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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 = {{1.03179-.049346*ii, 1.03179-.049346*ii, 1.03179-.049346*ii,
     ------------------------------------------------------------------------
     1.03179-.049346*ii, .169028+.301221*ii, .169028+.301221*ii,
     ------------------------------------------------------------------------
     .169028+.301221*ii, .169028+.301221*ii}, {-1.03179+.049346*ii,
     ------------------------------------------------------------------------
     1.03179-.049346*ii, -1.03179+.049346*ii, 1.03179-.049346*ii,
     ------------------------------------------------------------------------
     .169028+.301221*ii, -.169028-.301221*ii, .169028+.301221*ii,
     ------------------------------------------------------------------------
     -.169028-.301221*ii}, {1.03179-.049346*ii, -1.03179+.049346*ii,
     ------------------------------------------------------------------------
     1.03179-.049346*ii, -1.03179+.049346*ii, -.169028-.301221*ii,
     ------------------------------------------------------------------------
     .169028+.301221*ii, -.169028-.301221*ii, .169028+.301221*ii},
     ------------------------------------------------------------------------
     {-1.03179+.049346*ii, -1.03179+.049346*ii, -1.03179+.049346*ii,
     ------------------------------------------------------------------------
     -1.03179+.049346*ii, -.169028-.301221*ii, -.169028-.301221*ii,
     ------------------------------------------------------------------------
     -.169028-.301221*ii, -.169028-.301221*ii}, (1, -1, -1, -1,
     ------------------------------------------------------------------------
     -2.31741e-12+9.96347e-13*ii, 2.3174e-12-9.96309e-13*ii,
     ------------------------------------------------------------------------
     2.31741e-12-9.96347e-13*ii, 2.31741e-12-9.96384e-13*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     9.21774e-13-1.59955e-13*ii, 2.4216e-13+6.77759e-13*ii,
     ------------------------------------------------------------------------
     -9.21774e-13+1.59955e-13*ii, -2.08571e-12-3.57851e-13*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     9.21774e-13-1.59955e-13*ii, -2.4216e-13-6.77759e-13*ii,
     ------------------------------------------------------------------------
     -9.21774e-13+1.59955e-13*ii, 2.08571e-12+3.57851e-13*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     -2.61409e-12+1.18427e-12*ii, 1.48385e-12-5.30025e-13*ii,
     ------------------------------------------------------------------------
     -3.53616e-13-1.24224e-13*ii, -1.48385e-12+5.30025e-13*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     8.86774e-13+1.42355e-12*ii, 6.2027e-13+7.7389e-13*ii,
     ------------------------------------------------------------------------
     3.53768e-13+1.24226e-13*ii, -6.2027e-13-7.7389e-13*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     1.88021e-12+8.42497e-13*ii, 1.88012e-12+8.42607e-13*ii,
     ------------------------------------------------------------------------
     -1.88021e-12-8.42497e-13*ii, 1.8803e-12+8.42386e-13*ii), (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     -2.81449e-13+1.49875e-13*ii, -2.81376e-13+1.49819e-13*ii,
     ------------------------------------------------------------------------
     -2.81303e-13+1.49763e-13*ii, 2.81376e-13-1.49819e-13*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     1.08097e-12+4.29372e-13*ii, -1.08091e-12-4.2934e-13*ii,
     ------------------------------------------------------------------------
     -1.08097e-12-4.29372e-13*ii, -1.08102e-12-4.29405e-13*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     -9.64807e-13+4.90995e-13*ii, 9.64883e-13-4.91099e-13*ii,
     ------------------------------------------------------------------------
     -9.64959e-13+4.91204e-13*ii, -9.64883e-13+4.91099e-13*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     -4.21505e-13+5.89521e-13*ii, -4.21491e-13+5.89518e-13*ii,
     ------------------------------------------------------------------------
     -4.21476e-13+5.89515e-13*ii, 4.2149e-13-5.89518e-13*ii), (-1, -1, 1, -1,
     ------------------------------------------------------------------------
     1.18969e-12-4.28364e-13*ii, 1.18976e-12-4.28365e-13*ii,
     ------------------------------------------------------------------------
     -1.18969e-12+4.28364e-13*ii, 1.18962e-12-4.28362e-13*ii), (-1, 1, -1,
     ------------------------------------------------------------------------
     -1, -1.11842e-12-5.84427e-13*ii, 1.1186e-12+5.84344e-13*ii,
     ------------------------------------------------------------------------
     -1.11879e-12-5.84261e-13*ii, -1.1186e-12-5.84344e-13*ii), (1, -1, -1,
     ------------------------------------------------------------------------
     -1, -1.15563e-12-5.42919e-13*ii, 1.15563e-12+5.42922e-13*ii,
     ------------------------------------------------------------------------
     1.15563e-12+5.42919e-13*ii, 1.15564e-12+5.42917e-13*ii), (1, -1, -1, -1,
     ------------------------------------------------------------------------
     -9.21774e-13+1.59955e-13*ii, -2.4216e-13-6.77759e-13*ii,
     ------------------------------------------------------------------------
     9.21774e-13-1.59955e-13*ii, 2.08571e-12+3.57851e-13*ii), (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     -8.86774e-13-1.42355e-12*ii, -6.2027e-13-7.7389e-13*ii,
     ------------------------------------------------------------------------
     -3.53768e-13-1.24226e-13*ii, 6.2027e-13+7.7389e-13*ii), (-1, -1, 1, -1,
     ------------------------------------------------------------------------
     -9.21774e-13+1.59955e-13*ii, 2.4216e-13+6.77759e-13*ii,
     ------------------------------------------------------------------------
     9.21774e-13-1.59955e-13*ii, -2.08571e-12-3.57851e-13*ii), (-1, 1, -1,
     ------------------------------------------------------------------------
     -1, 8.86774e-13+1.42355e-12*ii, -6.2027e-13-7.7389e-13*ii,
     ------------------------------------------------------------------------
     3.53768e-13+1.24226e-13*ii, 6.2027e-13+7.7389e-13*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     -4.73249e-13-7.45879e-13*ii, 4.7325e-13+7.45843e-13*ii,
     ------------------------------------------------------------------------
     4.73249e-13+7.45879e-13*ii, 4.73248e-13+7.45915e-13*ii),
     ------------------------------------------------------------------------
     {1.65543e-12-1.57518e-12*ii, -1.41421, -1.65544e-12+1.57518e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -1, ii, 1, ii}, {-1.41421, 2.6882e-12+3.26404e-12*ii, 1.41421,
     ------------------------------------------------------------------------
     -2.68819e-12-3.26404e-12*ii, -ii, -1, -ii, 1}, (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     -3.48596e-13-1.51486e-13*ii, -3.48603e-13-1.51461e-13*ii,
     ------------------------------------------------------------------------
     -3.48609e-13-1.51438e-13*ii, 3.48603e-13+1.51461e-13*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     -2.36558e-12+6.25094e-13*ii, -5.91301e-12-3.93476e-13*ii,
     ------------------------------------------------------------------------
     2.36558e-12-6.25094e-13*ii, 1.18184e-12+1.64367e-12*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     4.89981e-13-1.22448e-12*ii, 6.81502e-14-6.74303e-13*ii,
     ------------------------------------------------------------------------
     -3.53688e-13-1.24134e-13*ii, -6.81502e-14+6.74303e-13*ii), (-1, 1, -1,
     ------------------------------------------------------------------------
     1, -3.46945e-18+6.28837e-18*ii, 3.46945e-18-6.28837e-18*ii,
     ------------------------------------------------------------------------
     -3.46945e-18+6.28837e-18*ii, 3.46945e-18-6.28837e-18*ii), [RF], (-1, 1,
     ------------------------------------------------------------------------
     1, -1, -1.2697e-12+1.18394e-13*ii, -8.92033e-13+8.57351e-14*ii,
     ------------------------------------------------------------------------
     1.2697e-12-1.18394e-13*ii, 8.92033e-13-8.57351e-14*ii), (1, -1, -1, 1,
     ------------------------------------------------------------------------
     -5.2042e-18+5.72459e-17*ii, 1.73472e-18+6.93889e-18*ii,
     ------------------------------------------------------------------------
     -1.7347e-18-5.81132e-17*ii, 1.21431e-17-6.93889e-18*ii), (-1, 1, -1, -1,
     ------------------------------------------------------------------------
     4.89981e-13-1.22448e-12*ii, -6.81502e-14+6.74303e-13*ii,
     ------------------------------------------------------------------------
     -3.53688e-13-1.24134e-13*ii, 6.81502e-14-6.74303e-13*ii), (1, -1, -1,
     ------------------------------------------------------------------------
     -1, 2.36558e-12-6.25094e-13*ii, -5.91301e-12-3.93476e-13*ii,
     ------------------------------------------------------------------------
     -2.36558e-12+6.25094e-13*ii, 1.18184e-12+1.64367e-12*ii),
     ------------------------------------------------------------------------
     {4.9295e-13+2.86246e-12*ii, -1.41421, -4.92959e-13-2.86244e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 1, ii, -1, ii}, {1.41421, -5.30848e-12+3.76398e-13*ii,
     ------------------------------------------------------------------------
     -1.41421, 5.30849e-12-3.76398e-13*ii, -ii, -1, -ii, 1},
     ------------------------------------------------------------------------
     {-1.65543e-12+1.57518e-12*ii, -1.41421, 1.65544e-12-1.57518e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -1, -ii, 1, -ii}, {-1.41421, -1.59458e-12+5.92897e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 1.59459e-12-5.92899e-12*ii, -ii, 1, -ii, -1}, {-1.41421,
     ------------------------------------------------------------------------
     -2.6882e-12-3.26404e-12*ii, 1.41421, 2.68819e-12+3.26404e-12*ii, ii, -1,
     ------------------------------------------------------------------------
     ii, 1}, {4.9295e-13+2.86246e-12*ii, 1.41421,
     ------------------------------------------------------------------------
     -4.92959e-13-2.86244e-12*ii, -1.41421, -1, ii, 1, ii}, (-1, -1, 1, -1,
     ------------------------------------------------------------------------
     -1.18196e-12-4.23438e-13*ii, -1.18196e-12-4.23441e-13*ii,
     ------------------------------------------------------------------------
     1.18196e-12+4.23438e-13*ii, -1.18196e-12-4.23435e-13*ii), (-1, 1, 1, -1,
     ------------------------------------------------------------------------
     -2.50589e-14-1.48453e-14*ii, 2.50592e-14+1.48432e-14*ii,
     ------------------------------------------------------------------------
     2.50589e-14+1.48453e-14*ii, -2.50592e-14-1.48432e-14*ii), (1, 1, 1, 1,
     ------------------------------------------------------------------------
     -3.46945e-18+6.28837e-18*ii, -3.46945e-18+6.28837e-18*ii,
     ------------------------------------------------------------------------
     -3.46945e-18+6.28837e-18*ii, -3.46945e-18+6.28837e-18*ii), (-1, -1, 1,
     ------------------------------------------------------------------------
     1, 6.28642e-15+3.68312e-15*ii, 6.28642e-15+3.68241e-15*ii,
     ------------------------------------------------------------------------
     -6.28642e-15-3.68312e-15*ii, -6.28642e-15-3.68241e-15*ii), (1, -1, -1,
     ------------------------------------------------------------------------
     1, 2.16609e-13+7.60264e-14*ii, -2.16646e-13-7.60438e-14*ii,
     ------------------------------------------------------------------------
     -2.16609e-13-7.60264e-14*ii, 2.16646e-13+7.60438e-14*ii), (-1, -1, -1,
     ------------------------------------------------------------------------
     -1, 3.46945e-18-6.28837e-18*ii, 3.46945e-18-6.28837e-18*ii,
     ------------------------------------------------------------------------
     3.46945e-18-6.28837e-18*ii, 3.46945e-18-6.28837e-18*ii), (1, 1, -1, -1,
     ------------------------------------------------------------------------
     3.46945e-18+6.72205e-18*ii, 3.46945e-18+1.30104e-17*ii,
     ------------------------------------------------------------------------
     -3.46945e-18-6.72205e-18*ii, -3.46945e-18-1.30104e-17*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     -1.49538e-12+2.09281e-12*ii, -1.49545e-12+2.09285e-12*ii,
     ------------------------------------------------------------------------
     1.49538e-12-2.09281e-12*ii, -1.49532e-12+2.09276e-12*ii),
     ------------------------------------------------------------------------
     {1.16375e-12+7.44722e-12*ii, 1.41421, -1.16376e-12-7.4472e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, 1, ii, -1, ii}, {1.41421, -3.7873e-13-1.13301e-11*ii,
     ------------------------------------------------------------------------
     -1.41421, 3.7872e-13+1.13302e-11*ii, ii, -1, ii, 1}, {1.41421,
     ------------------------------------------------------------------------
     5.25106e-12-2.75318e-12*ii, -1.41421, -5.25105e-12+2.75316e-12*ii, -ii,
     ------------------------------------------------------------------------
     1, -ii, -1}, {-4.9295e-13-2.86246e-12*ii, -1.41421,
     ------------------------------------------------------------------------
     4.92959e-13+2.86244e-12*ii, 1.41421, 1, -ii, -1, -ii},
     ------------------------------------------------------------------------
     {-4.9295e-13-2.86246e-12*ii, 1.41421, 4.92959e-13+2.86244e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, -1, -ii, 1, -ii}, {-1.41421, 1.59458e-12-5.92897e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -1.59459e-12+5.92899e-12*ii, ii, 1, ii, -1}, (-1, 1, -1, -1,
     ------------------------------------------------------------------------
     2.60858e-12+3.16334e-13*ii, -2.60858e-12-3.16195e-13*ii,
     ------------------------------------------------------------------------
     2.60857e-12+3.16056e-13*ii, 2.60858e-12+3.16195e-13*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     2.88081e-14-8.58896e-13*ii, -8.44207e-13-8.42506e-13*ii,
     ------------------------------------------------------------------------
     -2.88081e-14+8.58896e-13*ii, 7.8659e-13+2.5603e-12*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     -4.89981e-13+1.22448e-12*ii, 6.81502e-14-6.74303e-13*ii,
     ------------------------------------------------------------------------
     3.53688e-13+1.24134e-13*ii, -6.81502e-14+6.74303e-13*ii), (1, 1, -1, -1,
     ------------------------------------------------------------------------
     -1.62725e-14-3.8168e-14*ii, -4.98391e-15+2.55688e-14*ii,
     ------------------------------------------------------------------------
     1.62725e-14+3.81689e-14*ii, 4.98386e-15-2.55688e-14*ii), (1, -1, 1, -1,
     ------------------------------------------------------------------------
     3.46945e-18-6.28837e-18*ii, -3.46945e-18+6.28837e-18*ii,
     ------------------------------------------------------------------------
     3.46945e-18-6.28837e-18*ii, -3.46945e-18+6.28837e-18*ii), (-1, -1, 1,
     ------------------------------------------------------------------------
     -1, 2.36558e-12-6.25094e-13*ii, 5.91301e-12+3.93476e-13*ii,
     ------------------------------------------------------------------------
     -2.36558e-12+6.25094e-13*ii, -1.18184e-12-1.64367e-12*ii), (-1, -1, -1,
     ------------------------------------------------------------------------
     1, -4.89981e-13+1.22448e-12*ii, -6.81502e-14+6.74303e-13*ii,
     ------------------------------------------------------------------------
     3.53688e-13+1.24134e-13*ii, 6.81502e-14-6.74303e-13*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     2.43728e-12+1.5587e-13*ii, 2.43742e-12+1.55775e-13*ii,
     ------------------------------------------------------------------------
     2.43757e-12+1.55679e-13*ii, -2.43742e-12-1.55775e-13*ii), {1.41421,
     ------------------------------------------------------------------------
     -2.6882e-12-3.26404e-12*ii, -1.41421, 2.68819e-12+3.26404e-12*ii, ii, 1,
     ------------------------------------------------------------------------
     ii, -1}, {-1.16375e-12-7.44722e-12*ii, 1.41421,
     ------------------------------------------------------------------------
     1.16376e-12+7.4472e-12*ii, -1.41421, 1, -ii, -1, -ii}}

o6 : List
i7 : #oo

o7 = 63

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

i8 : findRealSolutions I
warning: some solutions are not regular: {8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 43, 44, 45, 46, 47, 48, 49, 56, 57, 58, 59, 60, 61, 62, 63, 64}

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: {12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, 45, 46, 47, 48, 49, 50, 51, 52, 59, 60, 61, 62, 63, 64, 65, 66, 67}

      +---+---+---+
o10 = |135|0  |0  |
      +---+---+---+
      |45 |0  |180|
      +---+---+---+
      |45 |180|0  |
      +---+---+---+
      |315|0  |0  |
      +---+---+---+
      |135|180|180|
      +---+---+---+
      |135|0  |0  |
      +---+---+---+
      |45 |180|0  |
      +---+---+---+
      |315|0  |180|
      +---+---+---+
      |315|0  |0  |
      +---+---+---+
      |225|0  |180|
      +---+---+---+
      |315|180|180|
      +---+---+---+
      |135|180|0  |
      +---+---+---+
      |315|180|180|
      +---+---+---+
      |135|180|180|
      +---+---+---+
      |225|180|0  |
      +---+---+---+
      |225|0  |180|
      +---+---+---+
      |135|180|180|
      +---+---+---+
      |45 |0  |180|
      +---+---+---+
      |315|0  |0  |
      +---+---+---+
      |135|180|0  |
      +---+---+---+
      |225|0  |0  |
      +---+---+---+
      |45 |180|180|
      +---+---+---+
      |225|0  |180|
      +---+---+---+
      |315|180|180|
      +---+---+---+
      |45 |180|0  |
      +---+---+---+
      |225|180|0  |
      +---+---+---+
      |45 |180|180|
      +---+---+---+
      |225|0  |0  |
      +---+---+---+
      |135|180|0  |
      +---+---+---+
      |225|0  |0  |
      +---+---+---+
      |45 |180|180|
      +---+---+---+
      |315|0  |180|
      +---+---+---+
      |45 |0  |180|
      +---+---+---+
      |225|0  |180|
      +---+---+---+
      |135|0  |0  |
      +---+---+---+
      |45 |180|0  |
      +---+---+---+
      |45 |180|180|
      +---+---+---+
      |315|0  |180|
      +---+---+---+
      |315|0  |180|
      +---+---+---+
      |225|180|0  |
      +---+---+---+
      |135|180|180|
      +---+---+---+
      |315|0  |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 Oscillators/Documentation.m2:263:0.