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oscSystem -- the ideal of the reduced equilibrium points of a dynamical system of oscillators

Description

$R$ should be a ring created with oscRing. The dynamical system involved is the oscillator system associated to $G$: one angle per vertex. If $a_{ij} = 1$ if $(i,j)$ is an edge of the undirected graph $G$, and is zero otherwise, then the system is $d\theta_i/dt = \sum_j a_{ij} \sin(\theta_j - \theta_i)$ where we consider only reduced equilibrium solutions $\theta_0 = 0$.

This function returns the ideal of equilibrium points, where angles $(0, \theta_1, ..., \theta_{n-1})$ are represented via cosines and sines of the angles.

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 : R = oo

o3 = R

o3 : PolynomialRing
i4 : I = oscSystem(G,R)

                                                                            
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  
     ------------------------------------------------------------------------
                                                       2    2       2    2  
     x y  + x y  - x y , - x y  - x y  + x y  + x y , x  + y  - 1, x  + y  -
      1 2    3 2    2 3     3 0    3 2    0 3    2 3   0    0       1    1  
     ------------------------------------------------------------------------
         2    2       2    2
     1, x  + y  - 1, x  + y  - 1)
         2    2       3    3

o4 : Ideal of R
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|
     +---------------------------+
     | 2    2                    |
     |x  + y  - 1                |
     | 0    0                    |
     +---------------------------+
     | 2    2                    |
     |x  + y  - 1                |
     | 1    1                    |
     +---------------------------+
     | 2    2                    |
     |x  + y  - 1                |
     | 2    2                    |
     +---------------------------+
     | 2    2                    |
     |x  + y  - 1                |
     | 3    3                    |
     +---------------------------+

We can find approximations to the 26 complex solutions to this system. If the system has positive dimension (not the case here), the idea is that this set of points should contain at least one on each component.

i6 : solveSystem I_*

o6 = {{-.992345+.03175*ii, .992345-.03175*ii, -.992345+.03175*ii,
     ------------------------------------------------------------------------
     .992345-.03175*ii, -.201665-.156233*ii, .201665+.156233*ii,
     ------------------------------------------------------------------------
     -.201665-.156233*ii, .201665+.156233*ii}, {.996984-.001895*ii,
     ------------------------------------------------------------------------
     .996984-.001895*ii, .996984-.001895*ii, .996984-.001895*ii,
     ------------------------------------------------------------------------
     -.0810557-.0233023*ii, -.0810557-.0233023*ii, -.0810557-.0233023*ii,
     ------------------------------------------------------------------------
     -.0810557-.0233023*ii}, {-.996984+.001895*ii, .996984-.001895*ii,
     ------------------------------------------------------------------------
     -.996984+.001895*ii, .996984-.001895*ii, -.0810557-.0233023*ii,
     ------------------------------------------------------------------------
     .0810557+.0233023*ii, -.0810557-.0233023*ii, .0810557+.0233023*ii},
     ------------------------------------------------------------------------
     {.992345-.03175*ii, .992345-.03175*ii, .992345-.03175*ii,
     ------------------------------------------------------------------------
     .992345-.03175*ii, -.201665-.156233*ii, -.201665-.156233*ii,
     ------------------------------------------------------------------------
     -.201665-.156233*ii, -.201665-.156233*ii}, {.996984-.001895*ii,
     ------------------------------------------------------------------------
     -.996984+.001895*ii, .996984-.001895*ii, -.996984+.001895*ii,
     ------------------------------------------------------------------------
     .0810557+.0233023*ii, -.0810557-.0233023*ii, .0810557+.0233023*ii,
     ------------------------------------------------------------------------
     -.0810557-.0233023*ii}, {-.992345+.03175*ii, -.992345+.03175*ii,
     ------------------------------------------------------------------------
     -.992345+.03175*ii, -.992345+.03175*ii, .201665+.156233*ii,
     ------------------------------------------------------------------------
     .201665+.156233*ii, .201665+.156233*ii, .201665+.156233*ii},
     ------------------------------------------------------------------------
     {-.996984+.001895*ii, -.996984+.001895*ii, -.996984+.001895*ii,
     ------------------------------------------------------------------------
     -.996984+.001895*ii, .0810557+.0233023*ii, .0810557+.0233023*ii,
     ------------------------------------------------------------------------
     .0810557+.0233023*ii, .0810557+.0233023*ii}, {.992345-.03175*ii,
     ------------------------------------------------------------------------
     -.992345+.03175*ii, .992345-.03175*ii, -.992345+.03175*ii,
     ------------------------------------------------------------------------
     .201665+.156233*ii, -.201665-.156233*ii, .201665+.156233*ii,
     ------------------------------------------------------------------------
     -.201665-.156233*ii}, (-1, 1, 1, 1, 1.18306e-12-1.29026e-12*ii,
     ------------------------------------------------------------------------
     -3.29597e-17-7.45931e-17*ii, 1.18307e-12-1.29022e-12*ii,
     ------------------------------------------------------------------------
     4.68375e-17+1.29237e-16*ii), (1, 1, -1, 1, 1.18306e-12-1.29026e-12*ii,
     ------------------------------------------------------------------------
     3.29597e-17+7.45931e-17*ii, 1.18307e-12-1.29022e-12*ii,
     ------------------------------------------------------------------------
     -4.68375e-17-1.29237e-16*ii), (1, -1, 1, 1, 1.31991e-12+4.4264e-13*ii,
     ------------------------------------------------------------------------
     4.75125e-14+2.70308e-14*ii, -1.41494e-12-4.96702e-13*ii,
     ------------------------------------------------------------------------
     -4.75125e-14-2.70309e-14*ii), (1, 1, 1, -1, -1.31991e-12-4.4264e-13*ii,
     ------------------------------------------------------------------------
     4.75125e-14+2.70308e-14*ii, 1.41494e-12+4.96702e-13*ii,
     ------------------------------------------------------------------------
     -4.75125e-14-2.70309e-14*ii), (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     6.92389e-14+4.46618e-14*ii, 6.9238e-14+4.46689e-14*ii,
     ------------------------------------------------------------------------
     6.9238e-14+4.46557e-14*ii, -6.9238e-14-4.46689e-14*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     -3.53415e-14-2.04511e-14*ii, 3.53433e-14+2.04539e-14*ii,
     ------------------------------------------------------------------------
     -3.53454e-14-2.04576e-14*ii, -3.5345e-14-2.04575e-14*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     -6.92389e-14-4.46618e-14*ii, -6.9238e-14-4.46689e-14*ii,
     ------------------------------------------------------------------------
     -6.9238e-14-4.46557e-14*ii, 6.9238e-14+4.46689e-14*ii), (-1, 1, -1, -1,
     ------------------------------------------------------------------------
     3.50926e-14+2.09739e-14*ii, -3.50943e-14-2.09776e-14*ii,
     ------------------------------------------------------------------------
     3.50948e-14+2.09807e-14*ii, 3.50943e-14+2.0981e-14*ii), (1, -1, -1, -1,
     ------------------------------------------------------------------------
     -1.18306e-12+1.29026e-12*ii, 3.29597e-17+7.45931e-17*ii,
     ------------------------------------------------------------------------
     -1.18307e-12+1.29022e-12*ii, -4.68375e-17-1.29237e-16*ii), (-1, -1, -1,
     ------------------------------------------------------------------------
     1, 1.31991e-12+4.4264e-13*ii, -4.75125e-14-2.70308e-14*ii,
     ------------------------------------------------------------------------
     -1.41494e-12-4.96702e-13*ii, 4.75125e-14+2.70309e-14*ii), (-1, -1, 1,
     ------------------------------------------------------------------------
     -1, -1.18306e-12+1.29026e-12*ii, -3.29597e-17-7.45931e-17*ii,
     ------------------------------------------------------------------------
     -1.18307e-12+1.29022e-12*ii, 4.68375e-17+1.29237e-16*ii), (-1, 1, -1,
     ------------------------------------------------------------------------
     -1, -1.31991e-12-4.4264e-13*ii, -4.75125e-14-2.70308e-14*ii,
     ------------------------------------------------------------------------
     1.41494e-12+4.96702e-13*ii, 4.75125e-14+2.70309e-14*ii),
     ------------------------------------------------------------------------
     {-2.04165e-12+4.20039e-12*ii, -1.41421, 2.04165e-12-4.20037e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -1, ii, 1, ii}, {-1.41421, 1.9403e-13+1.98468e-13*ii, 1.41421,
     ------------------------------------------------------------------------
     -1.94022e-13-1.98477e-13*ii, -ii, -1, -ii, 1}, (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     -1.86483e-17*ii, -1.86483e-17*ii, -1.82146e-17*ii, 1.86483e-17*ii), (1,
     ------------------------------------------------------------------------
     1, -1, 1, -4.6557e-15+3.22738e-14*ii, -1.09181e-14-9.90581e-15*ii,
     ------------------------------------------------------------------------
     4.65592e-15-3.2274e-14*ii, 1.5979e-15+7.44515e-14*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     -3.26995e-13-8.62092e-13*ii, -3.40408e-13-4.93195e-13*ii,
     ------------------------------------------------------------------------
     -3.5382e-13-1.24295e-13*ii, 3.40408e-13+4.93195e-13*ii), (-1, -1, 1, 1,
     ------------------------------------------------------------------------
     -6.91634e-14-4.4691e-14*ii, 6.92762e-14+4.4755e-14*ii,
     ------------------------------------------------------------------------
     6.91643e-14+4.46904e-14*ii, -6.92762e-14-4.4755e-14*ii), (-1, -1, 1, 1,
     ------------------------------------------------------------------------
     -4.33681e-17+3.72966e-17*ii, 1.21431e-17-1.05384e-16*ii,
     ------------------------------------------------------------------------
     4.33681e-17-3.59955e-17*ii, -1.21431e-17+1.02782e-16*ii), (1, -1, -1, 1,
     ------------------------------------------------------------------------
     -6.91374e-14-4.50054e-14*ii, -6.93213e-14-4.47682e-14*ii,
     ------------------------------------------------------------------------
     6.91409e-14+4.50083e-14*ii, 6.93291e-14+4.4768e-14*ii), (-1, 1, -1, -1,
     ------------------------------------------------------------------------
     -3.26995e-13-8.62092e-13*ii, 3.40408e-13+4.93195e-13*ii,
     ------------------------------------------------------------------------
     -3.5382e-13-1.24295e-13*ii, -3.40408e-13-4.93195e-13*ii), (1, -1, -1,
     ------------------------------------------------------------------------
     -1, 4.6557e-15-3.22738e-14*ii, -1.09181e-14-9.90581e-15*ii,
     ------------------------------------------------------------------------
     -4.65592e-15+3.2274e-14*ii, 1.5979e-15+7.44515e-14*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     6.93178e-14+4.46583e-14*ii, -6.93178e-14-4.4665e-14*ii,
     ------------------------------------------------------------------------
     6.93178e-14+4.46511e-14*ii, 6.93178e-14+4.4665e-14*ii),
     ------------------------------------------------------------------------
     {5.1318e-12+3.74496e-12*ii, -1.41421, -5.13181e-12-3.74495e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 1, ii, -1, ii}, {1.41421, -5.38406e-12+5.14141e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, 5.38407e-12-5.1414e-12*ii, -ii, -1, -ii, 1},
     ------------------------------------------------------------------------
     {2.04165e-12-4.20039e-12*ii, -1.41421, -2.04165e-12+4.20037e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -1, -ii, 1, -ii}, {-1.41421, -5.38406e-12+5.14141e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 5.38407e-12-5.1414e-12*ii, -ii, 1, -ii, -1}, {-1.41421,
     ------------------------------------------------------------------------
     5.9116e-12-3.61391e-12*ii, 1.41421, -5.91161e-12+3.61393e-12*ii, ii, -1,
     ------------------------------------------------------------------------
     ii, 1}, {-6.09525e-12-1.32367e-12*ii, 1.41421,
     ------------------------------------------------------------------------
     6.09525e-12+1.3237e-12*ii, -1.41421, -1, ii, 1, ii}, (-1, 1, 1, -1,
     ------------------------------------------------------------------------
     -2.16634e-13-7.6029e-14*ii, 2.16695e-13+7.61439e-14*ii,
     ------------------------------------------------------------------------
     2.16634e-13+7.6029e-14*ii, -2.16695e-13-7.61439e-14*ii), (-1, -1, 1, 1,
     ------------------------------------------------------------------------
     4.33238e-13+1.52107e-13*ii, 4.33263e-13+1.52094e-13*ii,
     ------------------------------------------------------------------------
     -4.33238e-13-1.52107e-13*ii, -4.33263e-13-1.52094e-13*ii), (-1, 1, 1,
     ------------------------------------------------------------------------
     -1, -1.38778e-17-3.46945e-18*ii, 1.38778e-17+5.20417e-18*ii,
     ------------------------------------------------------------------------
     1.38778e-17+3.03577e-18*ii, -1.38778e-17-4.77049e-18*ii), (1, -1, -1, 1,
     ------------------------------------------------------------------------
     4.44517e-15+2.60321e-15*ii, -4.44512e-15-2.60359e-15*ii,
     ------------------------------------------------------------------------
     -4.44517e-15-2.60321e-15*ii, 4.44512e-15+2.60359e-15*ii), (1, 1, -1, -1,
     ------------------------------------------------------------------------
     -4.44517e-15-2.60322e-15*ii, -4.44512e-15-2.60359e-15*ii,
     ------------------------------------------------------------------------
     4.44517e-15+2.60322e-15*ii, 4.44512e-15+2.60359e-15*ii),
     ------------------------------------------------------------------------
     {-1.31172e-12+2.43682e-12*ii, 1.41421, 1.31171e-12-2.43681e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, 1, ii, -1, ii}, {1.41421, 5.38406e-12-5.14141e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, -5.38407e-12+5.1414e-12*ii, ii, -1, ii, 1}, {1.41421,
     ------------------------------------------------------------------------
     1.9403e-13+1.98468e-13*ii, -1.41421, -1.94022e-13-1.98477e-13*ii, -ii,
     ------------------------------------------------------------------------
     1, -ii, -1}, {6.09525e-12+1.32367e-12*ii, -1.41421,
     ------------------------------------------------------------------------
     -6.09525e-12-1.3237e-12*ii, 1.41421, 1, -ii, -1, -ii},
     ------------------------------------------------------------------------
     {6.09525e-12+1.32367e-12*ii, 1.41421, -6.09525e-12-1.3237e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, -1, -ii, 1, -ii}, {-1.41421, 5.38406e-12-5.14141e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -5.38407e-12+5.1414e-12*ii, ii, 1, ii, -1}, (-1, 1, -1, -1,
     ------------------------------------------------------------------------
     -6.93178e-14-4.46583e-14*ii, 6.93178e-14+4.4665e-14*ii,
     ------------------------------------------------------------------------
     -6.93178e-14-4.46511e-14*ii, -6.93178e-14-4.4665e-14*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     -4.6557e-15+3.22738e-14*ii, 1.09181e-14+9.90581e-15*ii,
     ------------------------------------------------------------------------
     4.65592e-15-3.2274e-14*ii, -1.5979e-15-7.44515e-14*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     3.26995e-13+8.62092e-13*ii, -3.40408e-13-4.93195e-13*ii,
     ------------------------------------------------------------------------
     3.5382e-13+1.24295e-13*ii, 3.40408e-13+4.93195e-13*ii), (1, -1, -1, 1,
     ------------------------------------------------------------------------
     5.5258e-13-9.09342e-12*ii, -4.7102e-13-7.4734e-12*ii,
     ------------------------------------------------------------------------
     -5.5258e-13+9.09342e-12*ii, 4.7102e-13+7.4734e-12*ii), (1, 1, -1, -1,
     ------------------------------------------------------------------------
     -3.16495e-13+7.53139e-13*ii, -2.32767e-12-2.29808e-13*ii,
     ------------------------------------------------------------------------
     3.16495e-13-7.53139e-13*ii, 2.32767e-12+2.29807e-13*ii), (-1, -1, 1, -1,
     ------------------------------------------------------------------------
     4.6557e-15-3.22738e-14*ii, 1.09181e-14+9.90581e-15*ii,
     ------------------------------------------------------------------------
     -4.65592e-15+3.2274e-14*ii, -1.5979e-15-7.44515e-14*ii), (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     3.26995e-13+8.62092e-13*ii, 3.40408e-13+4.93195e-13*ii,
     ------------------------------------------------------------------------
     3.5382e-13+1.24295e-13*ii, -3.40408e-13-4.93195e-13*ii), {1.41421,
     ------------------------------------------------------------------------
     -1.9403e-13-1.98468e-13*ii, -1.41421, 1.94022e-13+1.98477e-13*ii, ii, 1,
     ------------------------------------------------------------------------
     ii, -1}, {2.04165e-12-4.20039e-12*ii, 1.41421,
     ------------------------------------------------------------------------
     -2.04165e-12+4.20037e-12*ii, -1.41421, 1, -ii, -1, -ii}}

o6 : List
i7 : #oo

o7 = 57

We can find approximations to the 6 real solutions to this system.

i8 : findRealSolutions I
warning: some solutions are not regular: {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60}

o8 = {{-1, 1, 1, 1, 0, 0, 0, 0}, {1, 1, -1, 1, 0, 0, 0, 0}, {1, -1, 1, 1, 0,
     ------------------------------------------------------------------------
     0, 0, 0}, {1, 1, 1, -1, 0, 0, 0, 0}, {1, -1, 1, 1, 0, 0, 0, 0}, {1, 1,
     ------------------------------------------------------------------------
     -1, -1, 0, 0, 0, 0}, {1, 1, 1, -1, 0, 0, 0, 0}, {-1, 1, -1, -1, 0, 0, 0,
     ------------------------------------------------------------------------
     0}, {-1, -1, 1, 1, 0, 0, 0, 0}, {1, -1, -1, -1, 0, 0, 0, 0}, {-1, -1,
     ------------------------------------------------------------------------
     -1, 1, 0, 0, 0, 0}, {-1, -1, 1, -1, 0, 0, 0, 0}, {-1, 1, -1, -1, 0, 0,
     ------------------------------------------------------------------------
     0, 0}, {-1, -1, -1, 1, 0, 0, 0, 0}, {1, 1, -1, 1, 0, 0, 0, 0}, {1, 1, 1,
     ------------------------------------------------------------------------
     -1, 0, 0, 0, 0}, {-1, -1, 1, 1, 0, 0, 0, 0}, {-1, -1, 1, 1, 0, 0, 0, 0},
     ------------------------------------------------------------------------
     {-1, 1, 1, -1, 0, 0, 0, 0}, {1, -1, -1, 1, 0, 0, 0, 0}, {-1, 1, -1, -1,
     ------------------------------------------------------------------------
     0, 0, 0, 0}, {1, -1, -1, -1, 0, 0, 0, 0}, {1, -1, 1, 1, 0, 0, 0, 0},
     ------------------------------------------------------------------------
     {-1, 1, -1, 1, 0, 0, 0, 0}, {1, -1, -1, 1, 0, 0, 0, 0}, {-1, 1, 1, -1,
     ------------------------------------------------------------------------
     0, 0, 0, 0}, {-1, -1, 1, 1, 0, 0, 0, 0}, {-1, 1, 1, -1, 0, 0, 0, 0}, {1,
     ------------------------------------------------------------------------
     -1, -1, 1, 0, 0, 0, 0}, {1, 1, -1, -1, 0, 0, 0, 0}, {-1, -1, -1, -1, 0,
     ------------------------------------------------------------------------
     0, 0, 0}, {-1, 1, -1, -1, 0, 0, 0, 0}, {-1, 1, 1, 1, 0, 0, 0, 0}, {1,
     ------------------------------------------------------------------------
     -1, 1, 1, 0, 0, 0, 0}, {-1, 1, 1, -1, 0, 0, 0, 0}, {1, -1, -1, 1, 0, 0,
     ------------------------------------------------------------------------
     0, 0}, {1, 1, -1, -1, 0, 0, 0, 0}, {1, 1, -1, -1, 0, 0, 0, 0}, {-1, -1,
     ------------------------------------------------------------------------
     1, -1, 0, 0, 0, 0}, {-1, -1, -1, 1, 0, 0, 0, 0}, {1, 1, 1, -1, 0, 0, 0,
     ------------------------------------------------------------------------
     0}}

o8 : List
i9 : #oo

o9 = 41

The angles of these solutions (in degrees, not radians, and the 3 refers to the numbner of oscillators).

i10 : netList getAngles(3, findRealSolutions I, Radians=>false)
warning: some solutions are not regular: {8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69}

      +---+---+---+
o10 = |315|180|180|
      +---+---+---+
      |45 |0  |180|
      +---+---+---+
      |135|180|180|
      +---+---+---+
      |135|0  |0  |
      +---+---+---+
      |45 |180|0  |
      +---+---+---+
      |315|0  |180|
      +---+---+---+
      |315|0  |0  |
      +---+---+---+
      |225|180|0  |
      +---+---+---+
      |225|0  |180|
      +---+---+---+
      |315|180|180|
      +---+---+---+
      |135|180|0  |
      +---+---+---+
      |225|0  |180|
      +---+---+---+
      |45 |0  |180|
      +---+---+---+
      |135|180|180|
      +---+---+---+
      |135|180|0  |
      +---+---+---+
      |135|180|0  |
      +---+---+---+
      |225|0  |0  |
      +---+---+---+
      |45 |180|180|
      +---+---+---+
      |45 |180|0  |
      +---+---+---+
      |225|0  |180|
      +---+---+---+
      |135|0  |0  |
      +---+---+---+
      |315|0  |0  |
      +---+---+---+
      |45 |0  |180|
      +---+---+---+
      |315|180|180|
      +---+---+---+
      |135|180|180|
      +---+---+---+
      |225|180|0  |
      +---+---+---+
      |45 |180|180|
      +---+---+---+
      |225|0  |0  |
      +---+---+---+
      |135|180|0  |
      +---+---+---+
      |45 |180|180|
      +---+---+---+
      |315|0  |180|
      +---+---+---+
      |45 |0  |180|
      +---+---+---+
      |135|180|180|
      +---+---+---+
      |225|180|0  |
      +---+---+---+
      |135|0  |0  |
      +---+---+---+
      |315|0  |0  |
      +---+---+---+
      |45 |180|0  |
      +---+---+---+
      |315|180|180|
      +---+---+---+
      |225|0  |180|
      +---+---+---+
      |225|0  |0  |
      +---+---+---+
      |45 |180|180|
      +---+---+---+
      |315|0  |180|
      +---+---+---+
      |315|0  |180|
      +---+---+---+
      |315|0  |0  |
      +---+---+---+
      |225|180|0  |
      +---+---+---+
      |45 |180|0  |
      +---+---+---+

See also

Ways to use oscSystem:

  • oscSystem(Graph)
  • oscSystem(Graph,Ring)

For the programmer

The object oscSystem is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:263:0.