Modal Extraction in Spatially Extended Systems
Modal Extraction in Spatially Extended Systems
Kapilanjan Krishan
Department of Physics and Astronomy, University of California - Irvine, Irvine, California 92697
Andreas Handel
Department of Biology, Emory University, Atlanta, Georgia 30322
Roman O. Grigoriev and Michael F. Schatz
Center for Nonlinear Science and School of Physics,
Georgia Institute of Technology, Atlanta, Georgia 30332
(Dated: August 10, 2021)
We describe a practical procedure for extracting the spatial structure and the growth rates of
slow eigenmodes of a spatially extended system, using a unique experimental capability both to
impose and to perturb desired initial states. The procedure is used to construct experimentally the
spectrum of linear modes near the secondary instability boundary in Rayleigh-Bénard convection.
This technique suggests an approach to experimental characterization of more complex dynamical
states such as periodic orbits or spatiotemporal chaos.
PACS numbers:
Numerous nonlinear nonequilibrium systems in nature
and in technology exhibit complex behavior in both space
and time ; understanding and characterizing such behav-
ior (spatiotemporal chaos) is a key unsolved problem in
nonlinear science [1]. Many such systems are modelled
by partial differential equations; hence, in principle, their
dynamics takes place in an infinite dimensional phase
space. However, dissipation often acts to confine these
systems’ asymptotic behavior to finite-dimensional sub-
spaces known as invariant manifolds [2]. Knowledge of
the invariant manifolds provides a wealth of dynamical
information; thus, devising methodologies to determine
invariant manifolds from experimental data would signif-
icantly advance understanding of spatiotemporal chaos.
In this Letter, we describe experiments in Rayleigh-
Bénard convection where several slow eigenmodes and
their growth rates associated with instability of roll states
are extracted quantitatively. Rayleigh-Bénard convec-
tion (RBC) serves well as a model spatially extended sys-
tem; in particular, the spiral defect chaos (SDC) state in
RBC is considered an outstanding example of spatiotem-
poral chaos. In SDC the spatial structure is primarily
composed of curved but locally parallel rolls, punctuated
by defects (Fig. 1) [3, 4]. The recurrent formation and
drift of defects in SDC is believed to play a key role in
driving spatiotemporal chaos; moreover, many aspects of
defect nucleation in SDC are related to defect formation
observed at the onset of instability in patterns of straight,
parallel rolls in RBC [10]. We obtain experimentally a
low-dimensional description of the modes responsible for
the nucleation of one important class of defects (disloca-
tions), by first imposing reproducibly a linearly stable,
straight roll state (stable fixed point) near instability on-
set. This state is subsequently subjected to a set of dis-
tinct, well-controlled perturbations, each of which initi-
FIG. 1: Shadowgraph visualization reveals spontaneous de-
fect nucleation in the spiral defect chaos state of Rayleigh-
Benard convection. Two convection rolls are compressed to-
gether (higher contrast region in left image). (b.) A short
time later (right image), one of the rolls pinches off and two
dislocations form.
ates a relaxational trajectory from the disturbed state to
the (same) fixed point. An ensemble of such trajectories
is used to construct a suitable basis for describing the em-
bedding space by means of a modified Karhunen-Loeve
decomposition. The dynamical evolution of small distur-
bances is then characterized by computing both finite-
time Lyapunov exponents and the spatial structure of
the associated eigenmodes (a similar approach was car-
ried out numerically by Egolf et al. [5]). This capability
is an important step toward developing a systematic way
of characterizing and, perhaps, controlling, spatiotempo-
rally chaotic states like SDC where localized “pivotal”
events like defect formation play a central role in driving
complex behavior.
The convection experiments are performed with
gaseous CO2 at a pressure of 3.2 MPa. A 0.697±0.06
mm-thick gas layer is contained in a 27 mm square cell,
which is confined laterally by filter paper. The layer
http://arxiv.org/abs/0704.1011v1
FIG. 2: Experimental images illustrate the flow response to
two different perturbations applied, in turn, to the same state
of straight convection rolls. Each image represents the dif-
ference between the perturbed and unperturbed convection
states and therefore, each image highlights the effect of a
given perturbation on the flow. In the two cases shown, the
localized perturbation is applied directly on a region of either
downflow (left image) or upflow (right image). In all cases,
the disturbance created by the perturbation decays away and
the flow returns to the original unperturbed state.
is bounded on top by a sapphire window and on the
bottom by a sheet of 1 mm-thick glass neutral den-
sity filter(NDF). The neutral density filter is bonded to
a heated metal plate with heat sink compound. The
temperature of the sapphire window held constant at
21.3 ◦C by water cooling. The temperature difference
between the top and bottom plates ∆T is held fixed
at 5.50 ± 0.01 ◦C by computer control of a thin film
heater attached to the bottom metal plate. These condi-
tions correspond to a dimensionless bifurcation parame-
ter ǫ=(∆T−∆Tc)/∆Tc = 0.41, where ∆Tc is the temper-
ature difference at the onset of convection. The vertical
thermal diffusion time, computed to be 2.1 s at onset,
represents the characteristic timescale for the system.
We use laser heating to alter the convective patterns
that occur spontaneously. A focused beam from an Ar-
ion laser is directed through the sapphire window at
a spot on the NDF. Absorption of the laser light by
the NDF increases the local temperature of the bottom
boundary and hence that of the gas, thereby inducing
locally a convective upflow. The convection pattern may
be modifed, either locally or globally, by rastering the
hot spot created by the laser beam. The beam is steered
using two galvanometric mirrors rotating about axes or-
thogonal to each other under computer control. The in-
tensity of the beam is modulated using an acousto-optic
modulator. This technique of optical actuation is used
to impose convection patterns with desired properties,
to perturb these convection patterns and to change the
boundary conditions. Similar approaches for manipulat-
ing convective flows were explored earlier using a high
intensity lamp and masks [11] in RBC and a rastered
infrared laser in Bénard-Marangoni convection [12].
The experiments begin by using laser heating to im-
pose a well-specified basic state of stable straight rolls.
The basic state is typically arranged to be near the on-
set of instability by imposing a sufficiently large pattern
wavenumber such that at fixed ǫ the system’s parame-
ters are near the skew-varicose stability boundary [10].
In this regime, the modes responsible for the instability
are weakly damped and, therefore, can be easily excited.
The linear stability of the basic state is probed by ap-
plying brief pulses of spatially localized laser heating. For
stable patterns, all small disturbances eventually relax.
To excite all modes governing the disturbance evolution,
we apply a set of localized perturbations consistent with
symmetries of the (ideal) straight roll pattern – continu-
ous translation symmetry in the direction along the rolls
and discrete translation symmetry in the perpendicular
direction plus the reflection symmetry in both directions.
Therefore, localized perturbations applied across half a
wavelength of the pattern form a ”basis” for all such per-
turbations – any other localized perturbation at a differ-
ent spatial location is related by symmetry. Localized
perturbations are produced in the experiment by aiming
the laser beam to create a “hot spot” whose location is
stepped from the center of a (cold) downflow region to
the center of an adjacent (hot) upflow region in differ-
ent experimental runs. The perturbations typically last
approximately 5 s and have a lateral extent of approxi-
mately 0.1 mm, which is less than 10 % of the pattern
wavelength.
The evolution of the perturbed convective flow is mon-
itored by shadowgraph visualization. A digital camera
with a low-pass filter (to filter out the reflections from the
Ar-ion laser) is used to capture a sequence of 256× 256
pixel images recorded with 12 bits of intensity resolution
at a rate of 41 images per second. A background im-
age of the unperturbed flow is subtracted from each data
image; such sequences of difference images comprise the
time series representing the evolution of the perturbation
(Fig 2).
The total power for each (difference) image in a time
series is obtained from 2-D spatial Fourier transforms.
The resulting time series of total power shows a strong
transient excursion (corresponding to the initial response
of the convective flow to a localized perturbation by laser
heating) followed by exponential decay as the system re-
laxes back to the stable state of straight convection rolls.
We restrict further analysis to the region of exponential
decay, which typically represents about 3.5 seconds of
data for each applied perturbation.
The dimensionality of the raw data is too high to
permit direct analysis, so each difference image is first
windowed (to avoid aliasing effects) and Fourier filtered
by discarding the Fourier modes outside a 31 × 31 win-
dow centered at the zero frequency. The discarded high-
frequency modes are strongly damped and contain less
than 1% of the total power. The basis of 312 Fourier
modes still contains redundant information, so we fur-
KL mode 1 KL mode 2 
KL mode 3 KL mode 4 
FIG. 3: The first four Karhunen-Loeve eigenvectors are shown
for a perturbed roll state near the skew-varicose boundary of
Rayleigh-Bénard convection. The eigenvectors are ordered by
their eigenvalues (largest to smallest), which are propotional
to the amount of power contained in the corresponding eigen-
vector.
ther reduce the dimensionality of the embedding space
by projecting the disturbance trajectories onto the “opti-
mal” basis constructed using a variation of the Karhunen-
Loeve (KL) decomposition [6, 7]. The correlation matrix
C is computed using the Fourier filtered time series xs(t),
(xs(t)− 〈xs(t)〉t)(x
s(t)− 〈xs(t)〉t)
†, (1)
where the index s labels different initial conditions and
the origin of time t = 0 corresponds to the time when
the perturbation applied by the laser is within the linear
neighbourhood of the statioary state. The angle brack-
ets with the subscript t indicate a time average. The
eigenvectors of C are the KL basis vectors. It is worth
noting that the average performed to compute C rep-
resents an ensemble average over different initial condi-
tions (obtained by applying different perturbations); this
is distinctly different from the standard implementation
of KL decomposition where statistical time averages are
typically employed.
The spatial structures of the first four KL vectors are
shown in Fig. 3. We find that the first 24 basis vectors
capture over 90% of the total power, so an embedding
space spanned by these vectors represents well the relax-
ational dynamics about the straight roll pattern. In our
convection experiments, the KL eigenvectors show two
distinct length scales. The first two dominant vectors
KL basis vector 1
FIG. 4: A two-dimensional projection of the experimental
time series (symbols) and the least squares fits (continuous
curves). The time series have been shifted such that the fixed
point is at the origin.
are spatially localized, while the remaining vectors are
spatially extended. This is consistent with earlier work
as suggested in [4].
More quantitative information can be obtained by find-
ing the eigenmodes of the system, excited by the pertur-
bation, and their growth rates. These can be extracted
from a nonlinear least squares fit with the cost function
i,s,t
i (t)−
i (∞) +
, (2)
where xsi (t) is a projection of the perturbation at time t
in the time series s onto the ith KL basis vector. In the fit
k and λk are the kth eigenmode and its growth rate and
Ask is the initial amplitude of the kth eigenmode excited
in the experimental time series s. The fixed points xs(∞)
are chosen to be different for the differing time series in
the ensemble to account for a slow drift in the parameters
and we assume that only n eigenmodes are excited.
The results for an ensemble of time series correspond-
ing to seven point perturbations applied across a wave-
length of the pattern with n = 6 are shown in Figs. 4-5.
(With seven different initial conditions we cannot hope
to distinguish more than seven different modes). In par-
ticular, Fig. 4 shows the projection of the experimental
time series and the least squares fit on the plane spanned
by the first two KL basis vectors. Such extraction of
the linear manifold in experiments on spatially extended
systems without the knowledge of the dynamical equa-
tions of the system aids in the application of techniques
that are well developed for low dimensional systems. The
manifolds of fixed points and periodic orbits are of par-
ticular interest in chaotic systems.
The extracted growth rates λk are shown in Fig. 5.
Not surprisingly, since the pattern is stable the growth
rates are negative. The leading eigenmode (see Fig. 6) is
spatially extended and shows a diagonal structure charac-
1 2 3 4 5 6
Mode number
FIG. 5: The growth rates of the six dominant eigenmodes
and the error bars extracted from the least squares fit. The
growth rates have been non-dimensionalized by the vertical
thermal diffusion time.
teristic of the skew-varicose instability in an unbounded
system. This is also expected as the pattern is near the
skew-varicose instability boundary. The second eigen-
mode is spatially localized and has no analog in spatially
unbounded systems. The subsequent modes are again
spatially delocalized and likely correspond to the Gold-
stone modes of the unbounded system (e.g., overall trans-
lation of the pattern) which are made weakly stable due
to confinement by the lateral boundaries of the convec-
tion cell.
If the system is brought across the stability boundary,
one of the modes is expected to become unstable (with-
out significant change in its spatial structure), thereby
determining further (nonlinear) evolution of the system
towards a state with a pair of dislocation defects. We
would also expect the spatially localized eigenmodes (like
the second one in Fig. 6) to preserve their structure if
the base state is smoothly distorted (as it would be, e.g.,
in the SDC state shown in Fig. 1), indicating the same
type of a spatially localized instability. Our further ex-
perimental studies will aim to confirm these expectations.
Defects represent a type of “coherent structure” in
spiral defect chaos. Previous efforts have used coher-
ent structures to characterize spatiotemporally chaotic
extended systems in both models [7] and experiments
[8]; the use of coherent structures to parametrize the in-
variant manifold was pioneered by Holmes et al. [6] in
the context of turbulence. In practice coherent struc-
tures are usually extracted using the Karhunen-Loéve
(or proper orthogonal) decomposition of time series of
system states, which picks out the statistically impor-
tant patterns. This prior work has met with only limited
success – indeed, it is unclear whether statistically im-
portant patterns are dynamically important. An alterna-
tive approach has been proposed by Christiansen et al.
Eigenmode 1 Eigenmode 2 
Eigenmode 3 Eigenmode 4 
FIG. 6: Four dominant eigenmodes extracted from the least
squares fit.
[9], who suggested instead to use the recurrent patterns
corresponding to the low-period unstable periodic orbits
(UPO) of the system, which are dynamically more impor-
tant. Our work sets the stage for attempting the more
ambitious task of extraction of UPOs and their stability
properties from experimental data.
Summing up, we have developed an experimental tech-
nique which allows extraction of quantitative information
describing the dynamics and stability of a pattern form-
ing system near a fixed point. This technique should be
applicable to a broad class of patterns, including unstable
fixed points, periodic orbits and segments of chaotic tra-
jectories. Moreover, we expect that a similar approach
could be applied to other pattern forming systems, con-
vective or otherwise, as long as a method of spatially
distibuted actuation of their state can be devised.
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