# Ian StewartThe University of Warwick · Mathematics Institute

Ian Stewart

PhD, DSc (hon) x4

## About

789

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18,281

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Citations since 2017

Introduction

## Publications

Publications (789)

Balanced colorings of networks classify robust synchrony patterns — those that are defined by subspaces that are flow-invariant for all admissible ODEs. In symmetric networks, the obvious balanced colorings are orbit colorings, where colors correspond to orbits of a subgroup of the symmetry group. All other balanced colorings are said to be exotic....

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals. Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras. We study conditions on prime ideals relating these properties. We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals, a...

Balanced colorings of networks correspond to flow-invariant synchrony spaces. It is known that the coarsest balanced coloring is equivalent to nodes having isomorphic infinite input trees, but this condition is not algorithmic. We provide an algorithmic characterization: two nodes have the same color for the coarsest balanced coloring if and only i...

Balanced colorings of networks classify robust synchrony patterns -- those that are defined by subspaces that are flow-invariant for all admissible ODEs. In symmetric networks the obvious balanced colorings are orbit colorings, where colors correspond to orbits of a subgroup of the symmetry group. All other balanced colorings are said to be exotic....

Homeostasis is a regulatory mechanism that keeps some specific variable close to a set value as other variables fluctuate, and is of particular interest in biochemical networks. We review and investigate a reformulation of homeostasis in which the system is represented as an input-output network, with two distinguished nodes ‘input’ and ‘output’, a...

Pattern formation, dynamics and bifurcations for lattice models are strongly influenced by the symmetry of the lattice. However, network structure introduces additional constraints, which sometimes affect the resulting behavior. We compute the automorphism groups of all doubly periodic quotient networks of the hexagonal lattice with nearest-neighbo...

Center manifold reduction is a standard technique in bifurcation theory, reducing the essential features
of local bifurcations to equations in a small number of variables corresponding to critical
eigenvalues. This method can be applied to admissible differential equations for a network, but it
bears no obvious relation to the network structure. A...

Patterns of synchrony in networks of coupled dynamical systems can be represented as colorings of the nodes, in which nodes of the same color are synchronous. Balanced colorings, where nodes of the same color have color-isomorphic input sets, correspond to dynamically invariant subspaces, which can have a significant effect on the typical bifurcati...

Multistable illusions occur when the visual system interprets the same image in two different ways. We model illusions using dynamic systems based on Wilson networks, which detect combinations of levels of attributes of the image. In most examples presented here, the network has symmetry, which is vital to the analysis of the dynamics. We assume th...

Patterns of dynamical synchrony that can occur robustly in networks of coupled dynamical systems are associated with balanced colorings of the nodes of the network. In symmetric networks, the orbits of any group of symmetries automatically determine a balanced orbit coloring. Balanced colorings not of this kind are said to be exotic. Exotic colorin...

This is a chapter of the book Complex Analysis by Stewart and Tall, on sale from Cambridge University Press. (c) CUP.

Cambridge Core - Real and Complex Analysis - Complex Analysis - by Ian Stewart

A Lie algebra (over any field and of any dimension) is Noetherian if it satisfies the maximal condition on ideals. We introduce a new and more general class of quasi-Noetherian Lie algebras that possess several of the main properties of Noetherian Lie algebras. This class is shown to be closed under quotients and extensions. We obtain conditions un...

Relations between conceptual maps and reality are widespread in mathematics. The nature of mathematics itself can be phrased in those terms. Applied mathematicians build mathematical maps of reality, but they normally call them models. We examine the process of mathematical modelling from several distinct directions, examining how a simplified mode...

Homeostasis is a regulatory mechanism whereby some output variables of a system are kept approximately constant as input parameters vary over some region. Important applications include biological and chemical systems. In [M. Golubitsky and I. Stewart, J. Math. Biol., 74 (2017), pp. 387-407], we reformulated homeostasis in the context of singularit...

Spontaneous symmetry-breaking proves a mechanism for pattern generation in legged locomotion of animals. The basic timing patterns of animal gaits are produced by a network of spinal neurons known as a Central Pattern Generator (CPG). Animal gaits are primarily characterized by phase differences between leg movements in a periodic gait cycle. Many...

Reuben Hersh has argued, persuasively, that mathematics is not a collection of eternal truths existing in some ideal but nebulous world—the Platonist viewpoint—but is instead a shared human mental construct [11]. It seems difficult to maintain that mathematics is not a shared human mental construct, since it has been developed by mathematicians com...

The internal state of a cell is affected by inputs from the extra-cellular environment such as external temperature. If some output, such as the concentration of a target protein, remains approximately constant as inputs vary, the system exhibits homeostasis. Special sub-networks called motifs are unusually common in gene regulatory networks (GRNs)...

Cell metabolism is an extremely complicated dynamical system that maintains important cellular functions despite large changes in inputs. This "homeostasis" does not mean that the dynamical system is rigid and fixed. Typically, large changes in external variables cause large changes in some internal variables so that, through various regulatory mec...

Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter \(\lambda \) varies over some interval. We discuss two main aspects of homeostasis, both related to the effect of coordinate changes on the input–output map. The first is a reformulation of homeostasis in the context...

We provide a geometric explanation, based on symmetry, for why the moduli space of all triangles up to similarity is itself a triangle. Symmetries occur because the lengths of the sides define triples in so are acted on by the symmetric group , which is isomorphic to the symmetry group of an equilateral triangle. The moduli space for triangles is a...

This chapter is an introduction to coupled cell networks, a formal setting in which to analyse general features of dynamical systems that are coupled together in a network. Such networks are common in many areas of application. The nodes (‘cells’) of the network represent system variables, and directed edges (‘arrows’) represent how variables influ...

Consider networks in which all arrows are distinct and all cells are distinct. In this context we obtain complete descriptions of the groups of diffeomorphisms that preserve network dynamics, in the following sense: changing coordinates via the diffeomorphism transforms the space of admissible maps to itself. Five distinct actions are considered: l...

We survey general results relating patterns of synchrony to network topology, applying the formalism of coupled cell systems. We also discuss patterns of phase-locking for periodic states, where cells have identical waveforms but regularly spaced phases. We focus on rigid patterns, which are not changed by small perturbations of the differential eq...

Science fiction novel exploring the ecological and human consequences of a feature of alien plant life.
"Classic science fiction: a startling and original premise, a character-driven plot exploring that premise with great imagination and ingenuity... Recalls Niven at his best — with better science."
—Stephen Baxter

Sir Erik Christopher Zeeman FRS 1925–2016 - Volume 100 Issue 548 - Ian Stewart

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 s...

Many biological systems have aspects of symmetry. Symmetry is formalized using group theory. This theory applies not just to the geometry of symmetric systems, but to their dynamics. The basic ideas of symmetric dynamics and bifurcation theory are applied to speciation, animal locomotion, the visual cortex, pattern formation in animal markings and...

Spiral patterns have been observed experimentally, numerically, and theoretically in a variety of systems. It is often believed that these spiral wave patterns can occur only in systems of reaction-diffusion equations. We show, both theoretically (using Hopf bifurcation techniques) and numerically (using both direct simulation and continuation of r...

First published in 1979 and written by two distinguished mathematicians with a special gift for exposition, this book is now available in a completely revised third edition. It reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat’s Last Theorem. Intended as a upper level textbook, it...

The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with Z(2) x Z(2) symmetry. They used symmetric bifurcation theory to predict the existen...

‘Structure of groups’ discusses the basic concepts of group theory. Isomorphism occurs when two technically distinct groups share the same abstract structure and order. Groups may also contain smaller subgroups — the orders of subgroups are divisions of the order of the whole group. Two symmetries of some objects may be essentially the same, but ap...

‘Nature's laws’ investigates instances of symmetry in the laws of nature. Noether's theorem states that wherever a Hamiltonian system has a continuous symmetry, there is a conserved quantity. The Lie theory is the standard landscape for studying continuous symmetries. Lie groups have a geometric structure analogous to that of a smooth surface, but...

There is a large body of written materials, available online, that are easily accessed and which recount aspects of Benoît’s life, times, research, quotations, and opinions. But here we try to capture afresh the fact that he was one of us, a mathematician, and to give a glimpse and feeling, for the time that you read this, of the real and amazing m...

modes. In contrast, if the boundary conditions are Neumann or Dirichlet, then the eigenfunctions are defined by Bessel functions
of real argument, and take the form of body modes filling the interior of the domain. Body modes typically do not exhibit pronounced spiral structure. We argue that
the wall modes are important for understanding the forma...

Behind the scenes, equations rule our everyday lives. Mathematician Ian Stewart goes in search of the most influential

Martin Gardner was fond of mathematically-based magic tricks using simple apparatus—in particular, pennies. Chapter 2 of his Mathematical Carnival [2, pages 12–26] is entirely about penny puzzles. About five years later, Gardner and Karl Fulves invented a delightfully simple trick with three pennies (see Demaine [1]). Its explanation involves an im...

The discovery of chaotic dynamics implies that deterministic systems may not be predictable in any meaningful sense. The best-known source of unpredictability is sensitivity to initial conditions (popularly known as the butterfly effect), in which small errors or disturbances grow exponentially. However, there are many other sources of uncertainty...

S. A. Marvel et al. [Chaos 19, No. 4, Article ID 043104, 11 p. (2009)] studied sinusoidally coupled phase oscillators, generalizing coupled Josephson junctions. They obtained an explicit reduction of the dynamics to a parametrised family of ODEs on the three-dimensional Möbius group. This differs from the usual reduction on to the orbit space of a...

Bifurcation problems with the symmetry group Z2⊕Z2 of the rectangle are common in applied science, for example, whenever a Euclidean invariant PDE is posed on a rectangular domain. In this work we derive normal forms for one-parameter bifurcations of steady states with symmetry of the group Z2⊕Z2. We study degeneracies of Z2⊕Z2-codimension 3 and mo...

It is now well known that bifurcation problems arising from elliptic PDEs on finite domains may possess translational symmetries, even though translations cannot leave a finite domain invariant. These "hidden symmetries" are well understood when the domain is a multidimensional rectangle, a square, and a hemisphere. Hidden symmetries have two effec...

In [Parker et al., 2008a] group theory was employed to prove the existence of homoclinic cycles in forced symmetry-breaking of simple (SC), face-centered (FCC), and body-centered (BCC) cubic planforms. In this paper we extend this classification demonstrating that more elaborate heteroclinic cycles and networks can arise through the same process. O...

We study a dynamical system modelling six coupled identical oscillators, introduced by Collins and Stewart in connection with hexapod gaits. The system has dihedral group symmetry D6, and they suggest that symmetric chaos may be present at some parameter values. We confirm this by employing the method of “detectives” introduced by Barany, Dellnitz,...

## Projects

Projects (2)

A network dynamical system can be described by a collection of individual interacting differential equations. Networks are used as models in a variety of applications such as neuronal networks, speciation, arrays of Josephson junctions, and gene regulatory dynamics. In this project, the investigators develop a mathematical theory for network dynamical systems and explore implications of that theory in applications. The network architecture is a graph that shows the couplings between nodes, and which nodes and couplings are identical. Symmetries in the network architecture have been used previously to explore certain properties of solutions to cell systems, such as synchrony or traveling waves. Symmetry, however, applies directly only to the most regular of networks. For a larger class of networks, local symmetries defined on part of the network can replace symmetries as a predictor of interesting and important dynamics. These local symmetries form a groupoid and it is this groupoid structure that is used to analyze properties of solutions and transitions between solutions in network dynamical systems. Symmetry and symmetry-breaking have been used widely by scientists and mathematicians to investigate a variety of physically and biologically interesting topics, including important types of fluid flows, crystal lattices, the existence of elementary particles, and the characteristic markings of the skins of tigers and leopards. The crucial feature of this approach is that it is model-independent in the sense that in appropriate situations symmetry permits the development of a menu of possible outcomes and the physics or chemistry or biology chooses from this menu. Relaxing the requirement that the network under consideration has symmetries extends the applicability of this approach to a wider range of problems, in particular to models of gene regulatory networks, as well as other network models from neuroscience. In these applications it is rare that the exact model equations are known. It is therefore important to develop mathematical tools that study model-independent features of solutions, tools that focus on solution properties that are determined by the general structure of the equations (such as network architecture) rather than by the details of the equations. This approach benefits researchers outside of mathematics by describing a menu of possible dynamics that a network can be expected to exhibit.

Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter varies over some region. This can be formulated in the context of singularity theory by replacing "approximately constant over an interval" by "zero derivative with respect to the input at a point". General coordinate changes need not leave this condition invariant, but in network dynamics there is a natural class of right network-preserving coordinate changes, which preserve homeostasis in a cell if the cell forms a block in the right core of the network, a combinatorial condition that often occurs. (Resources: https://www.asc.ohio-state.edu/golubitsky.4/reprintweb-0.5/output/entireyr.html)