaged over the x-direction, and f¯=∑i(Pifi)fi/∑iPifi.
Since K is highly sensitive to the portion of the
Fourier power spectrum lying outside a main central peak, K provides a quantitative measure of the loss of
temporal periodicity content within the pattern. To keep the analysis as simple
as possible, we did not attempt to measure moments higher than fourth
order. In Figure 4, we plot K as a function of
the laser cross-section length, which controls the pattern size in the y-direction. Error bars are obtained by
measuring K using space-frequency plots obtained from five
different horizontal strips within the pattern. The kurtosis shows a clear
systematic increase, by over an order of magnitude, with system size. The
dashed line in Figure 4 shows the value for K obtained when an identical liquid crystal
displaying STC over the entire sample (about 1 cm2), with no laser excitation present, is also driven at a control
parameter of ϵ=0.75.
Our measurements at the largest laser-controlled pattern sizes approach this
value, lending consistency to the idea that periodicity content is
systematically quenched to a small final value as the system approaches an
“infinite” size. It should be noted that the large-system-size
measurement was treated in the same way as the laser-induced patterns: we study
the variation of the pattern intensity over a long strip, parallel to the
director (x-direction), which has the same width as the
strips studied in the laser-induced  patterns.Figure 4
The fourth moment, the kurtosisK is plotted as a function of the
laser-cross-section-size control parameter, which sets the system size. The
dashed line shows the kurtosis obtained for a much larger 1-cm2 region driven at the same voltage-control parameter in the absence of
laser light.The plot of the kurtosis for different pattern sizes
can now be used in a simple way to provide a new experimental chaotic length
scalelc.
A natural length scale in terms of a pattern size required for the quenching of
the periodicity and the onset of complete nonperiodic STC is the pattern size
at which the kurtosis obtains one half of its infinite-system-size value. From
Figure 4, this length scale is lc=171⁢μm.
A vertical white bar corresponding to this length has been placed in Figure 1. It
is apparent from this figure that lc is the length of two or three attenuated
rolls. We conclude from this that the STC is apparently controlled by the
interaction between attenuated rolls, and that the attenuated rolls themselves
are the fundamental building blocks of the STC that we observe. For pattern
sizes (measured along y) that are comparable to lc,
two or three attenuated-roll building blocks will be highly correlated as
coupled oscillators and can maintain temporally periodic dynamics. For large
pattern sizes compared to lc,
distant rolls are completely decorrelated and temporal periodicity is lost, so
that the pattern displays chaotic dynamics with a much wider frequency
spectrum, as seen in the transition to chaos in classic single-particle systems
such as the damped driven pendulum. Our experimental conclusions appear to be
very consistent with the CGL computer simulations of Fishman and Egolf
[1], who found that the increase in fractal dimension D was quantized as the size of a small system
was increased from zero to include two or three building blocks, but that as
the system size was further increased to contain many more building blocks,
quantized increases in D could not be detected. Our main point is that
in both our experiments and in the computer simulations of Fishman and Egolf
[1], the STC is composed of well-defined building blocks, but one
must work with pattern sizes that are only large enough to contain two or three
building blocks in order to detect the presence of the building blocks and
measure their sizes.



---

*Source: 891324-2008-06-09.xml*