ENUMERATION OF HOMOGENEOUS COUPLED CELL NETWORKS

A coupled cell network is a schematic diagram employed to define a class of differential equations, and can be thought of as a directed graph whose nodes (cells) represent dynamical systems and whose edges (arrows) represent couplings. Often the nodes and edges are labeled to distinguish different types of system and coupling. The associated differential equations reflect this structure in a natural manner. The network is homogeneous if there is one type of cell and one type of arrow, and moreover, every cell lies at the head end of the same number r of arrows. This number is the valency of the network. We use a group-theoretic formula usually but incorrectly attributed to William Burnside to enumerate homogeneous coupled cell networks with N cells and valency r, in both the disconnected and connected cases. We compute these numbers explicitly when N, r ≤ 6.

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CONSTRUCTION OF LATTICES OF BALANCED EQUIVALENCE RELATIONS FOR REGULAR HOMOGENEOUS NETWORKS USING LATTICE GENERATORS AND LATTICE INDICES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409025067 ◽

2009 ◽

Vol 19 (11) ◽

pp. 3691-3705 ◽

Cited By ~ 6

Author(s):

HIROKO KAMEI

Keyword(s):

Special Kind ◽

Building Blocks ◽

Equivalence Relations ◽

Lattice Structures ◽

Internal Dynamics ◽

One Dimensional ◽

Cell Network ◽

Coupled Cell Networks ◽

Cell Networks ◽

Number Of Cells

Regular homogeneous networks are a class of coupled cell network, which comprises one type of cell (node) with one type of coupling (arrow), and each cell has the same number of input arrows (called the valency of the network). In coupled cell networks, robust synchrony (a flow-invariant polydiagonal) corresponds to a special kind of partition of cells, called a balanced equivalence relation. Balanced equivalence relations are determined solely by the network structure. It is well known that the set of balanced equivalence relations on a given finite network forms a complete lattice. In this paper, we consider regular homogeneous networks in which the internal dynamics of each cell is one-dimensional, and whose associated adjacency matrices have simple eigenvalues (real or complex). We construct explicit forms of lattices of balanced equivalence relations for such networks by introducing key building blocks, called lattice generators, along with integer numbers called lattice indices. The properties of lattice indices allow construction of all possible lattice structures for balanced equivalence relations of regular homogeneous networks of any number of cells with any valency. As an illustration, we show all 14 possible lattice structures of balanced equivalence relations for four-cell regular homogeneous networks.

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SYMMETRY AND SYNCHRONY IN COUPLED CELL NETWORKS 1: FIXED-POINT SPACES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127406015167 ◽

2006 ◽

Vol 16 (03) ◽

pp. 559-577 ◽

Cited By ~ 23

Author(s):

FERNANDO ANTONELI ◽

IAN STEWART

Keyword(s):

Dynamical System ◽

Fixed Point ◽

Dynamical Systems ◽

Symmetry Breaking ◽

Nearest Neighbor ◽

Invariant Subspaces ◽

Folk Theorem ◽

Coupled Cell Networks ◽

Equivariant Dynamical Systems ◽

Cell Networks

Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Γ of permutations of the nodes ("cells"). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals — ones that are not fixed-point spaces — can occur for such networks. We also prove the "folk theorem" that in any Γ-equivariant dynamical system on Rk the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.

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Coupled cell networks: Boolean perspective

BIOMATH ◽

10.11145/j.biomath.2017.03.227 ◽

2017 ◽

Vol 6 (1) ◽

pp. 1703227

Author(s):

Katarzyna Swirydowicz

Keyword(s):

Dynamical System ◽

Differential Equations ◽

Boolean Functions ◽

Individual Cell ◽

Important Concept ◽

Coupled Cell Networks ◽

Entire Theory ◽

Cell Networks ◽

Discrete Networks ◽

Cell Graph

During the 1980s and early 1990s, Martin Golubitsky and Ian Stewart formulated and developed a theory of "coupled cell networks" (CCNs). Their research was primarily focused onquadrupeds' gaits and they applied the framework of differential equations. Golubitsky and Stewart were particularly interested in change of synchrony between $4$ legs of an animal. For example what happens when the animal speeds up from walk to gallop. The most important concept of their theory is a {\it cell}. The cell captures the dynamics of one unit and a dynamical system consists of many identical (governed by the same principles) cells influencing (coupling to) each other. Models based on identical cooperating units are fairly common in many areas, especially in biology, ecology and sociology. The goal of investigation in Coupled Cell Networks theory is understanding the dependencies and interplay between dynamics of an individual cell, graph of connections between cells, and the nature of couplings. \vspace*{0.2em}In this paper, I redefine Coupled Cell Networks using framework of Boolean functions. This moves the entire theory to a new setting. Some phenomena proved to be very similar as for continuous networks and some are completely different. Also, for discrete networks we ask questions differently and study different phenomena. The paper presents two examples: networks that bring 2-cell bidirectional ring as a quotient and networks that bring 3-cell bidirectional ring as a quotient.

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ELIMINATION OF MULTIPLE ARROWS AND SELF-CONNECTIONS IN COUPLED CELL NETWORKS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127407017197 ◽

2007 ◽

Vol 17 (01) ◽

pp. 99-106 ◽

Cited By ~ 5

Author(s):

IAN STEWART

Keyword(s):

Directed Graphs ◽

Formal Theory ◽

Coupled Systems ◽

Inductive Step ◽

Combinatorial Objects ◽

Cell Network ◽

Coupled Cell Networks ◽

Multiple Edges ◽

Cell Networks ◽

Identical Edge

A coupled cell network is a finite directed graph in which nodes and edges are classified into equivalence classes. Such networks arise in a formal theory of coupled systems of differential equations, as a schematic indication of the topology of the coupling, but they can be studied independently as combinatorial objects. The edges of a coupled cell network are "identical" if they are all equivalent, and the network is "homogeneous" if all nodes have isomorphic sets of input edges. Golubitsky et al. [2005] proved that every homogeneous identical-edge coupled cell network is a quotient of a network that has no multiple edges and no self-connections. We generalize this theorem to any coupled cell network by removing the conditions of homogeneity and identical edges. The problem is a purely combinatorial assertion about labeled directed graphs, and we give two combinatorial proofs. Both proofs eliminate self-connections inductively. The first proof also eliminates multiple edges inductively, the main feature being the specification of the inductive step in terms of a complexity measure. The second proof obtains a more efficient result by eliminating all multiple edges in a single construction.

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Some Properties Involving q-Hermite Polynomials Arising from Differential Equations and Location of Their Zeros

Mathematics ◽

10.3390/math9111168 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1168

Author(s):

Cheon Seoung Ryoo ◽

Jung Yoog Kang

Keyword(s):

Differential Equations ◽

Hermite Polynomials ◽

Real Numbers ◽

Different Types

Hermite polynomials are one of the Apell polynomials and various results were found by the researchers. Using Hermit polynomials combined with q-numbers, we derive different types of differential equations and study these equations. From these equations, we investigate some identities and properties of q-Hermite polynomials. We also find the position of the roots of these polynomials under certain conditions and their stacked structures. Furthermore, we locate the roots of various forms of q-Hermite polynomials according to the conditions of q-numbers, and look for values which have approximate roots that are real numbers.

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