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

2006 ◽  
Vol 16 (03) ◽  
pp. 559-577 ◽  
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.

SYMMETRY AND SYNCHRONY IN COUPLED CELL NETWORKS 3

2008 ◽  
Vol 18 (02) ◽  
pp. 363-373 ◽  
Author(s):  
FERNANDO ANTONELI ◽  
IAN STEWART
Keyword(s):  
Fixed Point ◽  
General Theorem ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Two Dimensions ◽  
Point Pattern ◽  
Coupled Cell Networks ◽  
Cell Networks ◽  
Balanced Coloring

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flow-invariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Our aim is to provide a group-theoretic explanation of the "exotic" balanced coloring previously discussed in Part 2. Here we show that the pattern can be obtained as a projection into two dimensions of a fixed-point pattern in a three-dimensional lattice. We prove a general theorem giving sufficient conditions for such a construction to lead to a balanced coloring, for an arbitrary direct product of group networks.

PERIOD-DOUBLING SCENARIO WITHOUT FLIP BIFURCATIONS IN A ONE-DIMENSIONAL MAP

2005 ◽  
Vol 15 (04) ◽  
pp. 1267-1284 ◽  
Author(s):  
V. AVRUTIN ◽  
M. SCHANZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
One Dimensional ◽  
Poincaré Return Map ◽  
Smooth Dynamical System

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

STATISTICAL LAWS FOR COMPUTATIONAL COLLAPSE OF DISCRETIZED CHAOTIC MAPPINGS

1996 ◽  
Vol 06 (12a) ◽  
pp. 2389-2399 ◽  
Author(s):  
PHIL DIAMOND ◽  
ALEXEI POKROVSKII
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Close Agreement ◽  
Computer Experiments ◽  
Continuum State ◽  
Chaotic Dynamical Systems ◽  
The Family ◽  
Random Mappings ◽  
Statistical Laws

When a dynamical system is realized on a computer, the computation is of a discretization, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretizations often have collapsing effects to a fixed point or to short cycles. Statistical properties of this phenomenon can be modeled by random mappings with an absorbing center. The model gives results which are very much in line with computational experiments and there appears to be a type of universality summarised by an Arcsine Law. The effects are discussed with special reference to the family of mappings fl(x)=1−|1−2x|l,x∈[0, 1], 1<l≤2. Computer experiments show close agreement with predictions of the model.

SYMMETRY AND SYNCHRONY IN COUPLED CELL NETWORKS 2: GROUP NETWORKS

2007 ◽  
Vol 17 (03) ◽  
pp. 935-951 ◽  
Author(s):  
FERNANDO ANTONELI ◽  
IAN STEWART
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Symmetry Group ◽  
Extension Property ◽  
Square Lattice ◽  
Reduction Modulo ◽  
Coupled Cell Networks ◽  
A Finite Group ◽  
Cell Networks ◽  
Balance Property

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flow-invariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Let Γ be a permutation group acting on the set of cells. We define the group network [Formula: see text], whose architecture is entirely determined by the group orbits of Γ. We prove that if Γ has the "balanced extension property" then every balanced coloring of [Formula: see text] is a fixed-point coloring relative to the automorphism group of the group network. This theorem applies in particular when Γ is cyclic or dihedral, acting on cells as the symmetries of a regular polygon, and in these cases the automorphism group is Γ itself. In general, however, the automorphism group may be larger than Γ. Several examples of this phenomenon are discussed, including the finite simple group of order 168 in its permutation representation of degree 7. More dramatically, for some choices of Γ there exist balanced colorings of [Formula: see text] that are not fixed-point colorings. For example, there exists an exotic balanced 2-coloring when Γ is the symmetry group of the two-dimensional square lattice. This coloring is doubly periodic, and its reduction modulo 8 leads to a finite group with similar properties. Although these patterns do not arise from fixed-point spaces, we provide a group-theoretic explanation of their balance property in terms of a sublattice of index two.

scholarly journals Coupled cell networks: Boolean perspective

BIOMATH ◽  
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.

ENUMERATION OF HOMOGENEOUS COUPLED CELL NETWORKS

2005 ◽  
Vol 15 (08) ◽  
pp. 2361-2373 ◽  
Author(s):  
FALIH ALDOSRAY ◽  
IAN STEWART
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Directed Graph ◽  
Schematic Diagram ◽  
Natural Manner ◽  
Cell Network ◽  
Different Types ◽  
Coupled Cell Networks ◽  
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.

scholarly journals Repelling invariant curves in planar discrete dynamical systems

1994 ◽  
Vol 49 (3) ◽  
pp. 469-481 ◽  
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Continuous Dependence ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Invariant Curves ◽  
Dynamical Properties ◽  
Attracting Fixed Point

Constructive, simple proofs for the existence, regularity, continuous dependence and dynamical properties of a repelling invariant curve for a discrete dynamical system of the plane with an attracting fixed point with real eigenvalues are given. These proofs can be used to generate a numerical algorithm to find these curves and to compute explicitly the dependence of the curve with respect to the system.

Attracting invariant curves in planar discrete dynamical systems

1995 ◽  
Vol 51 (2) ◽  
pp. 273-286
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Dynamic Behaviour ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Invariant Curves ◽  
Attracting Fixed Point

We study the properties of an invariant attracting curve passing through an attracting fixed point of a planar discrete dynamical system. We compare these properties to the corresponding properties of the invariant repelling curve studied in [3] in order to determine the dynamic behaviour of the system near the fixed point.

Symmetries of Quotient Networks for Doubly Periodic Patterns on the Square Lattice

2019 ◽  
Vol 29 (10) ◽  
pp. 1930026 ◽  
Author(s):  
Ian Stewart ◽  
Dinis Gökaydin
Keyword(s):  
Dynamical Systems ◽  
Symmetry Group ◽  
Wreath Product ◽  
Nearest Neighbor ◽  
Automorphism Groups ◽  
Invariant Subspaces ◽  
Lattice Models ◽  
Square Lattice ◽  
Periodic Patterns ◽  
Nearest Neighbor Coupling

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 bifurcations of network dynamical systems. Orbit colorings for subgroups of the automorphism (symmetry) group of the network are always balanced, although the converse is false. We compute the automorphism groups of all doubly periodic quotient networks of the square lattice with nearest-neighbor coupling, and classify the “exotic” cases where this quotient network has extra automorphisms not induced by automorphisms of the square lattice. These comprise five isolated exceptions and two infinite families with wreath product symmetry. We also comment briefly on implications for bifurcations to doubly periodic patterns in square lattice models.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Sign in / Sign up

Export Citation Format

Share Document