Martin Golubitsky and Ian Stewart
Birkhäuser Verlag
Basel ● Boston ● Berlin
Preface
Pattern formation in physical systems is one of the major research frontiers of
mathematics. A tendency towards regular structure, often intricate, sometimes highly
complex, seems to be inherent in the natural world and in experimental science. These
structures — patterns — are often of considerable importance. The field is enormous, the
variety of techniques employed is huge, and the range of applications is extensive.
These are signs of a flourishing field of active research. However, in all branches of
science, there is a strong urge to seek general unifying principles and not merely to
continue elaborating a growing diversity of examples. This is especially the case in
mathematics. A central theme of The Symmetry Perspective is that many instances of
pattern formation can be understood within a single framework: the viewpoint of symmetry.
It has long been recognized that symmetry can be used to describe patterns — classical
crystallography, with its 230 distinct symmetry classes, was probably the first area of
applied science to make systematic use of such ideas. However, description alone is
rather a passive use of symmetry. We shall provide evidence that the role of symmetry in
pattern-formation can also be used to gain insight into the mechanisms by which patterns
form.
Over the past 15 years, we and other authors have been exploring a far more active role
for symmetry, in the context of nonlinear dynamical systems. It has become apparent that
the symmetries of a system of nonlinear ordinary or partial differential equations can be
used, in a systematic and unified way, to analyze, predict, and understand many general
mechanisms of pattern-formation. Specifically, the symmetries of a system can be used to
work out a ‘catalogue’ of typical forms of behavior. This catalogue is to a great extent
model-independent. By this we do not mean that the specific model involved is irrelevant,
but that a great deal can be deduced by knowing only the symmetries of that model. In a
sense, all models with a given symmetry explore the same range of pattern-forming
behaviors, and that range of behaviors can be studied in its own right without reference to
many details of the model.
Those details remain crucial, however, because they determine which behaviors the
system ‘chooses’ from its universal pattern-book. From this point of view, it often makes
sense to begin with the model-independent generalities, and only later to introduce the
specifics. This two-stage approach is not always the most sensible, of course. It runs into
difficulties when the model-independent analysis becomes too complicated by comparison
with a model-specific one. Why tackle a difficult general problem when some simpler
method, such as computer simulation, can provide an answer more quickly and more
concretely?
On the other hand, excessive focus on specifics is inefficient if the same general
phenomenon keeps occurring in many different examples. So the model-independent
approach is most useful when it really does provide insight into the observed phenomena,
and when a general framework helps with the understanding of the system and its
behavior.
Several other issues arise when we wish to compare experimental observations with
theoretical predictions, and we emphasize these issues throughout the book. One is the
distinction between phase space and physical space. The theory of non- linear dynamical
systems is largely discussed in terms of trajectories in an abstract phase space — a space
of variables that determines the state of the system. How- ever, the theory had its origins
in qualitative questions — such as whether a periodic solution exists, or a chaotic one —
rather than quantitative questions. As a result, the theoretical work often involves some
unspecified change of coordinates in phase space, and the connection between the
variables in the theory and the variables that are being observed can be lost. The situation
is especially glaring when the system is a PDE. For example, a complex fluid flow-pattern
in physical space is usually encoded as a single point in phase space. The interpretation
of that point (for example, its coordinates might be a series of Fourier coefficients) is highly
important in such circumstances.
Moreover, experimentalists typically observe a time series of measurements; most
commonly of a single variable, but possibly of several, or of a spatially distributed set of
variables (such as a video recording of patterns that change over time). The abstract
theory, on the other hand, works with geometric objects such as attractors, homoclinic
orbits, or invariant measures. The link between these abstract concepts and feasible
observations is often ignored. This is unfortunate, because the link is subtle and indirect.
In this connection, symmetric dynamical systems possess at least one significant
advantage. In most (though not all) cases, the symmetry of the system has a well-defined
physical interpretation and a well-defined interpretation in phase space. For example, if the
experimental apparatus has circular symmetry, then a group of rotations acts both in
physical space and in phase space. These actions need not be the same — usually they
are not — but the elements that act are the same. Thus a symmetry property of a solution
in phase space usually translates into a symmetry property in physical space, and
conversely. For example, the fact that a periodic solution is a rotating wave, or a standing
wave, can be deduced from its symmetries in space and time. Model-dependent specifics
then tell us what these waves actually look like. Thus symmetries provide an important
route between the abstract theory and experimental observations.
The above discussion raises another issue. Knowing that a system of differential
equations has a particular symmetry group is seldom very informative. The crucial
information is how that group acts on the appropriate region of phase space. For example,
we shall describe six different systems of PDEs in the plane, each being symmetric under
the planar Euclidean group. Moreover, all six systems exhibit some common types of
solution, for example parallel ‘rolls’. Nevertheless, all six systems have very different
pattern-forming properties. (This is even the case for isomorphic actions: the actual
meaning of the physical variables can be important, not just the abstract action on them.)
Again, we place emphasis on such considerations, because a failure to appreciate this
issue can be seriously misleading.
Traditionally, symmetry is mostly used to describe and classify static patterns. In the
context of dynamical systems, however, symmetry can usefully be employed in connection
with dynamic patterns too. We apply symmetry methods to increasingly complex kinds of
dynamic behavior as the book progresses: equilibria, period-doubling, time-periodic states,
homoclinic and heteroclinic orbits, and chaos. In each case we motivate the type of
dynamical behavior being studied by discussing one or more potential applications, drawn
from a wide variety of scientific disciplines ranging from theoretical physics to evolutionary
biology.
We now summarize the contents of the book.
Chapter 1 focuses on the simplest context for symmetry-based methods: equilibrium
states of ODEs. The theory is motivated through a model of speciation in evolution,
introducing the concept of a symmetry-breaking bifurcation. We define what we mean by a
symmettic dynamical system and discuss the symmetries of solutions — which, in the
nonlinear case, need not be the same as the symmetries of the system. Two key results
are introduced in this chapter. First, we discuss Liapunov-Schmidt reduction, which —
subject to suitable technical hypotheses — reduces the dimension of a bifurcation problem
to a size that can often be handled. Indeed, the analogous procedure for PDEs reduces
infinite-dimensional problems to finite-dimensional ones. The second key result is the
Equivariant Branching Lemma, which provides sufficient conditions for the existence of a
bifurcating branch of equilibria with a particular type of symmetry group (said to be ‘axial’).
We apply the Equivariant Branching Lemma to the speciation model. At this stage we raise
and discuss a number of modeling issues, concerning the relevance of symmetry
assumptions. We argue that even when the symmetry of the system being modeled is
imperfect or approximate, it is often useful to study an idealized, perfectly symmetric
model. This contention is supported by numerical experiments in which symmetry is
destroyed by introducing small imperfections and/or stochastic effects.
Chapter 2 remains in the equilibrium context, and examines the stability of equilibria,
especially those whose existence can be proved by using the Equivariant Branching
Lemma. Symmetry has a strong effect on stability because the structure of linear maps
that commute with a group action is highly restricted. We also consider the implications of
symmetry for nonlinear maps that commute with a group action, which leads into classical
invariant theory and the study of equivariant maps. One general principle emerges, which
has a strong effect on several applications later in the book: if the symmetry group action
has a nontrivial quadratic equivariant map, then generically all branches predicted by the
Equivariant Branching Lemma are unstable. Taking inspiration from singularity theory, we
argue that an effective way to deal with this difficulty is to add higher-order terms that
‘stabilize’ primary branches, and to treat these higher-order terms as small perturbations
that ‘unfold’ the degenerate case when such higher-order terms are absent.
Chapter 3 employs an application to legged locomotion in animals to motivate the
analysis of time-periodic patterns in symmetric systems. We show that time-periodic states
have additional ‘temporal’ symmetry, corresponding to a circle group S1 of phase shifts
(modulo the period). Thus the symmetry group of a time-periodic state involves ‘spatiotemporal’ symmetries. The central theoretical result is a classification of all possible spatiotemporal symmetry groups for time-periodic states of systems with a given spatial
symmetry group. We apply this theorem to deduce, from observations of animal gaits, the
architecture of the underlying Central Pattern Generator in the animal’s nervous system
(subject to a series of explicit assumptions about the symmetries of the Central Pattern
Generator, which we argue are plausible, and which lead to testable predictions). We end
by discussing multi-rhythms in coupled cell networks, where the presence of spatiotemporal symmetries can lead to surprising frequency-locked states.
One of the standard ways to generate time-periodic states of a system is through Hopf
bifurcation, and this is the topic of Chapter 4. Hopf bifurcation occurs when a stable
equilibrium loses stability to an oscillatory mode. In the symmetric case, the spatiotemporal symmetries of the bifurcating branches are highly influential. In particular there is
an analogue of the Equivariant Branching Lemma for equilibria, which we call the
Equivariant Hopf Theorem. We sketch the proof of this theorem by Liapunov-Schmidt
reduction from a suitable loop space. The closely related topic of Poincaré-Birkhoff normal
form, which sheds light on the stability of bifurcating branches of time-periodic states, is
introduced. We illustrate equivariant Hopf bifurcation with a discussion of spiral waves and
target patterns in reaction-diffusion systems on a circular domain in the plane. We also
apply our results to coupled cell systems. Chapters 3 and 4 both make use of an important
idea: certain kinds of dynamics can introduce a new symmetry — in this case the circle
groups of phase shifts. Chapter 4 ends with a discussion of other examples of these
‘dynamic symmetries’, including periodic orbits of maps, and the adding machine dynamics
that occurs in a period-doubling cascade at the Feigenbaum point.
In Chapter 5 we study lattice-symmetric equilibrium patterns in Euclidean- invariant
PDEs. Although the Eucidean group is non-compact, the assumption of lattice periodicity
reduces the problem to one with a compact symmetry group. In fact, we treat six different
applications with such symmetry: reaction-diffusion systems, the Navier-Stokes equations,
the primary visual cortex, liquid crystals, and Bénard convection with two different kinds of
boundary conditions. Each of these systems is symmetric under the Euclidean group in the
plane, but their pattern-forming behavior is different. The reason for these differences is
the main message of this chapter: knowing that a system is invariant under some
particular symmetry group is insufficient information for an accurate analysis. At least three
further pieces of information are necessary. First: are there additional symmetries
(possibly in some reformulation of the problem)? Second: what is the appropriate
representation of the symmetry group for the problem under study? Third: what is the
interpretation of the symmetry in terms of structures that might be observed
experimentally? Superficially, all six applications considered here appear to have the same
symmetries, so it might be thought that an analysis of the patterns formed in one of them
automatically transfers to all the others. In fact, the opposite happens: all six problems
have significant differences from the others. We end the chapter with a brief discussion of
stability issues.
In a symmetric system of ODEs or PDEs, solutions come in group orbits. In particular, if
the group includes continuous symmetries, these solutions may not be isolated. For
example, even solutions as apparently straightforward as Taylor vortices in Couette-Taylor
flow occur as a continuous group orbit in the usual infinite-cylinder model with periodic
boundary conditions, the relevant part of the group being axial translations. Chapter 6
addresses the consequences of this phenomenon. It begins with a model-independent
discussion of expected bifurcations in the Couette-Taylor system, which provides a
surprisingly detailed and reasonably complete account of the patterns that arise. With this
example as motivation, the main mathematical issues are examined. The main new
feature here is that when a solution loses stability and other solutions bifurcate, the
stability is lost across the entire group orbit. If the group has a continuous subgroup, these
orbits form manifolds, and the bifurcation behavior acquires novel features. For example,
solutions may ‘drift’ along a group orbit. One of the most striking applications of these
ideas is to meandering spirals in the Belousov-Zhabotinskii experiment. Here the tip of the
spiral wave can exhibit an epicyclic motion. There are two types of epicycle, and the
transition between them involves a state with unbounded linear drift. We are therefore
forced to face up to the non-compact nature of the Euclidean group. Our results do not
apply to this symmetry group, but our methods do — subject to suitable technical
modifications.
The main topic of Chapter 7 is the question: which symmetries of a system should be
taken into account when applying the methods of this book? At first sight this is a modeling
issue: the symmetries of real systems are generally only approximate (as is apparent in
both the speciation model and the infinite cylinder model for Couette-Taylor flow), but it is
often appropriate to idealize approximate symmetries into exact ones. However, there is a
second issue: having chosen a model, it may possess more symmetries than might be
anticipated from its construction. In particular, we show that simple PDEs often possess
more symmetry than is apparent from the symmetries of the domain, and show how to turn
these ‘hidden symmetries’ to advantage. Applications include the Faraday experiment, in
which a shallow dish of liquid is vibrated vertically, and convection in a porous medium.
Equilibrium and time-periodic states can arise by local bifurcation. In Chapter 8 we begin
the study of more ‘global’ dynamics by looking at the effects of symmetry on heteroclinic
cycles. Indeed, such cycles are much more common in symmetric systems than they are
in asymmetric ones. We discuss the related concept of a ‘pipe system’ in connection with
bursting in neurons. We also discuss ‘cycling chaos’ in systems of coupled cells, which is
related to heteroclinic cycling between chaotic states rather than equilibria.
Chapter 9 opens up the connection between symmetry and chaos. A good motivating
example is the Faraday experiment in a square container, which experimentally possesses
chaotic states which display no obvious patterns. Where has the symmetry gone? The
mathematical answer is that chaotic attractors, being extended objects, have two kinds of
symmetry. One, pointwise symmetry, is easily observed: every state looks symmetric but
the time-evolution is chaotic. The second, setwise symmetry, is more subtle: now the
symmetry of the state cannot be observed in a instantaneous snapshot — an effect often
known as spatio-temporal chaos. However, a state corresponding to a symmetric attractor
does possess observable symmetries that correspond to those of its attractor. These
symmetries become apparent as ‘symmetries on average’ in a time-series of observations.
The main theoretical concerns of this chapter are to characterize the possible symmetry
groups of chaotic attractors, and to develop some of the novel types of bifurcation that can
occur when chaotic attractors are present, notably the phenomenon of ‘bubbling’. This
phenomenon occurs, for example, in the much-studied problem of the synchronization of
chaotic oscillators. We also discuss how the symmetry of a chaotic attractor can be
observed in a practical experiment. One possible answer is the concept of a ‘detective’.
We discuss the formation of detectives by ergodic sums, and apply these ideas to chaotic
solutions of the Brusselator model on a square domain.
Up to this point, generically all systems studied have been dissipative. The final Chapter
10 demonstrates that the methods can also be applied to Hamiltonian systems. The basic
principles remain unchanged, but details vary considerably: the main reason is that it is
necessary to respect the symplectic structure that is present in Hamiltonian systems. This
chapter merely scratches the surface of a deep and beautiful area of mathematics and
physics. The main exhibit is an equivariant analogue of the Moser-Weinstein Theorem on
the existence of families of periodic solutions near an equilibrium. We discuss applications
to many-body problems, the resonant spring pendulum, and molecular vibrations.
We end with an extensive (though by no means comprehensive) bibliography.
How To Approach This Book. Our aim is to present a body of mathematics that we
consider to be central to the understanding of symmetry-breaking bifurcations and pattern
formation in equivariant dynamics. In a sense The Symmetry Perspective is the
mathematician’s version of our popular book Fearful Symmetry [495]. We use the word
‘perspective’ to emphasize the book’s stance: it is a point of view on the role of symmetry
as an explanation of pattern formation in nonlinear dynamics. This perspective has
evolved over the last twenty years, and it leads to a slightly unusual approach to applied
science. Instead of starting with very specific models and analysing them as individuals,
we begin by contemplating an entire class of models: all those with the appropriate
symmetries. We then ask: Which phenomena are typical in such classes? It turns out that
many important features of pattern formation and qualitative dynamics are typical, so this
question is a sensible one.
Only after establishing the typical properties of symmetry classes of models do we
proceed to more detailed analyses of specific models. The advantage of our perspective is
that it places the analysis in context, and avoids attributing special features to specific
models when in fact those features are universal. Moreover, symmetries of systems (often
approximate symmetries, which we idealize in order to carry out an analysis) are often
easier to discover and to justify than specific model equations. This is especially the case
for biology, and several of our examples are taken from this area of science.
Because our main focus is on a point of view, we have presented the mathematics on
several levels: simple examples to build intuition, applications to scientific questions to
illustrate the link to the real world, and more technical descriptions where we feel these
add to an understanding of the main issues. Thus the choice of what material to include is
a personal one, and we often refer to the existing literature for proofs — especially if the
source is reasonably accessible or if the area is a standard topic in mathematics. In
consequence, The Symmetry Perspective is not self-contained as a course text. It is
probably more useful for this purpose if it is supplemented by a formal textbook, for
example Singularities and Groups in Bifurcation Theory volume 2 [237]. We give
references to suitable sources for such material as it is needed, but we do not embed it in
the kind of logical development that would be required in a traditional postgraduate level
course. This book should therefore be considered as a guide to the subject area, not as a
comprehensive text.
Acknowledgements. These notes began with a series of lectures, organized by Jerry
Marsden at the Fields Institute in the Spring of 1993, then located at the University of
Waterloo. Nolan Evans did an excellent job of recording and TeXing the notes from these
lectures, including many of the figures. Due to a computer malfunction, the notes were
almost lost and would have been lost had it not been for the efforts of Bill Langford and
Nolan Evans to resurrect them. Eight years later, in a vastly expanded and changed form,
we again used these notes as the basis for a series of lectures — this time at the Summer
School on Bifurcations, Symmetry, and Patterns at Coimbra, Portugal in July, 2000
organized by Isabel Labouriau, Jorge Buescu, Sofia Castro, and Ana Dias, and supported
by the Centro de Matematica Aplicada, Universidade do Porto.
The research discussed here has been generously supported during the past decade by
the National Science Foundation, the Texas Advanced Research Program, the Office for
Naval Research, and the Engineering and Physical Sciences Research Council of the UK,
as well as by research supported visits to a number of institutions: the Fields Institute, the
Institute for Mathematics and its Applications (University of Minnesota), the University of
Houston Department of Mathematics, the Santa Fe Institute, the University of Warwick
Mathematics Institute, and the Boston University Center for Biodynamics.
We have had the pleasure to work with many people on the research described in these
notes. We wish to thank each of them: Don Aronson, Pete Ashwin, Ernie Barany, Paul
Bressloff, Jorge Buescu, Luciano Buono, Ernesto Buzano, Sofia Castro, David
Chillingworth, Pascal Chossat, Jack Cohen, Jim Collins, Jack Cowan, John David
Crawford, Gerhard Dangelmayr, Michael Dellnitz, Ana Dias, Benoit Dionne, Toby Elmhirst,
Mike Field, David Gillis, Gabriela Gomes, Herman Haaf, Andrew Hill, Andreas Hohmann,
Ed Ihrig, Mike Impey, Greg King, Edgar Knobloch, Michel Kroon, Martin Krupa, Bill
Langford, Victor LeBlanc, Jun Ma, Miriam Manoel, Jerry Marsden, Ian Melbourne, James
Montaldi, Matt Nicol, Antonio Palacios, Mark Roberts, David Schaeffer, Mary Silber, Jim
Swift, Harry Swinney, Peter Thomas, and André Vanderbauwhede.
Martin Golubitsky
Ian Stewart
Coventry and Houston, December 2000