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Attempts to apply mathematics to questions in science or technology often suffer from the problem that an analytical solution of such practical questions is usually either trivial (at least for anyone sufficiently well trained) or impossible (even for the most expert practitioners of mathematics of any given age). In applications of nonlinear dynamical… (More)

- Martin Golubitsky, Ian Stewart, Andrei Török
- SIAM J. Applied Dynamical Systems
- 2005

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. The architecture of a coupled cell network is a graph that indicates how cells are coupled and which cells are equivalent. Stewart, Golubitsky, and Pivato presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of network… (More)

A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even… (More)

- Ian Stewart, Martin Golubitsky, Marcus Pivato
- SIAM J. Applied Dynamical Systems
- 2003

A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can… (More)

- J. J. Collins, Ian Stewart
- Biological Cybernetics
- 1993

The general, model-independent features of different networks of six symmetrically coupled nonlinear oscillators are investigated. These networks are considered as possible models for locomotor central pattern generators (CPGs) in insects. Numerical experiments with a specific oscillator network model are briefly described. It is shown that some generic… (More)

Coupled cell systems are systems of ODEs, defined by ‘admissible’ vector fields, associated with a network whose nodes represent variables and whose edges specify couplings between nodes. It is known that non-isomorphic networks can correspond to the same space of admissible vector fields. Such networks are said to be ‘ODE-equivalent’. We prove that two… (More)

- Steven H. Strogatz, Ian Stewart
- Scientific American
- 1993

weight at the end of a string can take any of an infinite number of closed paths through phase space, depending on the height from which it is released. Biological systems (and clock pendulums), in contrast, tend to have not only a characteristic period hut also a characteristic amplitude. They trace a particular path through phase space, and if some… (More)

- Martin Golubitsky, Matthew Nicol, Ian Stewart
- J. Nonlinear Science
- 2004

We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that… (More)

We study genetic bifurcations of equifibtia in one-parameter Hamiltonian systems with symmetry group F where eigenvalues of the linearized system go through zero. Theorem 3.3 classifies expected actions of F on the generalized eigenspace of this zero eigenvalue. Genetic one degree of freedom symmetric systems are classified in section 4; remarks concerning… (More)

- J. J. Collins, Ian Stewart
- Biological Cybernetics
- 1994

In this paper, a general approach for studying rings of coupled biological oscillators is presented. This approach, which is group-theoretic in nature, is based on the finding that symmetric ring networks of coupled non-linear oscillators possess generic patterns of phaselocked oscillations. The associated analysis is independent of the mathematical details… (More)