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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 ..x , y ..y ]
       53  0   3   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, -1, 1, -1, -1.41095e-16-2.09622e-16*ii,
     ------------------------------------------------------------------------
     7.76722e-14+4.62065e-14*ii, 1.38753e-16+2.10576e-16*ii,
     ------------------------------------------------------------------------
     -7.79446e-14-4.66259e-14*ii), (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     -6.07153e-17-3.98986e-17*ii, -1.1831e-12+1.29022e-12*ii,
     ------------------------------------------------------------------------
     -3.1225e-17-7.80626e-18*ii, -1.18303e-12+1.29026e-12*ii), (-1, 1, -1,
     ------------------------------------------------------------------------
     -1, 6.07153e-17+3.98986e-17*ii, -1.1831e-12+1.29022e-12*ii,
     ------------------------------------------------------------------------
     3.1225e-17+7.80626e-18*ii, -1.18303e-12+1.29026e-12*ii), (1, -1, -1, -1,
     ------------------------------------------------------------------------
     -1.41095e-16-2.09622e-16*ii, -7.76722e-14-4.62065e-14*ii,
     ------------------------------------------------------------------------
     1.38753e-16+2.10576e-16*ii, 7.79446e-14+4.66259e-14*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     1.41095e-16+2.09622e-16*ii, 7.76722e-14+4.62065e-14*ii,
     ------------------------------------------------------------------------
     -1.38753e-16-2.10576e-16*ii, -7.79446e-14-4.66259e-14*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     6.07153e-17+3.98986e-17*ii, 1.1831e-12-1.29022e-12*ii,
     ------------------------------------------------------------------------
     3.1225e-17+7.80626e-18*ii, 1.18303e-12-1.29026e-12*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     1.41095e-16+2.09622e-16*ii, -7.76722e-14-4.62065e-14*ii,
     ------------------------------------------------------------------------
     -1.38753e-16-2.10576e-16*ii, 7.79446e-14+4.66259e-14*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     -6.07153e-17-3.98986e-17*ii, 1.1831e-12-1.29022e-12*ii,
     ------------------------------------------------------------------------
     -3.1225e-17-7.80626e-18*ii, 1.18303e-12-1.29026e-12*ii), (-1, -1, 1, -1,
     ------------------------------------------------------------------------
     1.21431e-17+4.3368e-19*ii, 1.56125e-17+8.6736e-19*ii,
     ------------------------------------------------------------------------
     -1.21431e-17-4.3368e-19*ii, 6.93889e-18+4.3368e-19*ii), (1, -1, -1, -1,
     ------------------------------------------------------------------------
     -1.21431e-17-1.30104e-18*ii, 1.21431e-17+8.6736e-19*ii,
     ------------------------------------------------------------------------
     1.21431e-17+1.30104e-18*ii, 1.38778e-17+8.6736e-19*ii),
     ------------------------------------------------------------------------
     {3.91751e-12+6.37933e-12*ii, -1.41421, -3.91752e-12-6.37931e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -1, ii, 1, ii}, {1.41421, -1.9512e-15-1.26753e-15*ii, -1.41421,
     ------------------------------------------------------------------------
     1.95136e-15+1.24277e-15*ii, ii, -1, ii, 1}, {3.91751e-12+6.37933e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -3.91752e-12-6.37931e-12*ii, -1.41421, 1, ii, -1, ii},
     ------------------------------------------------------------------------
     {1.41421, 3.15291e-12+4.15148e-12*ii, -1.41421,
     ------------------------------------------------------------------------
     -3.15291e-12-4.1515e-12*ii, -ii, 1, -ii, -1},
     ------------------------------------------------------------------------
     {2.55658e-13+3.94843e-12*ii, -1.41421, -2.55649e-13-3.94845e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 1, -ii, -1, -ii}, {-1.41421, -2.55854e-12-1.0541e-11*ii,
     ------------------------------------------------------------------------
     1.41421, 2.55854e-12+1.0541e-11*ii, ii, 1, ii, -1},
     ------------------------------------------------------------------------
     {2.55658e-13+3.94843e-12*ii, 1.41421, -2.55649e-13-3.94845e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, -1, -ii, 1, -ii}, {-1.41421, 3.15291e-12+4.15148e-12*ii,
     ------------------------------------------------------------------------
     1.41421, -3.15291e-12-4.1515e-12*ii, -ii, -1, -ii, 1}, (1, -1, 1, 1,
     ------------------------------------------------------------------------
     3.46945e-18-3.59955e-17*ii, 3.46945e-18+2.25514e-17*ii,
     ------------------------------------------------------------------------
     -5.20417e-18-3.64292e-17*ii, -3.46945e-18-2.29851e-17*ii), [RF], (1, -1,
     ------------------------------------------------------------------------
     -1, -1, 1.03285e-13-1.30887e-14*ii, 1.84867e-12+8.05562e-13*ii,
     ------------------------------------------------------------------------
     -1.03285e-13+1.30885e-14*ii, -2.05523e-12-7.79382e-13*ii), (-1, 1, -1,
     ------------------------------------------------------------------------
     -1, 3.99813e-14+3.33797e-14*ii, -1.6361e-14-1.45656e-14*ii,
     ------------------------------------------------------------------------
     -7.26074e-15-4.24828e-15*ii, 1.63611e-14+1.45657e-14*ii), (-1, -1, -1,
     ------------------------------------------------------------------------
     1, -3.99813e-14-3.33797e-14*ii, -1.6361e-14-1.45656e-14*ii,
     ------------------------------------------------------------------------
     7.26074e-15+4.24828e-15*ii, 1.63611e-14+1.45657e-14*ii), (-1, -1, 1, -1,
     ------------------------------------------------------------------------
     1.03285e-13-1.30887e-14*ii, -1.84867e-12-8.05562e-13*ii,
     ------------------------------------------------------------------------
     -1.03285e-13+1.30885e-14*ii, 2.05523e-12+7.79382e-13*ii), (1, -1, -1,
     ------------------------------------------------------------------------
     -1, -6.92857e-14-4.4461e-14*ii, 6.92857e-14+4.4461e-14*ii,
     ------------------------------------------------------------------------
     6.92857e-14+4.4461e-14*ii, 6.92875e-14+4.44605e-14*ii), (-1, -1, 1, -1,
     ------------------------------------------------------------------------
     -3.46945e-18-1.36609e-17*ii, -3.46945e-18+2.1684e-19*ii,
     ------------------------------------------------------------------------
     3.46945e-18+1.36609e-17*ii, -3.46945e-18-1.32273e-17*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     3.46945e-18+1.36609e-17*ii, 3.46945e-18-2.1684e-19*ii,
     ------------------------------------------------------------------------
     -3.46945e-18-1.36609e-17*ii, 3.46945e-18+1.32273e-17*ii), [RF], (-1, 1,
     ------------------------------------------------------------------------
     1, 1, 6.91504e-14+4.44946e-14*ii, -6.91504e-14-4.44941e-14*ii,
     ------------------------------------------------------------------------
     -6.91504e-14-4.44948e-14*ii, -6.91504e-14-4.44948e-14*ii), (-1, -1, 1,
     ------------------------------------------------------------------------
     1, -5.20417e-18-4.33681e-19*ii, -5.20417e-18-1.30104e-18*ii,
     ------------------------------------------------------------------------
     5.20417e-18+4.33681e-19*ii, 5.20417e-18+1.30104e-18*ii), (-1, -1, -1, 1,
     ------------------------------------------------------------------------
     -1.21431e-17-1.34441e-17*ii, -1.9082e-17+1.30104e-18*ii,
     ------------------------------------------------------------------------
     -1.9082e-17-1.30104e-17*ii, 1.9082e-17-1.30104e-18*ii), (1, 1, -1, 1,
     ------------------------------------------------------------------------
     -1.03285e-13+1.30887e-14*ii, 1.84867e-12+8.05562e-13*ii,
     ------------------------------------------------------------------------
     1.03285e-13-1.30885e-14*ii, -2.05523e-12-7.79382e-13*ii), (1, -1, 1, 1,
     ------------------------------------------------------------------------
     -3.99813e-14-3.33797e-14*ii, 1.6361e-14+1.45656e-14*ii,
     ------------------------------------------------------------------------
     7.26074e-15+4.24828e-15*ii, -1.63611e-14-1.45657e-14*ii), (-1, 1, 1, 1,
     ------------------------------------------------------------------------
     -1.03285e-13+1.30887e-14*ii, -1.84867e-12-8.05562e-13*ii,
     ------------------------------------------------------------------------
     1.03285e-13-1.30885e-14*ii, 2.05523e-12+7.79382e-13*ii), (1, 1, 1, -1,
     ------------------------------------------------------------------------
     3.99813e-14+3.33797e-14*ii, 1.6361e-14+1.45656e-14*ii,
     ------------------------------------------------------------------------
     -7.26074e-15-4.24828e-15*ii, -1.63611e-14-1.45657e-14*ii), [RF], (-1, 1,
     ------------------------------------------------------------------------
     -1, -1, 1.21431e-17+1.34441e-17*ii, -1.9082e-17+1.30104e-18*ii,
     ------------------------------------------------------------------------
     1.9082e-17+1.30104e-17*ii, 1.9082e-17-1.30104e-18*ii),
     ------------------------------------------------------------------------
     {-2.55658e-13-3.94843e-12*ii, 1.41421, 2.55649e-13+3.94845e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, -1, ii, 1, ii}, {-1.41421, -3.15291e-12-4.15148e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 3.15291e-12+4.1515e-12*ii, ii, -1, ii, 1},
     ------------------------------------------------------------------------
     {-2.55658e-13-3.94843e-12*ii, -1.41421, 2.55649e-13+3.94845e-12*ii,
     ------------------------------------------------------------------------
     1.41421, 1, ii, -1, ii}, {-1.41421, 2.55854e-12+1.0541e-11*ii, 1.41421,
     ------------------------------------------------------------------------
     -2.55854e-12-1.0541e-11*ii, -ii, 1, -ii, -1},
     ------------------------------------------------------------------------
     {-3.91751e-12-6.37933e-12*ii, 1.41421, 3.91752e-12+6.37931e-12*ii,
     ------------------------------------------------------------------------
     -1.41421, 1, -ii, -1, -ii}, {-3.91751e-12-6.37933e-12*ii, -1.41421,
     ------------------------------------------------------------------------
     3.91752e-12+6.37931e-12*ii, 1.41421, -1, -ii, 1, -ii}, {1.41421,
     ------------------------------------------------------------------------
     -3.15291e-12-4.15148e-12*ii, -1.41421, 3.15291e-12+4.1515e-12*ii, ii, 1,
     ------------------------------------------------------------------------
     ii, -1}, {1.41421, 1.9512e-15+1.26753e-15*ii, -1.41421,
     ------------------------------------------------------------------------
     -1.95136e-15-1.24277e-15*ii, -ii, -1, -ii, 1}, (1, 1, -1, 1,
     ------------------------------------------------------------------------
     6.91409e-14+4.44822e-14*ii, 6.91417e-14+4.44889e-14*ii,
     ------------------------------------------------------------------------
     -6.91409e-14-4.44822e-14*ii, 6.91409e-14+4.44835e-14*ii)}

o6 : List
 
i7 : #oo

o7 = 46

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

i8 : findRealSolutions I
warning: some solutions are not regular: {4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 44}

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}}

o8 : List
 
i9 : #oo

o9 = 27

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

i10 : netList getAngles(3, findRealSolutions I, Radians=>false)
warning: some solutions are not regular: {8, 9, 10, 11, 12, 13, 14, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 57, 58}

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