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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 = {{-.991627+.018641*ii, .991627-.018641*ii, -.991627+.018641*ii,
     ------------------------------------------------------------------------
     .991627-.018641*ii, -.169892-.108806*ii, .169892+.108806*ii,
     ------------------------------------------------------------------------
     -.169892-.108806*ii, .169892+.108806*ii}, {.997933-.002950*ii,
     ------------------------------------------------------------------------
     .997933-.002950*ii, .997933-.002950*ii, .997933-.002950*ii,
     ------------------------------------------------------------------------
     .0752831+.0391066*ii, .0752831+.0391066*ii, .0752831+.0391066*ii,
     ------------------------------------------------------------------------
     .0752831+.0391066*ii}, {-.997933+.002950*ii, .997933-.002950*ii,
     ------------------------------------------------------------------------
     -.997933+.002950*ii, .997933-.002950*ii, .0752831+.0391066*ii,
     ------------------------------------------------------------------------
     -.0752831-.0391066*ii, .0752831+.0391066*ii, -.0752831-.0391066*ii},
     ------------------------------------------------------------------------
     {.991627-.018641*ii, .991627-.018641*ii, .991627-.018641*ii,
     ------------------------------------------------------------------------
     .991627-.018641*ii, -.169892-.108806*ii, -.169892-.108806*ii,
     ------------------------------------------------------------------------
     -.169892-.108806*ii, -.169892-.108806*ii}, {.997933-.002950*ii,
     ------------------------------------------------------------------------
     -.997933+.002950*ii, .997933-.002950*ii, -.997933+.002950*ii,
     ------------------------------------------------------------------------
     -.0752831-.0391066*ii, .0752831+.0391066*ii, -.0752831-.0391066*ii,
     ------------------------------------------------------------------------
     .0752831+.0391066*ii}, {-.991627+.018641*ii, -.991627+.018641*ii,
     ------------------------------------------------------------------------
     -.991627+.018641*ii, -.991627+.018641*ii, .169892+.108806*ii,
     ------------------------------------------------------------------------
     .169892+.108806*ii, .169892+.108806*ii, .169892+.108806*ii},
     ------------------------------------------------------------------------
     {-.997933+.002950*ii, -.997933+.002950*ii, -.997933+.002950*ii,
     ------------------------------------------------------------------------
     -.997933+.002950*ii, -.0752831-.0391066*ii, -.0752831-.0391066*ii,
     ------------------------------------------------------------------------
     -.0752831-.0391066*ii, -.0752831-.0391066*ii}, {.991627-.018641*ii,
     ------------------------------------------------------------------------
     -.991627+.018641*ii, .991627-.018641*ii, -.991627+.018641*ii,
     ------------------------------------------------------------------------
     .169892+.108806*ii, -.169892-.108806*ii, .169892+.108806*ii,
     ------------------------------------------------------------------------
     -.169892-.108806*ii}, (1, -1, -1, -1, -1.38778e-17-1.30104e-18*ii,
     ------------------------------------------------------------------------
     1.73472e-17+1.73472e-18*ii, 1.38778e-17+1.30104e-18*ii,
     ------------------------------------------------------------------------
     1.38778e-17+3.03577e-18*ii), (-1, 1, 1, 1, 6.07566e-15+1.05738e-14*ii,
     ------------------------------------------------------------------------
     6.86407e-14-3.92009e-14*ii, -6.0756e-15-1.05739e-14*ii,
     ------------------------------------------------------------------------
     -8.07897e-14+1.80548e-14*ii), (1, 1, -1, 1, 6.07566e-15+1.05738e-14*ii,
     ------------------------------------------------------------------------
     -6.86407e-14+3.92009e-14*ii, -6.0756e-15-1.05739e-14*ii,
     ------------------------------------------------------------------------
     8.07897e-14-1.80548e-14*ii), (1, -1, 1, 1, 2.58758e-14+2.53814e-14*ii,
     ------------------------------------------------------------------------
     -9.31095e-15-1.05671e-14*ii, -7.25825e-15-4.25127e-15*ii,
     ------------------------------------------------------------------------
     9.31097e-15+1.05672e-14*ii), (1, 1, 1, -1, -2.58758e-14-2.53814e-14*ii,
     ------------------------------------------------------------------------
     -9.31095e-15-1.05671e-14*ii, 7.25825e-15+4.25127e-15*ii,
     ------------------------------------------------------------------------
     9.31097e-15+1.05672e-14*ii), (1, 1, -1, 1, -5.97965e-15+7.93004e-15*ii,
     ------------------------------------------------------------------------
     -5.97916e-15+7.93059e-15*ii, 5.97965e-15-7.93015e-15*ii,
     ------------------------------------------------------------------------
     -5.97867e-15+7.93126e-15*ii), (-1, -1, -1, 1, 1.5657e-14+8.87344e-15*ii,
     ------------------------------------------------------------------------
     1.5657e-14+8.87376e-15*ii, 1.5657e-14+8.87181e-15*ii,
     ------------------------------------------------------------------------
     -1.5657e-14-8.87544e-15*ii), (-1, 1, 1, 1, -4.33681e-18+4.33681e-19*ii,
     ------------------------------------------------------------------------
     7.80626e-18+1.38778e-17*ii, 4.33681e-18-4.33681e-19*ii,
     ------------------------------------------------------------------------
     4.33681e-18+6.07153e-18*ii), (1, 1, -1, -1, 6.92016e-14+4.44931e-14*ii,
     ------------------------------------------------------------------------
     6.92051e-14+4.44937e-14*ii, -6.92016e-14-4.44931e-14*ii,
     ------------------------------------------------------------------------
     -6.92051e-14-4.44937e-14*ii), (-1, -1, 1, -1, 4.1352e-15+6.34532e-15*ii,
     ------------------------------------------------------------------------
     4.13515e-15+6.34443e-15*ii, -4.1352e-15-6.34521e-15*ii,
     ------------------------------------------------------------------------
     4.13471e-15+6.34378e-15*ii), (-1, 1, -1, -1, 1.3457e-14+9.31381e-13*ii,
     ------------------------------------------------------------------------
     -1.3454e-14-9.31412e-13*ii, 1.3451e-14+9.31443e-13*ii,
     ------------------------------------------------------------------------
     1.3454e-14+9.31412e-13*ii), (1, -1, -1, -1, 4.33681e-18-4.33681e-19*ii,
     ------------------------------------------------------------------------
     -7.80626e-18-1.38778e-17*ii, -4.33681e-18+4.33681e-19*ii,
     ------------------------------------------------------------------------
     -4.33681e-18-6.07153e-18*ii), (-1, -1, 1, 1, -6.1271e-13-2.15095e-13*ii,
     ------------------------------------------------------------------------
     -6.12729e-13-2.15085e-13*ii, 6.1271e-13+2.15095e-13*ii,
     ------------------------------------------------------------------------
     6.12729e-13+2.15085e-13*ii), (1, -1, -1, -1,
     ------------------------------------------------------------------------
     -6.07566e-15-1.05738e-14*ii, -6.86407e-14+3.92009e-14*ii,
     ------------------------------------------------------------------------
     6.0756e-15+1.05739e-14*ii, 8.07897e-14-1.80548e-14*ii), (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     2.58758e-14+2.53814e-14*ii, 9.31095e-15+1.05671e-14*ii,
     ------------------------------------------------------------------------
     -7.25825e-15-4.25127e-15*ii, -9.31097e-15-1.05672e-14*ii), (-1, -1, 1,
     ------------------------------------------------------------------------
     -1, -6.07566e-15-1.05738e-14*ii, 6.86407e-14-3.92009e-14*ii,
     ------------------------------------------------------------------------
     6.0756e-15+1.05739e-14*ii, -8.07897e-14+1.80548e-14*ii), (-1, 1, -1, -1,
     ------------------------------------------------------------------------
     -2.58758e-14-2.53814e-14*ii, 9.31095e-15+1.05671e-14*ii,
     ------------------------------------------------------------------------
     7.25825e-15+4.25127e-15*ii, -9.31097e-15-1.05672e-14*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     -3.59517e-14-2.12935e-14*ii, 3.59513e-14+2.12885e-14*ii,
     ------------------------------------------------------------------------
     3.59513e-14+2.12932e-14*ii, 3.59461e-14+2.12918e-14*ii),
     ------------------------------------------------------------------------
     {-5.7252e-12-8.75496e-12*ii, -1.41421, 5.72519e-12+8.75497e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -1, ii, 1, ii}, {-1.41421, 2.47435e-12-2.19594e-12*ii, 1.41421,
     ------------------------------------------------------------------------
     -2.47434e-12+2.19592e-12*ii, -ii, -1, -ii, 1}, (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     -7.96086e-13-4.16023e-13*ii, -7.96077e-13-4.16026e-13*ii,
     ------------------------------------------------------------------------
     -7.96068e-13-4.16027e-13*ii, 7.96077e-13+4.16026e-13*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     2.18756e-13+6.35932e-14*ii, 1.21361e-12+4.65968e-13*ii,
     ------------------------------------------------------------------------
     -2.18756e-13-6.35932e-14*ii, -7.76096e-13-3.38778e-13*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     3.29597e-17+3.33934e-17*ii, 1.04422e-17+1.70288e-17*ii,
     ------------------------------------------------------------------------
     -3.46945e-17+5.55112e-17*ii, 2.40015e-17-5.63379e-17*ii), (-1, -1, 1, 1,
     ------------------------------------------------------------------------
     2.55511e-14+6.40585e-15*ii, 8.98641e-15+2.84444e-14*ii,
     ------------------------------------------------------------------------
     -2.55511e-14-6.40503e-15*ii, -8.98587e-15-2.8444e-14*ii), (1, -1, -1, 1,
     ------------------------------------------------------------------------
     -6.92788e-14-4.46015e-14*ii, -6.94106e-14-4.45256e-14*ii,
     ------------------------------------------------------------------------
     6.92788e-14+4.46047e-14*ii, 6.94141e-14+4.45288e-14*ii), (-1, 1, -1, -1,
     ------------------------------------------------------------------------
     1.29224e-14-2.4584e-16*ii, 7.8835e-16+4.36749e-15*ii,
     ------------------------------------------------------------------------
     -1.44993e-14-8.49256e-15*ii, -7.88288e-16-4.36781e-15*ii), (1, -1, -1,
     ------------------------------------------------------------------------
     -1, -2.18756e-13-6.35932e-14*ii, 1.21361e-12+4.65968e-13*ii,
     ------------------------------------------------------------------------
     2.18756e-13+6.35932e-14*ii, -7.76096e-13-3.38778e-13*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     1.66295e-14+1.21244e-14*ii, -1.66297e-14-1.21224e-14*ii,
     ------------------------------------------------------------------------
     1.66295e-14+1.21207e-14*ii, 1.66297e-14+1.21243e-14*ii),
     ------------------------------------------------------------------------
     {-3.32395e-12-5.99901e-12*ii, -1.41421, 3.32394e-12+5.99901e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 1, ii, -1, ii}, {1.41421, -6.41711e-13+4.54503e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, 6.4172e-13-4.54504e-12*ii, -ii, -1, -ii, 1},
     ------------------------------------------------------------------------
     {5.7252e-12+8.75496e-12*ii, -1.41421, -5.72519e-12-8.75497e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -1, -ii, 1, -ii}, {-1.41421, -6.41711e-13+4.54503e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 6.4172e-13-4.54504e-12*ii, -ii, 1, -ii, -1}, {-1.41421,
     ------------------------------------------------------------------------
     -2.47435e-12+2.19594e-12*ii, 1.41421, 2.47434e-12-2.19592e-12*ii, ii,
     ------------------------------------------------------------------------
     -1, ii, 1}, {-3.32395e-12-5.99901e-12*ii, 1.41421,
     ------------------------------------------------------------------------
     3.32394e-12+5.99901e-12*ii, -1.41421, -1, ii, 1, ii}, (-1, -1, 1, -1,
     ------------------------------------------------------------------------
     4.33681e-18-4.33681e-19*ii, 7.80626e-18+1.38778e-17*ii,
     ------------------------------------------------------------------------
     -4.33681e-18+4.33681e-19*ii, 4.33681e-18+6.07153e-18*ii), (1, -1, -1, 1,
     ------------------------------------------------------------------------
     -6.92016e-14-4.44931e-14*ii, 6.92051e-14+4.44937e-14*ii,
     ------------------------------------------------------------------------
     6.92016e-14+4.44931e-14*ii, -6.92051e-14-4.44937e-14*ii), (-1, -1, 1, 1,
     ------------------------------------------------------------------------
     6.92042e-14+4.44905e-14*ii, 6.92051e-14+4.44935e-14*ii,
     ------------------------------------------------------------------------
     -6.92042e-14-4.44905e-14*ii, -6.92051e-14-4.44935e-14*ii), (-1, 1, 1,
     ------------------------------------------------------------------------
     -1, 6.1271e-13+2.15095e-13*ii, -6.12729e-13-2.15085e-13*ii,
     ------------------------------------------------------------------------
     -6.1271e-13-2.15095e-13*ii, 6.12729e-13+2.15085e-13*ii), (1, -1, -1, 1,
     ------------------------------------------------------------------------
     -4.33681e-18+7.15573e-18*ii, 4.33681e-18-6.50521e-19*ii,
     ------------------------------------------------------------------------
     4.33681e-18-7.15573e-18*ii, -4.33681e-18+6.50521e-19*ii), (1, 1, -1, -1,
     ------------------------------------------------------------------------
     4.33681e-18-7.15573e-18*ii, 4.33681e-18-6.50521e-19*ii,
     ------------------------------------------------------------------------
     -4.33681e-18+7.15573e-18*ii, -4.33681e-18+6.50521e-19*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     -4.33681e-18+4.33681e-19*ii, -7.80626e-18-1.38778e-17*ii,
     ------------------------------------------------------------------------
     4.33681e-18-4.33681e-19*ii, -4.33681e-18-6.07153e-18*ii),
     ------------------------------------------------------------------------
     {6.11583e-12+5.40979e-12*ii, 1.41421, -6.11584e-12-5.4098e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, 1, ii, -1, ii}, {1.41421, 6.41711e-13-4.54503e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, -6.4172e-13+4.54504e-12*ii, ii, -1, ii, 1}, {1.41421,
     ------------------------------------------------------------------------
     2.47435e-12-2.19594e-12*ii, -1.41421, -2.47434e-12+2.19592e-12*ii, -ii,
     ------------------------------------------------------------------------
     1, -ii, -1}, {3.32395e-12+5.99901e-12*ii, -1.41421,
     ------------------------------------------------------------------------
     -3.32394e-12-5.99901e-12*ii, 1.41421, 1, -ii, -1, -ii},
     ------------------------------------------------------------------------
     {3.32395e-12+5.99901e-12*ii, 1.41421, -3.32394e-12-5.99901e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, -1, -ii, 1, -ii}, {-1.41421, 6.41711e-13-4.54503e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -6.4172e-13+4.54504e-12*ii, ii, 1, ii, -1}, (-1, 1, -1, -1,
     ------------------------------------------------------------------------
     -2.34188e-17-5.48606e-17*ii, 2.34188e-17+5.39933e-17*ii,
     ------------------------------------------------------------------------
     -3.03577e-17-5.22585e-17*ii, -3.03577e-17-5.39933e-17*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     2.18756e-13+6.35932e-14*ii, -1.21361e-12-4.65968e-13*ii,
     ------------------------------------------------------------------------
     -2.18756e-13-6.35932e-14*ii, 7.76096e-13+3.38778e-13*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     -1.29224e-14+2.4584e-16*ii, -7.8835e-16-4.36749e-15*ii,
     ------------------------------------------------------------------------
     1.44993e-14+8.49256e-15*ii, 7.88288e-16+4.36781e-15*ii), (-1, 1, 1, -1,
     ------------------------------------------------------------------------
     6.92788e-14+4.46015e-14*ii, 6.94106e-14+4.45256e-14*ii,
     ------------------------------------------------------------------------
     -6.92788e-14-4.46047e-14*ii, -6.94141e-14-4.45288e-14*ii), (1, 1, -1,
     ------------------------------------------------------------------------
     -1, -6.94991e-14-4.4327e-14*ii, 6.94366e-14+4.43328e-14*ii,
     ------------------------------------------------------------------------
     6.95078e-14+4.43272e-14*ii, -6.94366e-14-4.43345e-14*ii), (1, 1, -1, -1,
     ------------------------------------------------------------------------
     6.96995e-14+4.44902e-14*ii, -6.97723e-14-4.44564e-14*ii,
     ------------------------------------------------------------------------
     -6.97038e-14-4.44907e-14*ii, 6.97775e-14+4.44534e-14*ii), (-1, -1, 1,
     ------------------------------------------------------------------------
     -1, -2.18756e-13-6.35932e-14*ii, -1.21361e-12-4.65968e-13*ii,
     ------------------------------------------------------------------------
     2.18756e-13+6.35932e-14*ii, 7.76096e-13+3.38778e-13*ii), (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     -1.29224e-14+2.4584e-16*ii, 7.8835e-16+4.36749e-15*ii,
     ------------------------------------------------------------------------
     1.44993e-14+8.49256e-15*ii, -7.88288e-16-4.36781e-15*ii), {1.41421,
     ------------------------------------------------------------------------
     -2.47435e-12+2.19594e-12*ii, -1.41421, 2.47434e-12-2.19592e-12*ii, ii,
     ------------------------------------------------------------------------
     1, ii, -1}, {5.7252e-12+8.75496e-12*ii, 1.41421,
     ------------------------------------------------------------------------
     -5.72519e-12-8.75497e-12*ii, -1.41421, 1, -ii, -1, -ii}}

o6 : List
i7 : #oo

o7 = 65

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

i8 : findRealSolutions I
warning: some solutions are not regular: {12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 35, 36, 37, 44, 45, 52, 53, 54, 55, 56, 57, 58}

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}}

o8 : List
i9 : #oo

o9 = 32

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, 22, 23, 24, 25, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 53, 54, 55, 58, 59, 60}

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

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/macaulay2-1.25.06+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:263:0.