Macaulay2 » Documentation
Packages » Oscillators :: oscQuadrics
next | previous | forward | backward | up | index | toc

oscQuadrics -- find the homogeneous quadrics in the homogeneous Kuramoto ideal

Description

The ideal of quadrics of the oscillator system associated to the graph $G$. This is the ideal generated by the quadrics of the system, where the system is $d\theta_i/dt = \sum_j a_{ij} \sin(\theta_j - \theta_i)$ where we consider only reduced equilibrium solutions $\theta_0 = 0$.

i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}})

o1 = Graph{0 => {1, 3}}
           1 => {0, 2}
           2 => {1, 3}
           3 => {0, 2}

o1 : Graph
i2 : oscRing(G, CoefficientRing => CC)

o2 = CC  [x ..y ]
       53  0   3

o2 : PolynomialRing
i3 : S = oo

o3 = S

o3 : PolynomialRing
i4 : I = oscQuadrics(G,S)

o4 = ideal (x y  + x y  - x y  - x y , - x y  + x y  + x y  - x y , - x y  +
             1 0    3 0    0 1    0 3     1 0    0 1    2 1    1 2     2 1  
     ------------------------------------------------------------------------
     x y  + x y  - x y , - x y  - x y  + x y  + x y )
      1 2    3 2    2 3     3 0    3 2    0 3    2 3

o4 : Ideal of S
i5 : netList I_*

     +---------------------------+
o5 = |x y  + x y  - x y  - x y   |
     | 1 0    3 0    0 1    0 3  |
     +---------------------------+
     |- x y  + x y  + x y  - x y |
     |   1 0    0 1    2 1    1 2|
     +---------------------------+
     |- x y  + x y  + x y  - x y |
     |   2 1    1 2    3 2    2 3|
     +---------------------------+
     |- x y  - x y  + x y  + x y |
     |   3 0    3 2    0 3    2 3|
     +---------------------------+

Ways to use oscQuadrics:

  • oscQuadrics(Graph)
  • oscQuadrics(Graph,Ring)

For the programmer

The object oscQuadrics is a method function with options.


The source of this document is in Oscillators/Documentation.m2:409:0.