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Example 4.4: The square within a square -- example 4.4 in arXiv 2312.16069

The following is the one graph on 8 vertices with an exotic solution where the Jacobian has some off-diagonal extries negative.

i1 : needsPackage "Oscillators";
i2 : printingPrecision = 3;
i3 : G = graph {{0,1},{1,2},{2,3},{3,4},{4,5},{5,6},{6,0},{0,5},{0,2},{5,7},{2,7}}

o3 = Graph{0 => {1, 2, 5, 6}}
           1 => {0, 2}
           2 => {0, 1, 3, 7}
           3 => {2, 4}
           4 => {3, 5}
           5 => {0, 4, 6, 7}
           6 => {0, 5}
           7 => {2, 5}

o3 : Graph
i4 : stablesols = {{1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0},
         {.623, -.223, -.901, -.901, -.223, .623, -1, -.782, -.975, -.434, .434, .975, .782, 0},
         {.623, -.223, -.901, -.901, -.223, .623, -1, .782, .975, .434, -.434, -.975, -.782, 0}}

o4 = {{1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0}, {.623, -.223, -.901, -.901,
     ------------------------------------------------------------------------
     -.223, .623, -1, -.782, -.975, -.434, .434, .975, .782, 0}, {.623,
     ------------------------------------------------------------------------
     -.223, -.901, -.901, -.223, .623, -1, .782, .975, .434, -.434, -.975,
     ------------------------------------------------------------------------
     -.782, 0}}

o4 : List
elapsedTime stablesols = showExoticSolutions G

The Jacobian at each exotic solution has negative off diagonl entries.

i5 : Jac = oscJacobian(G, Reduced => true)

o5 = | -x_1-x_2-x_5-x_6 x_1               
     | x_1              -x_1x_2-y_1y_2-x_1
     | x_2              x_1x_2+y_1y_2     
     | 0                0                 
     | 0                0                 
     | x_5              0                 
     | x_6              0                 
     | 0                0                 
     ------------------------------------------------------------------------
     x_2                                           
     x_1x_2+y_1y_2                                 
     -x_1x_2-x_2x_3-x_2x_7-y_1y_2-y_2y_3-y_2y_7-x_2
     x_2x_3+y_2y_3                                 
     0                                             
     0                                             
     0                                             
     x_2x_7+y_2y_7                                 
     ------------------------------------------------------------------------
     0                            0                           
     0                            0                           
     x_2x_3+y_2y_3                0                           
     -x_2x_3-x_3x_4-y_2y_3-y_3y_4 x_3x_4+y_3y_4               
     x_3x_4+y_3y_4                -x_3x_4-x_4x_5-y_3y_4-y_4y_5
     0                            x_4x_5+y_4y_5               
     0                            0                           
     0                            0                           
     ------------------------------------------------------------------------
     x_5                                            x_6               
     0                                              0                 
     0                                              0                 
     0                                              0                 
     x_4x_5+y_4y_5                                  0                 
     -x_4x_5-x_5x_6-x_5x_7-y_4y_5-y_5y_6-y_5y_7-x_5 x_5x_6+y_5y_6     
     x_5x_6+y_5y_6                                  -x_5x_6-y_5y_6-x_6
     x_5x_7+y_5y_7                                  0                 
     ------------------------------------------------------------------------
     0                            |
     0                            |
     x_2x_7+y_2y_7                |
     0                            |
     0                            |
     x_5x_7+y_5y_7                |
     0                            |
     -x_2x_7-x_5x_7-y_2y_7-y_5y_7 |

                        8                 8
o5 : Matrix (QQ[x ..y ])  <-- (QQ[x ..y ])
                 1   7             1   7
i6 : sub(Jac, matrix{stablesols#1})

o6 = | -.8   .623  -.223 0     0     -.223 .623  0     |
     | .623  -1.25 .624  0     0     0     0     0     |
     | -.223 .624  -1.25 .624  0     0     0     .223  |
     | 0     0     .624  -1.25 .623  0     0     0     |
     | 0     0     0     .623  -1.25 .624  0     0     |
     | -.223 0     0     0     .624  -1.25 .624  .223  |
     | .623  0     0     0     0     .624  -1.25 0     |
     | 0     0     .223  0     0     .223  0     -.446 |

                8         8
o6 : Matrix RR    <-- RR
              53        53
i7 : sub(Jac, matrix{stablesols#2}) -- this is identical to the previous jacobian

o7 = | -.8   .623  -.223 0     0     -.223 .623  0     |
     | .623  -1.25 .624  0     0     0     0     0     |
     | -.223 .624  -1.25 .624  0     0     0     .223  |
     | 0     0     .624  -1.25 .623  0     0     0     |
     | 0     0     0     .623  -1.25 .624  0     0     |
     | -.223 0     0     0     .624  -1.25 .624  .223  |
     | .623  0     0     0     0     .624  -1.25 0     |
     | 0     0     .223  0     0     .223  0     -.446 |

                8         8
o7 : Matrix RR    <-- RR
              53        53
i8 : sub(Jac, matrix{stablesols#0}) -- negative of Laplacian of the graph

o8 = | -4 1  1  0  0  1  1  0  |
     | 1  -2 1  0  0  0  0  0  |
     | 1  1  -4 1  0  0  0  1  |
     | 0  0  1  -2 1  0  0  0  |
     | 0  0  0  1  -2 1  0  0  |
     | 1  0  0  0  1  -4 1  1  |
     | 1  0  0  0  0  1  -2 0  |
     | 0  0  1  0  0  1  0  -2 |

              8       8
o8 : Matrix ZZ  <-- ZZ

See also


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:936:0.