IDENTIFICATION OF INTERMITTENT ORDERED PATTERNS AS HETEROCLINIC CONNECTIONS
E. Stone
M. Gorman (gorman@uh.edu),
M. el-Hamdi
(el-hamdi@uhphys.phys.uh.edu)
Department of Physics
University of Houston
Houston, TX 77204-5506
K. A. Robbins (krobbins@runner.utsa.edu)
Division of Computer Science
The University of Texas at San Antonio
San Antonio, Texas 78249
Physical Review Letters 76 (1996) pp. 2061-2064.
Abstract
Cellular flames stabilized on a porous plug burner at low pressure form
ordered patterns consisting of concentric rings of cells. In certain regions
of the parameter space of propane-air mixtures, the ordered patterns appear
intermittently. In these states, patterns of concentric rings persist for
varying lengths of time, abruptly dissolve into highly irregular structures
with no discernible ring pattern, and then suddenly reappear as ordered
patterns with the same or slightly differently numbers of cells.
Experimental evidence and theoretical arguments are presented that these
intermittently ordered states correspond to a network of heteroclinic
connections among unstable equilibria.
Introduction
The phenomenon of intermittency plays a prominent role in
deterministic dynamics
(ref. 1,
ref. 2,
ref. 3,
ref. 4).
Pomeau and Manneville were among
the first to demonstrate that intermittent
dynamics could arise in time series generated from
one-dimensional maps or from the solutions of coupled
sets of ordinary differential equations.
There have been a number of experimental observations of such
dynamics
(ref. 5,
ref. 6).
In this paper we report the observation of intermittency
in a pattern-forming system in which concentric rings of propane-air
cellular flames appear and disappear at irregular intervals.
We present both quantitative and qualitative evidence to link these intermittently
ordered states to heteroclinic connections.
In intermittent regimes of this sort, the trajectory representing the
state of the system first approaches the neighborhood of an equilibrium
associated with a steady pattern, and then leaves it rapidly as a linearly
unstable mode for that pattern grows. The trajectory may then return to the
same equilibrium or possibly approach another unstable equilibrium pattern.
This process is repeated, so that the solution spends varying periods of
time hesitating near unstable equilibria between which excursions
(represented as spatial and temporal disorder in the overall pattern) occur.
The resulting motion is intermittent, with well-defined events occurring at
irregular intervals. The emergence of this type of intermittent state is
characteristic of the way in which some spatially extended systems undergo
transitions between two ordered states of the system.
The symmetry of the burner and
the small number of cells
play a crucial role
in our ability to observe this type of intermittent behavior.
Top
The Experiment
Premixed flames
(ref. 7) are
stabilized on a 5.62 cm circular porous
plug burner which is housed in a combustion chamber made from process glass
pipe. The working pressure, 1/2 atm, and the flow rate are controlled to
0.1. The control parameters are the total flow rate of the gases and the
equivalence ratio (the ratio of fuel to oxidizer relative to that at
stochiometry). The premixed propane-air flame is viewed from above by a
Silicon Intensified Target camera through a mirror mounted on the top of the
chamber. The motion is recorded on videotape with an embedded time code.
As the control parameters are adjusted, the steady uniform flame front breaks up
into (brighter, hotter) cells whose boundaries are demarked by (darker,
cooler) cusps and folds which curve away from the burner surface. The cells
organize into ordered states formed by two concentric rings of cells~\cite
{Gorman}. These ordered states are labeled according to the number of cells
in the each ring (e.g., 9/3 designates an ordered state with 9 cells in the
outer ring and 3 cells in the inner ring).
In propane-air
cellular flames the parameter range between consecutive ordered states
contains intermittently ordered states in which concentric rings
of cells appear at irregular intervals. In the example presented here a
9/3 ordered state is stable over a range of flow rates.
As the flow rate is increased beyond a critical value, this
9/3 state becomes unstable and is replaced by an intermittenly ordered state.
In the intermittent state, the 9/3 pattern appears for a
time, mostly for short times (approximately 1/3 second)
but occasionally for long
times (120 seconds), after which it dissolves into a very complicated
spatio-temporal sequence, finally resolving itself into another (or possibly
the same) quasi-stationary pattern.
A frame-by-frame analysis was performed of a two-hour run taken
at a flow rate of 8.27 liters/minute, an equivalence ratio of
1.85, and a pressure of 1/2 atmosphere.
The flow rate is just beyond the value at which
the 9/3 ordered state is stable.
Figure 1
shows the four principal
patterns of concentric rings (9/3, 9/4, 10/3, and 10/4)
which appear at various times. The analysis reveals that the system spends
59% of its total time in excursion between ordered patterns. The 9/3 pattern
receives 338 of the visits, 86% of the total, and is the only pattern for
which we have sufficiently good statistics to make a quantitative comparison
with theory. The other states---9/4, 10/3, and 10/4---receive
9%, 4% and 1% of the visits respectively.
Figure 2
shows a passage near an equilibrium. Frames from the videotape of
the experiment are numbered to correspond with a schematic diagram of an
orbit in a traverse near a saddle point in phase space shown in
Figure 3.
As the system moves along the unstable manifold (frames 1--3), the flame
front has a highly irregular shape with no discernible structure. As the
system passes near the saddle point (frames 4--6), the pattern of concentric
rings becomes visually discernible. As the traverse continues away from the
saddle point (frames 7--9), the flame resumes its irregular motion.
A qualitative argument that the intermittently ordered states correspond to
heteroclinic connections is provided by the observed sequence of
transitions. At low values of the flow rate the 9/3 state is stable; but, at
a larger value, it becomes unstable to a cycle in which the
ordered pattern repeatedly breaks up and reforms. At successively larger
values of the flow rate, a 9/4 ordered state becomes stable. The 9/4 state
is in turn destabilized at a higher value of the flow rate to a situation in
which the 10/4 cell state breaks up and reforms. This sequence of
transitions suggests global bifurcations in which cycles are formed and
broken as the unstable and stable manifolds of the saddle equilibria
associated with the stationary states intersect tangentially and then
separate. Such a sequence is typical for propane-air cellular
flames. Other fuels have bifurcation sequences with different kinds of
transitions. In isobutane-air and butane-air flames the predominant type of
transition involves ordered states which bifurcate directly
into other ordered states with different numbers of cells or to dynamic
states in which cells in a ring move collectively.
Top
Theoretical Analysis
Quantitative evidence of heteroclinic cycles comes from an analysis of the
time the system spends hesitating near unstable equilibria. In general,
homoclinic and heteroclinic orbits are structurally unstable, meaning that
infinitesimally small changes in the parameters can remove the connection.
In experimental systems this condition means that such orbits would be not
observed, since it is impossible to tune the parameters to the exact value.
It has been found, however, that the presence of symmetry
groups under which the vector field is equivariant can result in
structurally stable orbits
(ref. 9).
Symmetry-induced heteroclinic cycles are not generally associated with
chaotic behavior. Under suitable conditions on the eigenvalues of
the saddle point (ref. 10), the cycles are
asymptotically as well as structurally
stable; all orbits nearby approach them as t approaches infinity.
Solutions
spend increasing periods near the equilibria, and the heteroclinic
excursions become less frequent. There is no inherent time scale in the
system, and the period of the cycle approaches infinity.
Busse
(ref. 11) first noticed the importance
of small amplitude perturbations due to
round-off errors in his theoretical study of convection patterns
in which a set of three ordinary differential equations invariant
under cyclic permutations was shown to possess a heteroclinic cycle. Such
perturbations may be too small to significantly change the behavior of the
system on a heteroclinic excursion because the vector field is large
compared to the perturbation. Near an equilibrium where the field becomes
vanishingly small, the perturbations can keep an orbit from approaching the
equilibrium arbitrarily closely. The time spent by an orbit near a fixed
point is limited, and a ``statistical limit cycle'' is created from the
heteroclinic cycle
(ref. 11). The period of the cycle is
finite and random.
In the combustion experiment described in this paper,
a source of perturbations is the
low-amplitude irregular vibrations characteristic
of the ordered states
(ref. 8).
The switching between roll patterns with different orientations observed by
Ning and Ecke
(ref. 12)
example of symmetry-induced heteroclinic connections.
The dependence of the time scale of a statistical limit cycle on the largest
unstable eigenvalue can be worked out explicitly. In the interest of
brevity we refer the reader to
ref. 13 and
ref. 14
for more details on the
derivation and reproduce the main points only. Consider a point q in thephase space of the governing equations for the physical system. The point
q
is contained in a heteroclinic orbit to a pair of fixed points
(p+, p-)
if the orbit based at q
approaches p+ as t approaches plus infinity.
and p- as t approaches
minus infinity. The fixed points p+ and p- must
be saddle points with eigenvalues on both sides of the imaginary axis,
hence the term saddle connection.
The orbit is called homoclinic if
p- and p+ are the same point.
Referring to Figure 3,
the circulation time for trajectories near the cycle
will be dominated by the slow passage near the saddle point
p, i.e., the
time spent in U.
We approximate the total circulation time by the sum of a
constant ``cycle-time'', and time spent in U,
or ``passage time'', T. It
is within U that external perturbations are assumed to have a major
effect, so the variation in circulation time is determined by
T alone.
This calculation can be done by a linear analysis at
p, incorporating
added noise. Given an incoming distribution of initial data to
U, its
evolution can be computed and the distribution of passage times through
U can be calculated:
P(T) = 2 \lambda \Delta(T) e^{-\Delta^2(T)}/[\sqrt{\pi}(1-e^{-2 \lambda
T})]
where:
\Delta(t)=\delta [ (\epsilon^2/\lambda)(e^{2 \lambda t}-1)]^{-1/2},
\lambda is the unstable eigenvalue of p, \epsilon is the r.m.s. noise
level, and \delta is the size of the neighborhood in which noise is
assumed to have an appreciable effect.
P(T) is a skewed distribution, with its peak off the mean, and it can beshown
(ref. 13)
that it possesses an exponential tail
namely:
$P(T) \approx \frac{2 \delta}{\sqrt{\pi}\epsilon}
\lambda^{3/2} e^{-\lambda T}$
as T approaches infinity.
In (ref. 14)
it is shown that the derivation can be carried out
for systems of higher dimension
than two, and in that the details of the perturbing signal are
not critical in this derivation.
Top
Results
This theoretical result suggests an experimental technique
for analyzing heteroclinic connections. If the time the system spends in a
single quasi-stationary state is recorded for successive visits near that
equilibrium, a histogram of these times should coincide with the
passage-time distribution derived above. The principal experimental
difficulty is the identification of the entrance of the trajectory into the
``box'', U, shown schematically in
Figure 3.
We use the visual criterion
that the system point is in the box U around an
equilibrium if a pattern
of two complete concentric rings of cells can be distinguished.
The uncertainty in the
experimental identification of the entrance and
exit from the box U is +-2 frames.
The passage times are determined from a frame-by-frame analysis of the
videotape. The histogram of the passage times for the 9/3 pattern
is shown in
Figure 4.
We first fit the tail of the histogram to
a single exponential using times longer than 20 frames.
The resulting fit is shown in the inset of
Figure 4.
The fitted value, 70 +- 2, is the inverse of the unstable
eigenvalue. It changes very little (+- 1) as the bin size is
varied from 20 to 30 frames.
The quality of the fit is excellent for
this kind of experimental data.
The eigenvalue (1/70) associated with the 9/3 pattern
increases smoothly as the control parameters are varied away from the
transition point where the pattern first becomes unstable.
The main plot in
Figure 4 shows the full histogram of passage times.
The bin size of three
frames has been chosen to reveal the structure of the peak
at 8 +- 3 frames. The position of the peak varies slightly
as the bin size is changed from two to seven frames.
The analytic result for the distribution of passage times, P(T),
is superposed on the experimental distribution of
Figure 4. P(T) has
three parameters: $\lambda$, $\epsilon$, and $\delta$.
The graph
uses the experimentally determined value of $\lambda$ and was found
to be relatively insensitive to the selection of the other two parameters.
The small number of cells plays an important role in this
observation of intermittently ordered states.
A small number of cells forces a wider separation
in parameter space between the stable states of
the system, and therefore provides a significant range over which
the intermittently ordered states are observed.
The symmetry of the experiment creates a situation in which
the heteroclinic orbits persist over a finite range of parameters,
rather than a measure zero set.
This feature was noticed by Aubry et al. for O(2) symmetry
in a model for boundary layer dynamics
(ref. 16),
by Armbruster et al.
for the one-dimensional Kuramoto-Sivashinsky
equation
(ref. 17),
and by Proctor and Jones in truncated
models of Rayleigh-Benard convection
(ref. 18).
Top
Discussion
In this paper we have presented a discussion of the characteristics of
pattern-forming systems exhibiting heteroclinic connections, emphasizing
the importance of
the passage time of the intermittent pattern and its relationship to the
physics of unstable equilibria.
A skewed distribution of passage times with exponential
tail is a distinctive signature of systems in which
a trajectory is repeatedly injected near the stable
manifold of a saddle node in the presence of noise.
A specific
criterion for measuring the passage
times in a pattern-forming system was developed, and a quantitative
comparison was made between
the theoretical expression for the form of the distribution of
these times and experimental measurements.
The results presented here are representative of seven other
points we have investigated in propane-air cellular flames.
They provide substantial evidence that the intermittently
ordered states arise because of the presence
of a network of heteroclinic connections among unstable equilibria.
Top
Acknowledgements
The experimental work was supported by a grant,
NK-00014-K-0613 from the Office of Naval Research. We would like to thank
Dieter Armbruster and Eric Kostelich for helpful discussions. Many thanks to
Kevork Abazajian who analyzed the videotape. Gregory Sivashinsky, Bernard
Matkowsky, Alvin Bayliss and Stephen Margolis have provided useful guidance
on issues of combustion science. Gemunu Gunaratne and Marty Golubitsky
have provided suggestions on the analysis of the data.
World Wide Web:
http://vip.cs.utsa.edu/flames/overview.html.
References
[1]
Y. Pomeau and P. Manneville, Commun. Math. Phys.
74, 189 (1980).
[2]
N. Platt, E. A. Spiegel and C. Tresser,
Phys. Rev. Lett. 70, 279
(1993)
[3]
U. Frisch and R. Morf,
Phys. Rev. A. 23, 2673 (1981).
[4]
H. K. Moffatt, J. Fluid Mech. 166, 359 (1986).
[5]
P. H. Hammer, N. Platt, S. M. Hammel,
J. F. Heagy, and B. D. Lee, Phys. Rev. Lett.
73, 1095 (1994).
[6]
F. Argoul, J. Huth, P. Merzeau,
A. Arneodo, and H. L. Swinney,
Physica D62, 710 (1993).
[7]
M. el-Hamdi, M. Gorman, and K. A. Robbins, Comb. Sci.
Tech. 94, 83 (1993).
[8]
M. Gorman, M. El-hamdi and K. A. Robbins, Comb. Sci. Tech.
98, 37 (1994).
[9]
D. Armbruster, J. Guckenheimer and P. Holmes, Physica
D 29, 257 (1988).
[10]
L. P. Silnikov, Sov. Math. Dokl. 6, 163 (1965).
[11]
F. M. Busse, in \textit{Hydrodynamic Instabilities and the
Transition to Turbulence}, edited by H. L. Swinney and J. P. Gollub
(Springer-Verlag, 1981) p. 97.
[12]
L. Ning and R. E. Ecke, Phys. Rev. 47E, R2991
(1993).
[13]
E. Stone and P. Holmes, SIAM J. Appl. Math. 50, 726
(1990).
[14]
E. Stone and P. Holmes, Phys. Let. 155A, 29 (1991).
[15]
P. Holmes, \textit{Lecture Notes in Physics}, edited by J.
Lumley, (Springer-Verlag, 1990), 357, p. 195.
[16]
N. Aubry, P. Holmes, J. L. Lumley, and E. Stone, J. Fluid
Mech. 192, 115 (1988).
[17]
D. Armbruster, J. Guckenheimer and P. Holmes,
SIAM J. Applied Math 49, 676 (1989).
[18]
M. K. Proctor and C. Jones, J. Fluid Mech. 188,
301 (1988).
Abstract