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Oscillators : Index
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allUniquePrincipalMinors
-- Compute all unique principal minors of a given matrix
allUniquePrincipalMinors(...,Modulo=>...)
-- Compute all unique principal minors of a given matrix
allUniquePrincipalMinors(Matrix)
-- Compute all unique principal minors of a given matrix
Checking the codimension and irreducible decomposition of the IG ideal
-- generating all SCT graphs on n vertices
Example 4.1: unique graph on 8 vertices with exotic solutions and no induced cycle of length at least 5
-- example 4.1 in arXiv 2312.16069
Example 4.2: a K5 and pentagon glued along an edge
-- example 4.2 in arXiv 2312.16069
Example 4.3: examples of gluing two cycles along an edge
-- example 4.3 in arXiv 2312.16069
Example 4.4: The square within a square
-- example 4.4 in arXiv 2312.16069
findRealSolutions
-- find real solutions, at least one per component for well-conditioned systems
findRealSolutions(Graph)
-- find real solutions, at least one per component for well-conditioned systems
findRealSolutions(Ideal)
-- find real solutions, at least one per component for well-conditioned systems
Generation of all SCT (simple, connected, 2-connected) graphs on small numbers of vertices
-- generating all SCT graphs on n vertices
getAngles
-- Compute angles from a list of solutions
getAngles(...,Radians=>...)
-- Compute angles from a list of solutions
getAngles(ZZ,List)
-- Compute angles from a list of solutions
getExoticSolutions
-- Display exotic solutions: linearly stable solutions which are not all-in-phase solution
getExoticSolutions(Graph)
-- Display exotic solutions: linearly stable solutions which are not all-in-phase solution
getLinearlyStableSolutions
-- Compute linearly stable solutions for the Kuramoto oscillator system associated to a graph
getLinearlyStableSolutions(Graph)
-- Compute linearly stable solutions for the Kuramoto oscillator system associated to a graph
Harrington-Schenck-Stillman
-- Arxiv 2312.16069 reference
identifyStability
-- Identify the stability of a list of eigenvalues, or of potential solutions to the oscillator system
identifyStability(...,Tolerance=>...)
-- Identify the stability of a list of eigenvalues, or of potential solutions to the oscillator system
identifyStability(BasicList)
-- Identify the stability of a list of eigenvalues, or of potential solutions to the oscillator system
identifyStability(Matrix,List)
-- Identify the stability of a list of eigenvalues, or of potential solutions to the oscillator system
isStableSolution
-- Check if a given solution is stable for the Kuramoto oscillator system
isStableSolution(Matrix,List)
-- Check if a given solution is stable for the Kuramoto oscillator system
Oscillators
-- generation and analysis of oscillator steady states for small graphs
oscJacobian
-- create the Jacobian for the oscillator system associated to a graph
oscJacobian(...,Reduced=>...)
-- create the Jacobian for the oscillator system associated to a graph
oscJacobian(Graph)
-- create the Jacobian for the oscillator system associated to a graph
oscJacobian(Graph,Ring)
-- create the Jacobian for the oscillator system associated to a graph
oscJacobian(Ideal)
-- create the Jacobian for the oscillator system associated to a graph
oscQuadrics
-- find the homogeneous quadrics in the homogeneous Kuramoto ideal
oscQuadrics(...,Reduced=>...)
-- find the homogeneous quadrics in the homogeneous Kuramoto ideal
oscQuadrics(Graph)
-- find the homogeneous quadrics in the homogeneous Kuramoto ideal
oscQuadrics(Graph,Ring)
-- find the homogeneous quadrics in the homogeneous Kuramoto ideal
oscRing
-- create a polynomial ring for a given graph or number of oscillators
oscRing(...,CoefficientRing=>...)
-- create a polynomial ring for a given graph or number of oscillators
oscRing(...,Reduced=>...)
-- create a polynomial ring for a given graph or number of oscillators
oscRing(...,Symbols=>...)
-- create a polynomial ring for a given graph or number of oscillators
oscRing(Graph)
-- create a polynomial ring for a given graph or number of oscillators
oscRing(ZZ)
-- create a polynomial ring for a given graph or number of oscillators
oscSystem
-- the ideal of the reduced equilibrium points of a dynamical system of oscillators
oscSystem(...,Reduced=>...)
-- the ideal of the reduced equilibrium points of a dynamical system of oscillators
oscSystem(Graph)
-- the ideal of the reduced equilibrium points of a dynamical system of oscillators
oscSystem(Graph,Ring)
-- the ideal of the reduced equilibrium points of a dynamical system of oscillators
SCT graphs with exotic solutions
-- finding graphs of small size with exotic solutions
showExoticSolutions
-- Display exotic solutions: linearly stable solutions which are not all-in-phase solution
showExoticSolutions(Graph)
-- Display exotic solutions: linearly stable solutions which are not all-in-phase solution
standardSols
-- find the "standard solutions" for the oscillator system associated to a graph
standardSols(...,Reduced=>...)
-- find the "standard solutions" for the oscillator system associated to a graph
standardSols(Graph)
-- find the "standard solutions" for the oscillator system associated to a graph
standardSols(Graph,Ring)
-- find the "standard solutions" for the oscillator system associated to a graph
vertexSpanningPolynomial
-- computes the vertex spanning polynomial
vertexSpanningPolynomial(Graph)
-- computes the vertex spanning polynomial
vertexSpanningPolynomial(Graph,Ring)
-- computes the vertex spanning polynomial