Pattern formation steven strogatz ahlers

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Martin Golubitsky
Dan Luss
Steven H. Strogatz
Editors

Pattern Formation in
Continuous and
Coupled Systems
A Survey Volume

With 101 Illustrations

Springer
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Martin Golubitsky
Department of Mathematics
University of Houston

Houston, TX 77204-4792, USA

Dan Luss
Department of Chemical Engineering
University of Houston
Houston, TX 77204-4792, USA

Steven H. Strogatz
Theoretical and Applied Mechanics
Cornell University
Ithaca, NY 14853, USA

Series Editor:
Willard Miller, Jr.
Institute for Mathematics and its
Applications
University of Minnesota
Minneapolis, MN 55455, USA
Mathematics Subject Classifications (1991): 34Cxx, 35Bxx, 58Fxx, 76XX, 92Exx
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SPIN 10729787


FOREWORD

This IMA Volume in Mathematics and its Applications

PATTERN FORMATION IN CONTINUOUS
AND COUPLED SYSTEMS

is based on the proceedings of a workshop with the same title, but goes beyond the proceedings by presenting a series of mini-review articles that survey, and provide an introduction to, interesting problems in the field. The
workshop was an integral part of the 1997-98 IMA program on "EMERGING APPLICATIONS OF DYNAMICAL SYSTEMS."
I would like to thank Martin Golubitsky, University of Houston (Mathematics) Dan Luss, University of Houston (Chemical Engineering), and

Steven H. Strogatz, Cornell University (Theoretical and Applied Mechanics) for their excellent work as organizers of the meeting and for editing the
proceedings.
I also take this opportunity to thank the National Science Foundation
(NSF), and the Army Research Office (ARO), whose financial support made
the workshop possible.

Willard Miller, Jr., Professor and Director

v

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PREFACE
Pattern formation has been studied intensively for most of this century by both experimentalists and theoreticians, and there have been many
workshops and conferences devoted to the subject. In the IMA workshop
on Pattern Formation in Continuous and Coupled Systems held May 11-15,
1998 we attempted to focus on new directions in the patterns literature.
In particular, we stressed systems and phenomena that generate new types
of pattern (those that appear in discrete coupled systems, those that appear in systems with global coupling, and those that appear in combustion
experiments) and on well-known patterns where there has been significant
recent development (for example, spiral waves and superlattice patterns).
The participants at this meeting included, in more or less equal parts,
experimentalists and theoreticians. One goal was to continue communication between these groups, and we were pleased by the result. Another
goal was to familiarize a larger audience with some of the newer directions
in the field, and again the result was very satisfying.
With these goals in mind, we decided to produce a nonstandard workshop proceedings. We did not want to publish a collection of research
articles, which could have appeared elsewhere as refereed journal articles,
nor did we want to publish a list of abstracts. Instead, we attempted to collect a series of mini-review articles of at most 15 to 20 pages (with extensive
bibliographies) that would discuss why certain topics are interesting and

merit additional research. The response has been quite heartening and we
hope that readers will find these reviews a useful entry into the literature.
The articles that appear here have not been refereed - though they
have been read for comprehensibility. We thank the authors for their efforts
to produce these articles in a timely fashion. We also thank the IMA staff
- in particular Patricia V. Brick and Phong Nguyen - for their expert
help in producing this volume and doing so so expeditiously.

Marty Golubitsky
University of Houston (Mathematics)
Dan Luss
University of Houston (Chemical Engineering)
Steven H. Strogatz
Cornell University (Theoretical and Applied Mechanics)

vii

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Dedication
JOHN DAVID CRAWFORD 1954-1998

John David Crawford died on 23 August, 1998 from Burkitt's Lymphoma at the age of 44. This IMA workshop was the last workshop that
John David attended - at a time when it was thought that the disease
was in remission. John David's presence and active participation added a
special dimension to the workshop for those who knew him.
John David graduated with honors from Princeton University in 1977
and with a doctorate in Physics from the University of California at Berkeley in 1983. His thesis on Hopf Bifurcation and Plasma Instabilities was
written under the direction of Henry Abarbanel. During his career, John

David spent six years at the University of California at San Diego, at first in
the Physics Department working on non-neutral plasmas and subsequently
at the Institute for Nonlinear Science pursuing his interests in bifurcation
theory. In 1989 he held visiting positions at the Mathematics Institute at
the University of Warwick and at the Institute for Fusion Studies at the
University of Texas at Austin. He joined the faculty of the Department of
Physics and Astronomy at the University of Pittsburgh in 1990.
John David's interests ranged from the physics of collisionless plasmas
to the mathematics of pattern formation. However, there was a common
thread: understanding the development and equilibration of instabilities
in diverse systems, be they Hamiltonian or dissipative. This workshop
focused on pattern formation in continuous systems, a subject to which
John David contributed greatly. He worked on developing group-theoretic
methods for use in pattern formation studies of dissipative systems and
ix

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x

JOHN DAVID CRAWFORD 1954-1998

new techniques for studying bifurcation phenomena in Hamiltonian systems
associated with the emergence of an eigenvalue from a continuous spectrum.
The former area of research was motivated primarily by his interest in
parametrically driven water waves (the Faraday system) and the latter by
the beam-plasma instability in the Vlasov-Poisson system.
The Faraday instability is a subharmonic instability and is therefore
associated with a Floquet multiplier at -1. John David's early work with

Edgar Knobloch and Hermann Riecke [10]-[12] discussed mode interaction
in the Faraday experiment in a circular container, focusing on the dynamics
of discrete time-T maps with -1 Floquet multipliers of double multiplicity
as appropriate for modes that break the 0(2) symmetry of the container.
Of particular interest was the classification of the conditions under which
the mixed patterns resulting from such interaction drift azimuthally. Such
rotating patterns were observed in experiments by Sergio Ciliberto and
Jerry Gollub. His subsequent and classic work on the Faraday system in a
square container [18] was also motivated by Gollub's experiments. In this
work John David focused on understanding the hidden symmetries, both
translations and rotations, introduced into the Faraday system by Neumann boundary conditions [18, 21]. These depend on the modes excited
and on their degeneracy. John David's observation that as a result there is
a significant difference between the Faraday system in a square container
and one with D4 symmetry but in a nonsquare container [17] was confirmed in subsequent experiments by Gollub and David Lane [20]. Related
work on parametrically modulated Hopf bifurcation in systems with 0(2)
symmetry [4, 13] predicted that such modulation would stabilize standing
waves even in cases in which traveling waves were preferred in the absence
of modulation. This prediction was also confirmed in elegant experiments
by Victor Steinberg and David Andereck and their colleagues.
At the same time John David continued his studies of bifurcations in
collisionless plasmas. Using the technique of spectral deformation developed in landmark papers with Peter Hislop [8, 9] he was able to understand
in detail the appearance of a neutral eigenvalue (or mode) embedded in a
continuous spectrum at threshold for instability. In this problem, as in the
closely related shear flow problems for ideal fluids, the instability appears
when the electron distribution function or shear flow profile are gradually
changed, for example, by injecting a beam of fast electrons to create a bump
on the tail of the electron distribution or changing the pressure distribution
driving the flow. However, because of the presence of the continuum center
manifold theory cannot be used to study the resulting bifurcation. John
David's understanding of the structure of the linear problem led him to

consider the equilibration of the resulting instability using the instability
growth rate 'Y as the bifurcation parameter. In a remarkable paper [23] he
showed that in the limit of fixed (Le. heavy) ions the instability saturates
at O( 2 ) amplitude, in contrast to the Ob!) amplitude familiar from dissipative systems. This result is nonperturbative, and terms of all orders

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JOHN DAVID CRAWFORD 1954-1998

xi

contribute to the equilibration as "1 --+ 0 [24]. Thus not only do these
instabilities saturate at a much smaller amplitude but they do not have
to approach the equilibrium monotonically. The predicted "12 "trapping"
scaling agrees with numerical and experimental observations. Subsequent
work by John David's student Anand Jayaraman [27, 30] generalized these
conclusions to mobile ions showing that the scaling changes to 'Y~.
While engaged in this work John David realized that similar mathematics applies to the Kuramoto model of phase-coupled oscillators. This
model consists of many globally coupled oscillators with frequencies drawn
from a prescribed frequency distribution and exhibits a remarkable "phase
transition" as the strength K of the interaction increases in which the oscillators begin to phase-lock. As in the Vlasov-Poisson system the stability
problem for the incoherent state has a continuous spectrum and this state
loses stability at K = Kc when an unstable eigenvalue pops out of the
neutral continuum. As a result of a calculation to all orders similar to
the plasma one, John David showed [22] that the saturated amplitude (the
fraction of synchronized oscillators) scales like (K - Kc)! for the Kuramoto
model but scales like K - Kc for more general couplings than assumed by
Kuramoto [25, 29]. These results resolve analytically several long-standing
issues in both theoretical and numerical studies of this important model.

John David wrote two influential review articles, one on basic bifurcation theory [19] and one with Edgar Knobloch on the use of equivariant
bifurcation theory for studies of pattern formation in fluid dynamics [16].
A bibliography of John David's contributions to pattern formation and
bifurcation theory is included below.
John David was a consummate scholar, devoted to deep understanding
of important and challenging problems. His solutions to these problems
were always innovative offering a fresh perspective. At home both in physics
and mathematics John David was an invaluable colleague, generous with
his time and ideas, and a rare knack for explaining scientific principles to
friends, colleagues and students. His lectures were a model of clarity and
he was a much sought-after speaker. At the workshop his delight in being
back in the milieu he so loved was almost palpable. He will be greatly
missed by all of us.
Edgar Knobloch
October, 1998

REFERENCES
[1] J.D. Crawford and S. Omohundro. On the global structure of period doubling
flows, Physica D 13 (1984) 161-180.
[2] J.D. Crawford and E. Knobloch. Symmetry breaking bifurcations in 0(2) Maps,
Phys. Lett. A 128 (1988) 327-331.
[3] J.D. Crawford and E. Knobloch. Classification of 0(2) symmetric Hopfbifurcation:
no distinguished parameter, Physica D 31 (1988) 1-48.

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xii

JOHN DAVID CRAWFORD 1954-1998

[4] H. Riecke, J.D. Crawford and E. Knobloch. Time-modulated oscillatory convection,

Phys. Rev. Lett. 61 (1988) 1942-1945.
[5] J.D. Crawford and E. Knobloch. On degenerate Hopfbifurcation with broken 0(2)
symmetry, Nonlinearity 1 (1988) 617-652.
[6] J.D. Crawford, M. Golubitsky, and W. Langford. Modulated rotating waves in
0(2) mode interaction, Dynamics and Stability of Systems 3 (1988) 159-175.
[7] J.D. Crawford and P. Hislop. Vlasov equation on a symplectic leaf, Phys. Lett. A
134 (1988) 19-24.
[8] J.D. Crawford and P. Hislop. Application of the method of spectral deformation
to the Vlasov-Poisson model, Ann. Phys. 189 (1989) 265-317.
[9] P. Hislop and J.D. Crawford. Application of the method of spectral deformation
to the Vlasov-Poisson system II, J. Math. Phys. 189 (1989) 2819-2837.
[10] J.D. Crawford, E. Knobloch, and H. Riecke. Competing parametric instabilities
with circular symmetry, Phys. Lett. A 135 (1989) 20-24.
[11] J.D. Crawford, E. Knobloch, and H. Riecke. Mode interactions and symmetry, in
Proc. Intern. Conf. on Singular Behavior and Nonlinear Dynamics, vol. 1, S.
Pnevmatikos et al. (eds), World Scientific, 1989, 277-297.
[12] J.D. Crawford, E. Knobloch, and H. Riecke. Period-doubling mode interactions
with circular symmetry, Physica D 44 (1990) 340-396.
[13] H. Riecke, J.D. Crawford and E. Knobloch. Temporal modulation of a subcritical
bifurcation to travelling waves, in New Trends in Nonlinear Dynamics and
Pattern-Forming Phenomena: The Geometry of Nonequilibrium, P. Coullet
and P. Huerre (eds), NATO ASI Series B 237, Plenum Press, 1991, 61-64.
[14] J.D. Crawford, M. Golubitsky, M.G.M. Gomes, E. Knobloch. and I. Stewart,
Boundary conditions as symmetry constraints, in Singularity Theory and its
Applications, Warwick 1989 Part 2, R.M. Roberts and I.N. Stewart (eds) ,
Lecture Notes in Mathematics, Springer-Verlag, 1991, 63-79.
[15] J.D. Crawford. Amplitude equations on unstable manifolds: singular behavior from

neutral modes, in Modern Mathematical Methods in Transport Theory (Operator Theory: Advances and Applications, vol. 51), W. Greenberg and J.
Polewczak (eds), Birkhauser Verlag, 1991, 97-108.
[16) J.D. Crawford and E. Knobloch. Symmetry and symmetry-breaking bifurcations
in fluid dynamics, Annu. Rev. Fluid Mech. 23 (1991) 341-387.
[17] J.D. Crawford. Surface waves in non-square containers with square symmetry,
Phys. Rev. Lett. 67 (1991) 441-444.
[18] J.D. Crawford. Normal forms for driven surface waves: boundary conditions, symmetry, and genericity, Physica D 52 (1991) 429-457.
[19) J.D. Crawford. Introduction to bifurcation theory, Rev. Mod. Phys. 63 (1991)
991-1037.
[20] J.D. Crawford, J.P. Gollub, and David Lane. Hidden symmetries of parametrically
forced waves, Nonlinearity 6 (1993) 119-164.
[21] J.D. Crawford. D4+T2 Mode interactions and hidden rotational symmetry, Nonlinearity 7 (1994) 697-739.
[22] J.D. Crawford. Amplitude expansions for instabilities in populations of globallycoupled oscillators, J. Stat. Phys. 74 (1994) 1047-1084.
[23] J.D. Crawford. Universal trapping scaling on the unstable manifold of a collisionless
electrostatic mode, Phys. Rev. Lett. 73 (1994) 656-659.
[24] J.D. Crawford. Amplitude equations for electrostatic waves: universal singular
behavior in the limit of weak instability, Phys. Plasmas 2 (1995) 97-128.
[25) J.D. Crawford. Scaling and singularities in the entrainment of globally-coupled
oscillators, Phys. Rev. Lett. 74 (1995) 4341-4344.
[26] J.D. Crawford. D4-symmetric maps with hidden Euclidean symmetry, in Pattern Formation: Symmetry Methods and Applications, J. Chadam et al. (eds)
Amer. Math. Soc., 1995, 93-124.
[27] J.D. Crawford and A. Jayaraman. Nonlinear saturation of electrostatic waves: mobile ions modify trapping scaling, Phys. Rev. Lett. 77 (1996) 3549-3552.

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JOHN DAVID CRAWFORD 1954-1998

xiii

[28] J.D. Crawford and E. Knobloch. Amplitude equations for coupled electrostatic
waves in the limit of weak instability, J. Plasma Phys. 60 (1998) 159-180.
[29] J.D. Crawford and K.T.R. Davies. Phase dynamical models of globally coupled
oscillators: singularities and scaling with arbitrary coupling. Submitted to
Physica D, (preprint, patt-sol/9701006 at LANL archives).
[30] J.D. Crawford and A. Jayaraman. Amplitude equations for electrostatic waves:
Multiple species, J. Math. Phys., 39 (1998) 4546-4576.
[31] J.D. Crawford and A. Jayaraman. First principles justification of a "Single Wave
Model" for electrostatic instabilities. Submitted to Phys. Plasmas, (preprint,
physics/9804014 at LANL archives).

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CONTENTS

Foreword ............................................................. v
Preface ............................................................. vii
Dedication to John David Crawford 1954-1998 ........................ ix
Edgar Knobloch
Rayleigh-Benard convection with rotation at small
Prandtl numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
Guenter Ahlers and Kapil M.S. Bajaj
Chaotic intermittency of patterns in symmetric systems. . . . . . . . . . . . .. 11
Peter Ashwin
Heteroclinic cycles and phase turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25
F.H. Busse and R.M. Clever
Hopf bifurcation in anisotropic systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
Gerhard Dangelmayr and Michael Wegelin
Heteroclinic cycles in symmetrically coupled systems. . . . . . . . . . . . . . . .. 49

Michael Field
Symmetry and pattern formation in coupled cell networks. . . . . . . . . . .. 65
Martin Golubitsky and I an Stewart
Spatial hidden symmetries in pattern formation. . . . . . . . . . . . . . . . . . . . .. 83
M. Gabriela M. Gomes, Isabel S. Labouriau,
and Eliana M. Pinho
Stability boundaries of the dynamic states
in pulsating and cellular flames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 101
Michael Gorman
A quantitative description of the relaxation
of textured patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 111
Gemunu H. Gunaratne
Forced symmetry breaking: theory and applications ................. 121
Frederic Guyard and Reiner Lauterbach
xv

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xvi

CONTENTS

Spatiotemporal patterns in electrochemical systems. . . . . . . . . . . . . . . . .. 137
J.L. Hudson
Memory effects and complex patterns in a catalytic
surface reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 147
R.Imbihl
Bursting mechanisms for hydro dynamical systems ................... 157
E. Knobloch and J. M oehlis

Bifurcation from periodic solutions with
spatiotemporal symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175
Jeroen S. W. Lamb and Ian Melbourne
Resonant pattern formation in a spatially
extended chemical system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 193
Anna L. Lin, Valery Petrov, Harry L. Swinney,
Alexandre Ardelea, and Graham F. Carey
Time-dependent pattern formation for two-layer
convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 203
Y. Renardy and C. G. Stoltz
Localized structures in pattern-forming systems ..................... 215
Hermann Riecke
Pattern formation in a surface reaction
with global coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 231
Harm Hinrich Rotermund
Dynamical behavior of patterns with
Euclidean symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 249
Bjorn Sandstede, Arnd Scheel, and Claudia Wulff
Pattern selection in a diffusion-reaction system
with global or long-range interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 265
Moshe Sheintuch and Olga Nekhamkina
Dynamics of kinks and vortices in Josephson-junction
arrays .............................................................. 283
H.S.J. van der Zant and Shinya Watanabe
Josephson junction arrays: Puzzles and prospects. . . . . . . . . . . . . . . . . .. 303
Kurt Wiesenfeld
List of Participants ................................................. 311

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RAYLEIGH-BENARD CONVECTION WITH ROTATION AT
SMALL PRANDTL NUMBERS
GUENTER AHLERS" AND KAPIL M.S. BAJAJ"
Abstract. This paper reviews past results from and future prospects for experimental studies of Rayleigh-Benard convection with rotation about a vertical axis. At
dimensionless rotation rates 0 ::; n ::; 20 and for Prandtl numbers u '::::' 1, KiippersLortz-unstable patterns offered a unique opportunity to study spatio-temporal chaos
immediately above a supercritical bifurcation where weakly-nonlinear theories in the
form of Ginzburg-Landau (GL) or Swift-Hohenberg (SH) equations can be expected to
be valid. However, the dependence of the time and length scales of the chaotic state
on f == AT/ATe - 1 was found to be different from the expected dependence based on
the structure of GL equations. For n ~ 70 and 0.7 ~ u ~ 5 patterns were found to
be cellular near onset with local four-fold coordination. They differ from the theoretically expected Kiippers-Lortz-unstable state. Stable as well as intermittent defect-free
rotating square lattices exist in this parameter range.
Smaller Prandtl numbers ( 0.16 ~ u ~ 0.7 ) can only be reached in mixtures of
gases. These fluids are expected to offer rich future opportunities for the study of a line
of tricritical bifurcations, of supercritical Hopf bifurcations to standing waves, of a line
of co dimension-two points, and of a codimension-three point.

1. Introduction. Convection in a thin horizontal layer of a Huid
heated from below (Rayleigh-Benard convection or RBC) has become a
paradigm for the study of pattern formation [1). It evolves from the
spatially-uniform pure-conduction state via a super critical bifurcation when
the temperature difference flT is increased beyond a critical value flTc. It
reveals numerous interesting phenomena including spatio-temporal chaos
(STC) as to == flT / flTc - 1 grows [1]. Many of these phenomena have been
studied in detail recently [2-21), using primarily compressed gases as the
Huid, sensitive shadowgraph How-visualization, image analysis, and quantitative heat-Hux measurements [16]. However, as flTc is approached from
above and to becomes small enough for the pattern-formation problem to
become theoretically tractable by weakly-nonlinear methods, the system
becomes relatively simple and its behavior can be described in potential

(or variational) form. Then the steady-state pattern is time independent.
In the absence of perturbing boundaries it consists of parallel rolls as shown
in Fig. 1a [17) and as predicted theoretically [22) already 33 years ago.
The system becomes much more complex and interesting even near
onset when it is rotated about a vertical axis with an angular velocity
In that case the coriolis force proportional to
x acts on the Huid
(here is the Huid velocity field in the rotating frame) and renders the
system non-variational. Thus time dependent states can occur arbitrarily
close to onset. Since the bifurcation remains supercritical for n > 0, the
How amplitudes still grow continuously from zero and the usual weakly-

n.

n v

v

"Department of Physics and Center for Nonlinear Science, University of California,
Santa Barbara, CA 93106, USA.
1

M. Golubitsky et al. (eds.), Pattern Formation in Continuous and Coupled Systems
© Springer Science+Business Media New York 1999

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2

GUENTER AHLERS AND KAPIL M.S. BAJAJ

FIG. 1. Convection patterns for small €. (a) is for n = 0 and Ar gas with (]" = 0.69
and { = 0.07 (from Ref. [17)}. It shows the predicted [22} straight-roll pattern. (b) is for
n = 15.4 and C02 at a pressure of 32 bar with (]" = 1.0 and { = 0.05 (from Ref. [23)}.
It is a typical pattern in the K uppers- Lortz-unstable range. (c) is for Argon at 40 bar
with (]" = 0.7, n = 145, and { = 0.04 (from Ref. [24)}; it shows no evidence of the
Kuppers-Lortz instability, and instead consists of a slowly-rotating square lattice.

nonlinear theories, for instance in the form of GL or SH equations, should
remain applicable. Thus one may expect interesting new effects to occur
in a theoretically tractable parameter range.
Indeed it was predicted [25-27] that, for 0 > Oc, the primary bifurcation from the conduction state should be to parallel rolls which are
unstable. Although Oc depends on the Prandtl number (J (the ratio of the
kinematic viscosity v to the thermal diffusivity K), it has a value near 14
for the (J-values near unity which are characteristic of compressed gases (0
is made dimensionless by scaling time with the vertical viscous diffusion
time d2 Iv where d is the cell thickness). The instability is to plane-wave
perturbations which are advanced relative to the rolls at an angle e K L in
the direction of rotation. This phenomenon is known as the Kiippers-Lortz
instability. A snapshot [23] of the resulting nonlinear state of convection is
shown in Fig. lb. The pattern consists of domains of rolls which incessantly
replace each other, primarily by irregular domain-wall motion [5, 23, 2831]. The spatial and temporal behavior suggests the term "domain chaos"
for this state. We discuss this state in the next Section.
Theoretically, the KL instability is expected to persist near onset up
to large values of O. Thus it was a surprise that the patterns found in
experiments near onset changed dramatically when 0 was increased [24].
For 0 ;::: 70, there was no evidence of the characteristic domain chaos until
to was increased well above 0.1. At smaller to, slowly-rotating, aesthetically appealing, square lattices were encountered. Since these experimental observations are very new, it remains to be seen whether a reasonable
explanation can be offered. They will be discussed in Sect. 3.

Finally, we look forward to as yet unrealized experimental opportunities which this .system has to offer in the parameter range of Prandtl
numbers well below unity. Pure fluids (with rare exceptions [32]) have
(J ;::: 0.7.
Recently it was shown [17, 19] that smaller values of (J can be

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RAYLEIGH-BENARD CONVECTION WITH ROTATION

3

reached by mixing two gases, one with a large and the other with a small
atomic or molecular weight. The most extreme example readily available
is a mixture of H2 and Xe. Prandtl numbers as small as 0.16 can be
reached. In the range (J' ;S 0.6, a number of interesting new phenomena
are predicted to occur [27, 33]. In the (J' - n plane they include subcritical
bifurcations below a line of tricritical bifurcations [34], Hopf bifurcations to
standing waves, a line of codimension-two points where the Hopf bifurcation meets the stationary bifurcation, and a co dimension-three point where
the co dimension-two line and the tricritical line meet. The opportunities
for research in this parameter range are outlined in Sect. 4.
2. Kiippers-Lortz domain-chaos. For (J' ;::: 0.33, the bifurcation
to convection in the presence of rotation is expected to be super critical
both below and above nco Thus the KL instability offers a rare opportunity to study STC in a system where the average flow amplitude evolves
continuously from zero and where thus weakly-nonlinear theories might be
expected to be applicable. After receiving only limited attention for several
decades [25-29,35,36], the opportunity to study STC has led to a recent
increase in activity both theoretically and experimentally [16, 23, 37-43].
Indeed, as predicted theoretically [25], the straight rolls at the onset of
convection for dimensionless rotation rates n > nc are unstable to another

set of rolls oriented at an angle e K L with respect to the original rolls along
the direction of rotation. In the spatially extended system this leads to
the co-existence of domains of rolls of more or less uniform orientation
with other domains of a different orientation. A typical example is shown
in Fig. lb. Experiments by Heikes and Busse [28, 29] using shadowgraph
visualization rather far from onset (f ;::: 0.5) established qualitatively the
existence of the KL instability. The replacement of a given domain of rolls
proceeded via domain-wall propagation. More recently the KL instability
was investigated with shadowgraph flow-visualization very close to onset.
It was demonstrated that the bifurcation is indeed supercritical, and that
the instability leads to a continuous domain switching through a mechanism of domain-wall propagation also at small f [5, 23, 30, 31]. This
qualitative feature has been reproduced by Tu and Cross [39] in numerical
solutions of appropriate coupled GL equations, as well as by Neufeld et al.
[41] and Cross et al. [42] through numerical integration of a generalized SH
equation.
Of interest are the time and length scales of the KL instability near
onset. The GL model assumes implicitly a characteristic time dependence
which varies as c 1 and a correlation length which varies as C 1 / 2. We
measured a correlation length given by the inverse width of the square of the
modulus of the Fourier transform as well as a domain-switching frequency
as revealed in Fourier space, and obtained the data in Fig. 2 [23, 30].
These results seem to be inconsistent with GL equations since they show
that the time in the experiment scales approximately as C 1 / 2 and that the

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4

GUENTER AHLERS AND KAPIL M.S. BAJAJ

V/
,.

e

"
,t' •
,'x

" '1.':1

,'It·

..

~--~ +

.... +
x • ,,+

10'

•

x ••
xxx

,'ex

'

,,'-

,,'.

FIG. 2. The characteristic frequencies Wa (left) and lengths ~ (right) of the KL
state. The data were divided by n-dependent constants Wr and ~r so as to collapse
them onto single curves. The dashed lines are shown for reference and have the slopes
1 for Wa and -1/2 for ~ which correspond to the theoretically expected exponents of
the time and length scales near onset. The data sets cover approximately the range
14 n 20. See Refs. [23j, [30j, and [31j for details.

;s ;s

two-point correlation length scales approximately as C 1 / 4 . These results
also differ from numerical results based on a generalized SH equation [42]
although the range of to in the numerical work is rather limited. We regard
the disagreement between experiment and theory as a major problem in
our understanding of STC [44].
3. Square patterns at modest (J. Motivated by the unexpected
scaling of length and time with to for the KL state at n .:s 20, new investigations were undertaken recently in which the range of n was significantly
extended to larger values. Contrary to theoretical predictions [27, 33], it
was found [24] in preliminary work that for n ;:::, 70 the nature of the pattern near onset changed qualitatively although the bifurcation remained
supercritical. Square patterns like the one shown in Fig. lc were stable,
instead of typical KL patterns like the one in Fig. 1b. The squares occurred
both when Argon with (J ::: 0.7 was used and when the fluid was water with
(J ::: 5. The occurrence of squares in this system is completely unexpected
and not predicted by theory; according to the theory the KL instability
should continue to be found near onset also at these higher values of D.

Thus the preliminary work has uncovered a major disagreement with theoretical predictions in a parameter range where one might have expected
the theory to be reliable. We believe that this calls for a systematic experimental study over appropriate parameter ranges. We expect to explore
the range 0 .:s D .:s 400 and 0.7 .:s (J .:s 5 in the near future, and hope that
this will shed light on the extent and origin of the difference between the
physical system and the predictions.
4. The range 0.16 < (J < 0.7. When a RBC system is rotated about
a vertical axis, the critical Rayleigh number Rc(D) increases as shown in

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RAYLEIGH-BENARD CONVECTION WITH ROTATION

5

Fig.3a. Rc(n) is predicted to be independent of a, and experiment [30] and
theory [45] for it are in excellent agreement. For a > 0.33 the bifurcation
is expected to be super critical and to lead to KL chaos unless n is quite
large. As discussed above in Sect. 3, our recent preliminary experiments
have cast doubt upon this; for n ;::: 70 we found square patterns which are
clearly unrelated to the typical KL domains. For large n and a < 0.68, the
stationary bifurcation is predicted [33] to be preceded by a super critical
Hopf bifurcation; but for a > 0.33 we do not expect to reach values of n
sufficiently high to encounter this in the experiment.
The range 0.16 ;S a ;S 0.33 is truly remarkable because of the richness
of the bifurcation phenomena which occur there when the system is rotated.
For instance, for a = 0.26 there is a range from n ~ 16 to 190 over
which the bifurcation is predicted to be subcritical. This is shown by the
dashed section of the curve in Fig. 3c. The sub critical range depends on
a. In Fig. 3b it covers the area below the dashed curve. Thus, the dashed

curve is a line of tricritical bifurcations [34]. It has a maximum in the
n - a plane, terminating in a "tricritical endpoint". An analysis of the
bifurcation phenomena which occur near it in terms of Landau equations
may turn out to be interesting. One may expect path-renormalization [46]
of the classical exponents in the vicinity of the maximum. We are not
aware of equivalent phenomena in equilibrium phase transitions, although
presumably they exist in as yet unexplored parameter ranges.
At relatively large n, the stationary bifurcation (regardless of whether
it is super- or sub-critical) is predicted to be preceded by a supercritical
Hopf bifurcation which is expected to lead to standing waves of convection
rolls [33]. Standing waves are relatively rare; usually a Hopt bifurcation in
a spatially-extended system leads to traveling waves. An example is shown
by the dash-dotted line near the right edge of Fig. 3b. As can be seen
there, the Hopf bifurcation terminates at small n at a co dimension-two
point on the stationary bifurcation which, depending on a, can be superor sub-critical. The line of co dimension-two points is shown in Fig. 3a as a
dash-dotted line. One sees that the tricriticalline and the codimension-two
line meet at a co dimension-three point, located at n ~ 270 and a ~ 0.24.
We note that this is well within the parameter range accessible to our
experiments. We are not aware of any experimentally-accessible examples
of co dimension-three points. This particular case should be accessible to
analysis by weakly-nonlinear theories, and a theoretical description in terms
of G L equations would be extremely interesting and could be compared
with experimental measurements.
The a-range of interest is readily accessible to us by using mixtures
of a heavy and a light gas [19]. Values of a vs. the mole fraction x of the
heavy component for a typical pressure of 22 bar and at 25 0 C are shown
in Fig. 4. An important question in this relation is whether the mixtures

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www.pdfgrip.com

so

.I

1S0

/'

200

/

o

100

.Q

0.3

,"
'" ~."" ..:: .. :,-......................
",I

,,-~~~,

................ ' ..... ~,..........................

(b)

~

2S0

o

0.1

I

i

I

,
101

". I

,

,
,,

,"

!

i

.Q

,i

,.'

102

• I'!

.'.'

,.,.",

1., 0.2 .........../ ........................................../,/..........................

~

O'~--~--~--~----~--~

S~

(a)

-r:l
o

II:

10° r
101

1 1
Cl 0
II

o

"

(c)

"

!

"

!!

.Q

,,'

"

'1~2

""

, ,~

,

FIG. 3. The bifurcation diagram for RBG with rotation about a vertical axis. (a.) Experimental and theoretical results for Rc(n) obtained
with water (open circles) and Ar at three different pressures (triangles) on linear scales. After Ref. {24}. This curve is expected to be independent
of u. (b.) The dashed curve gives the predicted tricriticalline in the n - u plane (the n axis is logarithmic). The dash-dotted line is the predicted
codimension-two line where the Hopf bifurcation is expected to meet the stationary bifurcation {e.g. the solid circle for u = 0.26 in Fig. (c)j.
For u = 0.24 the codimension-two line is expected to intersect the tricritical line, leading to the codimension-three point shown by the open circle
in (c.). The upper dotted line in (b.) corresponds to the path represented in (c.). The lower dotted line in (b.) represents the lowest u-value
accessible to experiment using gas mixtures. (c.) The solid and dashed lines show the critical Rayleigh number as a function of n as in (a), but
on logarithmic scales. The dashed line shows the range over which the bifurcation is predicted to be subcritical for u = 0.26. The two plusses,
indicating the limits of this range, are the tricritical points for this u-value. The dash-dotted line at large n shows the predicted Hopf bifurcation
for u = 0.26.

9uII:

i' 10 ~

e:

1Sr

>
'-0
>

'-0

t:Jj

o

E!::

8

?;

~

t:I

>
Z

&l

tzj

~
~
~
t"'

o
c::

tzj

O'l


RAYLEIGH-BENARD CONVECTION WITH ROTATION

7

will behave in the same way as pure fluids with the same a. We believe
that to a good approximation this is the case because the Lewis numbers
are of order one. This means that heat diffusion and mass diffusion occur
on similar time scales. In that case, the concentration gradient will simply
contribute to the buoyancy force in synchrony with the thermally-induced
density gradient, and thus the critical Rayleigh number will be reduced.
Scaling bifurcation lines by Rc('lT) ('IT is the separation ratio of the mixture) will mostly account for the mixture effect. To a limited extent we
showed already that this is the case [17, 19]. In more recent work we have
begun to show that the bifurcation line Rc(n)/ Rc(O) is independent of'lT.
Nonetheless we recognize that a theoretical investigation of this issue will
be very important.

0.8

0.6

0.4

0.2

0.2

0.4

x

0.6

0.8

FIG. 4. The Prandtl number u as a function of the mole fraction x of the heavy
component for three gas mixtures at a pressure of 22 bar and at 25 0 C. From Ref. [19]'

Assuming that the mixtures behave approximately like pure fluids, we
see that the co dimension-three point can be reached using either H 2 -Xe or
He-Xe mixtures. The tricritical point can be reached also using He-SF 6 •
Acknowledgment. The contents of this review is based on the work
of many members of our groups, both at Santa Barbara and at Los Alamos.
These include Robert Ecke, Yu-Chou Hu, Jun Liu, Brian Naberhuis, and
others. We are also much indebted to discussions with a number of scientists elsewhere, including particularly Fritz Busse, Mike Cross, and Werner
Pesch. This work was supported by the Department of Energy through
Grant DE-FG03-87ER13738.

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8

GUENTER AHLERS AND KAPIL M.S. BAJAJ

REFERENCES

[1] For a recent review, see for instance, M.C. CROSS AND P.C. HOHENBERG, Rev.
Mod. Phys. 65, 851 (1993).
[2] V. CROQUETTE, Contemp. Phys. 30, 113 (1989).
[3] V. CROQUETTE, Contemp. Phys. 30, 153 (1989).
[4] E. BODENSCHATZ, J.R DE BRUYN, G. AHLERS, AND D.S. CANNELL, Phys. Rev.
Lett. 67, 3078 (1991).
[5] E. BODENSCHAT2;, D.S. CANNELL, J.R. DE BRUYN, R. ECKE, Y. Hu, K. LERMAN,
AND G. AHLERS, Physica D 61, 77 (1992).
[6] S.W. MORRIS, E. BODENSCHATZ, D.S. CANNELL, AND G. AHLERS, Phys. Rev. Lett.
7'1, 2026 (1993).
[7] M. ASSENHEIMER AND V. STEINBERG, Phys. Rev. Lett. 70, 3888 (1993).
[8] M. ASSENHEIMER AND V. STEINBERG, Nature 367, 345 (1994).
[9] Y. Hu, R.E. ECKE, AND G. AHLERS, Phys. Rev. E 48, 4399 (1993).
[10] L. NING, Y. Hu, R.E. ECKE, AND G. AHLERS, Phys. Rev. Lett. 71, 2216 (1993).
[11] Y. Hu, R.E. ECKE, AND G. AHLERS, Phys. Rev. Lett. 72, 2191 (1994).
[12] Y. Hu, R.E. ECKE, AND G. AHLERS, Phys. Rev. Lett. 74, 391 (1995).
[13] Y. Hu, R.E. ECKE, AND G. AHLERS, Phys. Rev. E 51, 3263 (1995).
[14] R.E. ECKE, Y. Hu, R. MAINIERI, AND G. AHLERS, Science 269, 1704 (1995).
[15] S.W. MORRIS, E. BODENSCHATZ, D.S. CANNELL, AND G. AHLERS, Physica D 97,
164 (1996).
[16] J.R DE BRUYN, E. BODENSCHATZ, S.W. MORRIS, S. TRAINOFF, Y. Hu, D.S.
CANNELL, AND G. AHLERS, Rev. Sci. Instrum. 67, 2043 (1996).
[17] J. LIU AND G. AHLERS, Phys. Rev. Lett. 77,3126 (1996).
[18] B. PLAPP AND E. BODENSCHATZ, Phys. Script. 67, 111 (1996).
[19] J. LIU AND G. AHLERS, Phys. Rev. E 55, 6950 (1997).
[20] K.M.S. BAJAJ, D. CANNELL, AND G. AHLERS, Phys. Rev. E 55, 4869 (1997).
[21] R. CAKMUR, D. EGOLF, B. PLAPP, AND E. BODENSCHATZ, Phys. Rev. Lett. 79,
1853 (1997).
[22] A. SCHLUTER, D. LORTZ, AND F.H. BUSSE, J. Fluid Mech. 23, 129 (1965).
[23] Y. Hu, RE. ECKE, AND G. AHLERS, Phys. Rev. Lett. 74, 5040 (1995).

[24] K.M.S. BAJAJ, J. LIU, B. NABERHUIS, AND G. AHLERS, Phys. Rev. Lett. 81, 806
(1998).
[25] G. KUPPERS AND D. LORTZ, J. Fluid Mech. 35, 609 (1969).
[26] G. KUPPERS, Phys. Lett. 32A, 7 (1970).
[27] RM. CLEVER AND F.H. BUSSE, J. Fluid Mech. 94, 609 (1979).
[28] F.H. BUSSE AND K.E. HEIKES, Science 208,173 (1980).
[29] K.E. HEIKES AND F.H. BUSSE, Ann. N. Y. Acad. Sci. 357,28 (1980).
[30] Y. Hu, R.E. ECKE, AND G. AHLERS, Phys. Rev. E 55, 6928 (1997).
[31] Y. Hu, W. PESCH, G. AHLERS, AND RE. ECKE, Phys. Rev. E, in print (1998).
[32] One exception is liquid helium. As the superfluid-transition temperature 2.176
K is approached from above, u vanishes. However, experiments are difficult
because u varies from a value of order one to zero over a narrow temperature
range of a few mK, and because of the problem of flow visualization, which
has only recently been achieved under the required cryogenic conditions (P.
Lucas, A. Woodcraft, R. Matley, and W. Wong, International Workshop on
Ultra-High Reynolds-Number Flows, Brookhaven National Laboratory, June
18 to 20, 1996). Other exceptions are liquid metals which have u
0(10- 2 )
because of the large electronic contribution to the conductivity. However, it
is not possible to explore the range 10- 2 ~ u ~ 0.7 with them. Since liquid
metals are not transparent to visible light, flow visualization is also a problem.
[33] T. CLUNE AND E. KNOBLOCH, Phys. Rev. E 47, 2536 (1993).
[34] Early theoretical evidence for the existence of a sub critical and tricritical bifurcation is contained in the work of Clever and Busse (CB) (Ref. [27]). More
recent calculations of the tricriticalline by Clune and Knobloch (Ref. [33]) are

=

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RAYLEIGH-BENARD CONVECTION WITH ROTATION

[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]

9

inconsistent with the result of CB. Using programs developed by W. Pesch, we
re-calculated the tricritical line and obtained the result shown in Fig. 3 which
is more detailed than, but agrees with that of CB.
K. BUHLER AND H. OERTEL, J. Fluid Meeh. 114, 261 (1982).
J.J. NIEMELA AND R.J. DONNELLY, Phys. Rev. Lett. 57,2524 (1986).
F. ZHONG, R. ECKE, AND V. STEINBERG, Physica D 51,596 (1991).
F. ZHONG AND R. ECKE, Chaos 2, 163 (1992).
Y. Tu AND M. CROSS, Phys. Rev. Lett. 69, 2515 (1992).
M. FANTZ, R. FRIEDRICH, M. BESTEHORN, AND H. HAKEN, Physiea D 61, 147
(1992).
M. NEUFELD, R. FRIEDRICH, AND H. HAKEN, Z. Phys. B. 92, 243 (1993).
M. CROSS, D. MEIRON, AND Y. Tu, Chaos 4, 607 (1994).
Y. PONTY, T. PASSOT, AND P. SULEM, Phys. Rev. Lett. 79, 71 (1997).

Recently it was shown in Ref. [31] that the data for and Wa can be fit with a
powerlaw and the expected theoretical leading exponents if large correction
terms are allowed in the analysis.
S. CHANDRASEKHAR, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford, 1961).
M.E. FISHER, Phys. Rev. 176,257 (1968).

e

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CHAOTIC INTERMITTENCY OF PATTERNS IN
SYMMETRIC SYSTEMS*
PETER ASHWINt
Abstract. We examine some properties of attractors for symmetric dynamical
systems that show what we refer to as 'chaotic intermittency'. These are attractors
that contain points with several different symmetry types, with the consequence that
attracted trajectories come arbitrarily close to possessing a variety of different symmetries. Such attractors include heteroclinic attractors, on-off and in-out intermittency and
cycling chaos. We indicate how they can be created at bifurcation, some open problems
and further reading.

1. Introduction. In a wide variety of dynamical systems that possess
symmetries, one can find complicated dynamics occurring with a range of
different symmetries [20]. Moreover, spatio-temporal symmetries of patterns in symmetric systems can appear to change as time progresses; a
pattern may appear to have more symmetry at some points in time than
at others.
In this article we discuss 'chaotic intermittency', and suggest how it
can drive certain features of spatiotemporal chaos. We take the view that
we are interested in dynamics that appears 'typically' given that a symmetry is present. Since chaotic behaviour in systems that occurs in systems
is not very well understood even without symmetry (especially due to lack

of structural stability of most chaotic attractors), we speculate as to what
will generically happen based on results for specific maps, numerical observations and general theory.
We address two questions. First, what sort of dynamics is generic for
a given class of systems with a symmetry. Second, how does dynamics
change on changing a parameter in the system; i.e. what sort of bifurcations are typical. Note that these questions have to a large extent been
answered for steady and periodic solutions by using the techniques of group
representation and invariant theory; see e.g. [29, 27, 28].

2. Generic dynamics with symmetries. Assume we have a dynamical system generated by an ODE
(2.1)

:i; =

f(x)

such that f commutes with a compact Lie group of symmetries r acting
orthogonally on the phase space Rn. Let Il>t be the flow of (2.1). Then
Il>t')' = .,.Il>t for all t and.,. E r. We refer to r as the (spatial) symmetries of
• Submitted to proceedings of IMA Workshop on pattern formation in coupled and
continuous systems, May 1998.
tDepartment of Mathematics and Statistics, University of Surrey, Guildford GU2
5XH, UK. Email:
11

M. Golubitsky et al. (eds.), Pattern Formation in Continuous and Coupled Systems
© Springer Science+Business Media New York 1999

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12

PETER ASHWIN

system. As noted in [28], one can characterise the set of points that have
a given symmetry ~ C r by the fixed point subspace
Fix(~)

= {x E R n

:

gx = x for all 9 E ~}.

These linear subspaces are flow-invariant for any ~ c r and we will see
that their invariance organises attractors in a number of ways. Given a
point x ERn we can define its symmetry by the isotropy subgroup

= {g E r :

~'"

Suppose A

~

gx

= x}.

Rn is a closed invariant set. The basin of attraction of A is

8(A)

= {x E R n

:

w(x)

~

A}

where w{x) is the set of limit points of x under the flow induced by the
vector field (2.1), i.e. w(x) = nt>ous>tattractor [40] if the Lebesgue meas~re of 8(A) is non-zero. It is a minimal
attractor if no proper invariant subset has the same Lebesgue measure basin
of attraction. In the case 8(A) has zero Lebesgue measure and 8(A) is not
just A, we say that A is a chaotic saddle.
2.1. Symmetries of attractors. Given an attractor or invariant set
A we can define the group of symmetries on average of A by
~(A)

= hEr:

'YA

= A}

where h(A, B) is the Hausdorff distance 1 between the sets A and B and
the group of instantaneous symmetries of A by

T(A) =

hEr : 'YX = x

for all x

E A}.

As shown in [39], T(A) is a normal subgroup of ~(A) (i.e. there can be
more symmetries on average than at any point in time). In the case that
A is periodic, one can say more: ~(A) is a cyclic extension of T(A). There
are typically restrictions on which groups can be the symmetry on average
of an attractor if A is asymptotically stable. These are caused by existence
of co dimension one invariant subspaces [11, 39]. For Milnor attractors, the
subgroups that can be realised as symmetries of attractors is in general not
known, but i't will include mOJ;e po~sibilities than for asymptotically stable
attractors [4].
A number of techniques have been developed to compute average symmetries of attractors in symmetric dynamical systems, based on taking
equivariant maps (so-called symmetry det~ctives) and averaging them in a
number of ways: see [12, 14, 16, 22, 25, 26, 48].
li.e.

h{A, B) = sUPaEA infbEB d{a, b)

+ sUPbEB infaEA d{a, b)

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CHAOTIC INTERMITTENCY OF PATTERNS IN SYMMETRIC SYSTEMS 13

If A is an attractor that is an asymptotically stable w-limit set with
a dense orbit one can conclude that every point in a neighbourhood of
A will have symmetry T(A). However, more general (Milnor) attractors
can contain points of several different symmetries, even if they have dense
orbits.
2.2. Heteroclinic attractors. Heteroclinic attractors are of interest
for symmetric systems because they can form robust attractors containing
points with a number of different symmetries. A heteroclinic cycle can
be robust if at least some of its equilibria lie in fixed point spaces with
more symmetry than the connections. Their stability can be determined by
examination of eigenvalues of linearisations at fixed points; see e.g. [35,36].
A prototype example of a robust heteroclinic cycle is that discussed by
Guckenheimer and Holmes [30] of a flow in R 3 that has symmetry group
generated by reflections in the three coordinate planes (Z2)3 and cyclic
permutation of the axes Z3' The fixed point subspaces and the heteroclinic connections between hyperbolic equilibria are shown schematically
in figure l(a).
The connections lie on coordinate planes and are robust to perturbations within these planes. If the network is an attractor (and it is an
open condition that this is the case) any nearby trajectory from an initial
condition with no symmetry will have an w-limit set S that consists of a
union of three equilibria with symmetry Z2 x Z2, and three connections
with symmetry Z2' Note that for this example we have
~(S) =

Z3,

T(S) = id,

and so even though T(S) is triviql, all points in S have some nontr~vial
symmetry.
Such attractors can be seen in the dynamics of symmetric systems, for
example the Kuramoto-Sivashinsky equation [34] or rotating convection
[19]; they are evident from timeseries where a trajectory seems to settle
down to a fixed point for a period of time, after which the traject9ry does
a quick 'flip' to another fixed point. After a number of, flips the trajectory
approaches the first fixed point again, but this time coming even closer; figure l(b) shows a schematic example of such an 'asymptotic slowing down'
on an approach to a heteroclinic attractor. Suppose that the time of residence at the nth visit to a given neighbourhood of the fixed point is Tn.
Then T:;-t typically converges to a constant, > 1 as n -4 00. The constant, is related to the eigenvalues of the linearisations at the fixed points
in the cycle [35].
These attractors show a form of intermittency of symmetry in that the
limiting set of the trajectory includes points with several possible symmetries. However, the form of approach is quite regular in that the time spent
near a particular fixed point subspace increases monotonically in the limit.
If we examine the time of residence of a trajectory in a neighbourhood of a

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