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.

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

2020 ◽  
Vol 30 (02) ◽  
pp. 2030004
Author(s):  
Ian Stewart ◽  
Dinis Gökaydin
Keyword(s):  
Pattern Formation ◽  
Network Structure ◽  
Nearest Neighbor ◽  
Automorphism Groups ◽  
Hexagonal Lattice ◽  
Lattice Models ◽  
Square Lattice ◽  
Periodic Patterns ◽  
Formation Dynamics ◽  
Additional Constraints

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-neighbor coupling, with emphasis on “exotic” cases where this quotient network has extra automorphisms not induced by automorphisms of the square lattice. These cases comprise three isolated networks and two infinite families with wreath product structure. We briefly discuss the implications for pattern formation, dynamics and bifurcations. This paper is a sequel to a similar analysis of the square lattice and uses similar methods.

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.

THE CRITICAL BEHAVIOR OF THE 2D SPIN-1 ISING MODEL WITH THE BILINEAR AND POSITIVE BIQUADRATIC NEAREST NEIGHBOR INTERACTIONS ON A CELLULAR AUTOMATON

2004 ◽  
Vol 15 (10) ◽  
pp. 1425-1438 ◽  
Author(s):  
A. SOLAK ◽  
B. KUTLU
Keyword(s):  
Cellular Automaton ◽  
Phase Boundary ◽  
Critical Behavior ◽  
Nearest Neighbor ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Creutz Cellular Automaton ◽  
Beg Model

The two-dimensional BEG model with nearest neighbor bilinear and positive biquadratic interaction is simulated on a cellular automaton, which is based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transitions of the model are presented for comparison with those obtained from other calculations. We confirm the existence of the tricritical points over the phase boundary for D/K>0. The values of static critical exponents (α, β, γ and ν) are estimated within the framework of the finite size scaling theory along D/K=-1 and 1 lines. The results are compatible with the universal Ising critical behavior except the points over phase boundary.

ON THE LOCATION AND CONTINUATION OF HOPF BIFURCATIONS IN LARGE-SCALE PROBLEMS

2008 ◽  
Vol 18 (05) ◽  
pp. 1589-1597 ◽  
Author(s):  
M. FRIEDMAN ◽  
W. QIU
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Orthonormal Basis ◽  
Invariant Subspaces ◽  
Hopf Bifurcations ◽  
Augmented System ◽  
Large Scale Problems ◽  
System Technique ◽  
Matlab Package ◽  
Parameter Dependent

CL_MATCONT is a MATLAB package for the study of dynamical systems and their bifurcations. It uses a minimally augmented system for continuation of the Hopf curve. The Continuation of Invariant Subspaces (CIS) algorithm produces a smooth orthonormal basis for an invariant subspace [Formula: see text] of a parameter-dependent matrix A(s). We extend a minimally augmented system technique for location and continuation of Hopf bifurcations to large-scale problems using the CIS algorithm, which has been incorporated into CL_MATCONT. We compare this approach with using a standard augmented system and show that a minimally augmented system technique is more suitable for large-scale problems. We also suggest an improvement of a minimally augmented system technique for the case of the torus continuation.

Engineering Nearest Neighbor Coupling in Huygens Metasurfaces

2021 ◽  
Author(s):  
Isaac O. Oguntoye ◽  
Siddharth Padmanabha ◽  
Brittany Simone ◽  
Adam Ollanik ◽  
Matthew D. Escarra
Keyword(s):  
Nearest Neighbor ◽  
Nearest Neighbor Coupling

SUPERCONDUCTING CLASSES IN THE EXTENDED HUBBARD MODEL

1996 ◽  
Vol 10 (12) ◽  
pp. 1397-1423 ◽  
Author(s):  
MASA-AKI OZAKI ◽  
EIJI MIYAI ◽  
TOMOAKI KONISHI ◽  
KAORU HANAFUSA
Keyword(s):  
Hubbard Model ◽  
Nearest Neighbor ◽  
Mean Field ◽  
Square Lattice ◽  
Extended Hubbard Model ◽  
S Wave ◽  
Ferromagnetic Order ◽  
Numerical Studies ◽  
The Mean

This paper describes group theoretical classification of superconducting states (SC) in the extended Hubbard model with on-site repulsion (U), nearest neighbor attraction (V) and nearest neighbour exchange interaction (J) on the two-dimensional square lattice using the mean field approach. By decomposing the pairing interaction into irreducible parts; A1g, B1g and Eu of D4h point symmetry, we have derived two singlet SCs (s-wave and d-wave) from A1g and B1g, eight triplet SCs from Eu. The first three types of triplet SC have pairing by electrons with antiparallel spin, the second two types have pairing by electrons with equal spin and the last three types are non-unitary and have pairing by only up-spin electrons. We showed that three non-unitary states have to be accompanied with a ferromagnetic order from the structure of the maximal little groups. We performed numerical studies for these SCs. For parameters and electron density favorable for the ferromagnetic order, a non-unitary SC coexistent with ferromagnetism is most stable.

A LATTICE STATISTICS MODEL DERIVED FROM THE TWO-LAYER ADSORPTION PROBLEM

1965 ◽  
Vol 43 (6) ◽  
pp. 980-985
Author(s):  
D. D. Betts ◽  
D. L. Hunter
Keyword(s):  
Nearest Neighbor ◽  
Chemical Potential ◽  
Physical Adsorption ◽  
Square Lattice ◽  
Lateral Interaction ◽  
Lattice Statistics ◽  
Regular Lattice ◽  
Adsorption Sites ◽  
Layer Adsorption ◽  
Gas Molecules

A model is proposed for the physical adsorption of two layers of gas molecules at the sites of a regular lattice with lateral interaction between nearest-neighbor molecules. The model is more complicated than the two-dimensional Ising model. However, for a particular relation among the three energy parameters and at a particular value of the chemical potential the model simplifies considerably. For the simplified model and a square lattice of adsorption sites, high- and low-temperature series expansions for the specific heat have been obtained and the transition temperature estimated.

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