Some Curious Phenomena in Coupled Cell Networks

Golubitsky, Martin; Nicol, Matthew; Stewart, Ian

doi:10.1007/s00332-003-0593-6

Cited by 99 publications

(161 citation statements)

References 11 publications

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“…Several examples of networks with interesting properties were presented in [7]. The simplest network with self-coupling is the feed-forward network shown in Figure 6.…”

Section: Examples Of Networkmentioning

confidence: 99%

“…In certain instances, such as with certain types of colorings [7,11], this classification can be completed, and these classifications provide interesting information about patterns of synchrony. The work in [7,9] shows that codimension-one synchrony-breaking bifurcations can be highly nonstandard (and complicated to analyze). A complete theory for synchrony-breaking will require a better understanding of the Jacobian matrices at synchronous equilibria (parallel to the representation theory of matrices commuting with a given group action).…”

Section: Corollary 93 Suppose That An Identical-edge Homogeneous Nementioning

confidence: 99%

“…This network has a surprising bifurcation structure. It is shown in [7] that synchronybreaking bifurcations occur with multiple eigenvalues (and nontrivial Jordan normal form) in codimension one. Suppose that λ is the bifurcation parameter and that Hopf bifurcation occurs at λ = 0.…”

Section: Examples Of Networkmentioning

confidence: 99%

“…For example, [7] shows that there exists an infinite family of balanced two colorings, almost all of which are not spatially periodic. Wang and Golubitsky [11] classify all balanced two-color patterns with nearest neighbor coupling (NN) and with both nearest and next nearest neighbor coupling (NNN).…”

Section: Examples Of Networkmentioning

confidence: 99%

“…To see that this coloring is balanced observe that each pink cell (cells 1, 5, 6, 10) receives an input from one pink cell and one white cell, each cyan cell (cells 3, 8) receives inputs from two white cells, and each white cell (cells 2, 4, 7, 9) receives inputs from one pink cell and one cyan cell. See also [7]. Structure of the paper.…”

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confidence: 99%

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Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

Golubitsky¹,

Stewart²,

Török³

2005

SIAM J. Appl. Dyn. Syst.

253

445

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Abstract. 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 architecture. They also studied the existence of other robust dynamical patterns using a concept of quotient network. There are two difficulties with their approach. First, there are examples of networks with robust patterns of synchrony that are not included in their class of networks; and second, vector fields on the quotient do not in general lift to vector fields on the original network, thus complicating genericity arguments. We enlarge the class of coupled systems under consideration by allowing two cells to be coupled in more than one way, and we show that this approach resolves both difficulties. The theory that we develop, the "multiarrow formalism," parallels that of Stewart, Golubitsky, and Pivato. In addition, we prove that the pattern of synchrony generated by a hyperbolic equilibrium is rigid (the pattern does not change under small admissible perturbations) if and only if the pattern corresponds to a balanced equivalence relation. Finally, we use quotient networks to discuss Hopf bifurcation in homogeneous cell systems with two-color balanced equivalence relations.

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“…Several examples of networks with interesting properties were presented in [7]. The simplest network with self-coupling is the feed-forward network shown in Figure 6.…”

Section: Examples Of Networkmentioning

confidence: 99%

Section: Corollary 93 Suppose That An Identical-edge Homogeneous Nementioning

confidence: 99%

Section: Examples Of Networkmentioning

confidence: 99%

Section: Examples Of Networkmentioning

confidence: 99%

mentioning

confidence: 99%

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Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

Golubitsky¹,

Stewart²,

Török³

2005

SIAM J. Appl. Dyn. Syst.

253

445

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Evolving dynamical networks: A formalism for describing complex systems

2011

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A note on fractional feed‐forward networks

Pinto

2016

Math Methods in App Sciences

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We simulate a fractional feed-forward network. This network consists of three coupled identical 'cells' (aka, oscillators). We study the behaviour of the associated coupled cell system for variation of the order of the fractional derivative, 0 < < 1. We consider the Caputo derivative, approximated by the Grünwald-Letnikov approach, using finite differences of fractional order. There is observed amplification of the small signals by exploiting the nonlinear response of each oscillator near its intrinsic Hopf bifurcation point for each value of˛. The value of the Hopf bifurcation point varies with the order of the fractional derivative˛.

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Some Curious Phenomena in Coupled Cell Networks

Cited by 99 publications

References 11 publications

Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

Evolving dynamical networks: A formalism for describing complex systems

A note on fractional feed‐forward networks

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