### Abstract

We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts René Thom's general viewpoint, one version of which became known as “catastrophe theory.” We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K
theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette–Taylor flow, flames, the Belousov–Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network.

Received 18 December 2014
Accepted 16 March 2015
Published online 20 April 2015

Lead Paragraph:
Symmetry is a feature of many systems of interest in applied science. Mathematically, a symmetry is a transformation that preserves structure; for example, a square looks unchanged if it is rotated through any multiple of a right angle, or reflected in a diagonal or a line joining the midpoints of oppose edges. These symmetries also appear in mathematical models of real-world systems, and their effect is often extensive. The last thirty to forty years has seen considerable advances in the mathematical understanding of the effects of symmetry on dynamical systems—systems of ordinary differential equations. In general, symmetry leads to pattern formation, via a mechanism called symmetry-breaking. For example, if a system with circular symmetry has a time-periodic state, repeating the same behavior indefinitely, then typically this state is either a standing wave or a rotating wave. This observation applies to the movement of a flexible hosepipe as water passes through it: in the standing wave, the hosepipe moves to and fro like a pendulum; in the rotating wave, it goes round and round with its end describing a circle. This observation also applies to how flame fronts move on a circular burner. We survey some basic mathematical ideas that have been developed to analyze effects of symmetry in dynamical systems, along with an extension of these techniques to networks of coupled dynamical systems. The underlying viewpoint goes back to ideas of René Thom on catastrophe theory which require models to be structurally stable; that is, predictions should not change significantly if the model is changed by a small amount, within an appropriate context. Applications include the movements of animals, such as the walk/trot/gallop of a horse; pattern formation in fluid flow; the Belousov–Zhabotinskii chemical reaction, in which expanding circular patterns or rotating spiral patterns occur; and rivalry in human visual perception, in which different images are shown to each eye—and what is perceived may be neither of them.

Acknowledgments:
The research of M.G. was supported in part by NSF Grant No. DMS-0931642 to the Mathematical Biosciences Institute.

Article outline:

I. INTRODUCTION
II. HISTORICAL CONTEXT
A. Nonlinearity and topological dynamics
B. Catastrophe theory
C. Hopf bifurcation as an elementary catastrophe
D. Symmetry
E. Networks
III. SYMMETRIES IN SPACE
A. Equivariant dynamics
B. Isotropy subgroups and fixed-point subspaces
C. Symmetry-breaking
D. Continuous symmetries and patterns
IV. SPATIOTEMPORAL SYMMETRIES
A. Experiments featuring rotating and standing waves
1. Laminar premixed flames
2. Flow through a hosepipe
3. Flow between counterrotating coaxial circular cylinders
B. Symmetries of periodic states in networks
1. A unidirectional ring of three coupled cells
2. Rigidity of phase shifts
V. THE HOPF BIFURCATION AND *H*/*K*
THEOREMS
A. Equivariant Hopf theorem
1. Poincaré–Birkhoff normal form
B. *H*/*K*
theorem: Motivation and statement
1. Application: Animal gaits
VI. MODE INTERACTIONS
A. Couette–Taylor flow
B. Other work on mode interactions
VII. NON-COMPACT EUCLIDEAN SYMMETRY
A. Heuristic description of unbounded tip motion
B. Reaction-diffusion systems
VIII. NETWORKS
A. Synchrony and phase relations
B. Balanced colorings
1. Application: Rivalry
C. Network *H*/*K*
theorems
IX. ANOMALOUS GROWTH IN NETWORK HOPF BIFURCATION
X. OTHER CONTEXTS

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