oscJacobian -- create the Jacobian for the oscillator system associated to a graph

Usage:

oscJacobian(G,R)

oscJacobian(I)

oscJacobian(G)
Inputs:
- G, a graph, an undirected, connected graph $G$
- S, a ring, created with oscRing
- I, an ideal, an ideal, intended to be ideal created by oscSystem
Optional inputs:
- Reduced => a Boolean value, default value false, if true, then the angles are reduced to the first angle being 0. That is, the number of oscillators will be one less than the number of vertices of $G$
Outputs:
- a matrix, the $n \times n$ Jacobian matrix of the system, as a matrix of polynomials involving the cosines and sines of angles $\theta_1, \ldots, \theta_{n-1}$, where we set $\theta_0 = 0$ if $\texttt{Reduced => true}$

Description

The matrix is a symmetric $n \times n$ matrix (with determinant zero).

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 : S = oo

o3 = S

o3 : PolynomialRing

i4 : I = oscSystem(G,S)

                                                                            
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 S

i5 : Jac = oscJacobian(G,S)

o5 = | -x_0x_1-x_0x_3-y_0y_1-y_0y_3 x_0x_1+y_0y_1               
     | x_0x_1+y_0y_1                -x_0x_1-x_1x_2-y_0y_1-y_1y_2
     | 0                            x_1x_2+y_1y_2               
     | x_0x_3+y_0y_3                0                           
     ------------------------------------------------------------------------
     0                            x_0x_3+y_0y_3                |
     x_1x_2+y_1y_2                0                            |
     -x_1x_2-x_2x_3-y_1y_2-y_2y_3 x_2x_3+y_2y_3                |
     x_2x_3+y_2y_3                -x_0x_3-x_2x_3-y_0y_3-y_2y_3 |

             4      4
o5 : Matrix S  <-- S

i6 : assert(det Jac == 0)

i7 : assert(Jac - transpose Jac == 0)

We can find the eigenvalues of the Jacobian at approximate points, and see if they are stable (all eigenvalues negative, except for the one required 0), unstable (a positive eigenvalue), or semistable (no positive eigenvalues, up to a certain tolerance).

i8 : realsols = findRealSolutions I
warning: some solutions are not regular: {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 61, 62}

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}, {-1, 1, 1, 1, 0, 0, 0, 0}}

o8 : List

i9 : jacs = for pt in realsols list sub(Jac, matrix{pt})

o9 = {| -2 1  0  1  |, | 0  1  0  -1 |, | 0  -1 0  1  |, | 2  -1 0  -1 |, |
      | 1  0  -1 0  |  | 1  -2 1  0  |  | -1 2  -1 0  |  | -1 0  1  0  |  |
      | 0  -1 2  -1 |  | 0  1  0  -1 |  | 0  -1 0  1  |  | 0  1  -2 1  |  |
      | 1  0  -1 0  |  | -1 0  -1 2  |  | 1  0  1  -2 |  | -1 0  1  0  |  |
     ------------------------------------------------------------------------
     0  -1 0  1  |, | 0  -1 0  1  |, | 2  -1 0  -1 |, | 2  -1 0  -1 |, | 0 
     -1 0  1  0  |  | -1 0  1  0  |  | -1 2  -1 0  |  | -1 2  -1 0  |  | 1 
     0  1  0  -1 |  | 0  1  0  -1 |  | 0  -1 2  -1 |  | 0  -1 2  -1 |  | 0 
     1  0  -1 0  |  | 1  0  -1 0  |  | -1 0  -1 2  |  | -1 0  -1 2  |  | -1
     ------------------------------------------------------------------------
     1  0  -1 |, | 2  -1 0  -1 |, | 0  1  0  -1 |, | -2 1  0  1  |, | 0  -1
     0  -1 0  |  | -1 0  1  0  |  | 1  -2 1  0  |  | 1  0  -1 0  |  | -1 2 
     -1 0  1  |  | 0  1  -2 1  |  | 0  1  0  -1 |  | 0  -1 2  -1 |  | 0  -1
     0  1  0  |  | -1 0  1  0  |  | -1 0  -1 2  |  | 1  0  -1 0  |  | 1  0 
     ------------------------------------------------------------------------
     0  1  |, | -2 1  0  1  |, | 2  -1 0  -1 |, | 0  1  0  -1 |, | 2  -1 0 
     -1 0  |  | 1  0  -1 0  |  | -1 0  1  0  |  | 1  -2 1  0  |  | -1 0  1 
     0  1  |  | 0  -1 2  -1 |  | 0  1  -2 1  |  | 0  1  0  -1 |  | 0  1  -2
     1  -2 |  | 1  0  -1 0  |  | -1 0  1  0  |  | -1 0  -1 2  |  | -1 0  1 
     ------------------------------------------------------------------------
     -1 |, | 0  -1 0  1  |, | 0  1  0  -1 |, | -2 1  0  1  |, | 0  1  0  -1
     0  |  | -1 2  -1 0  |  | 1  -2 1  0  |  | 1  0  -1 0  |  | 1  0  -1 0 
     1  |  | 0  -1 0  1  |  | 0  1  0  -1 |  | 0  -1 2  -1 |  | 0  -1 0  1 
     0  |  | 1  0  1  -2 |  | -1 0  -1 2  |  | 1  0  -1 0  |  | -1 0  1  0 
     ------------------------------------------------------------------------
     |, | 2  -1 0  -1 |, | -2 1  0  1  |, | 0  -1 0  1  |, | 0  1  0  -1 |, |
     |  | -1 0  1  0  |  | 1  -2 1  0  |  | -1 2  -1 0  |  | 1  -2 1  0  |  |
     |  | 0  1  -2 1  |  | 0  1  -2 1  |  | 0  -1 0  1  |  | 0  1  0  -1 |  |
     |  | -1 0  1  0  |  | 1  0  1  -2 |  | 1  0  1  -2 |  | -1 0  -1 2  |  |
     ------------------------------------------------------------------------
     -2 1  0  1  |, | 0  -1 0  1  |, | -2 1  0  1  |, | 2  -1 0  -1 |, | -2
     1  0  -1 0  |  | -1 0  1  0  |  | 1  0  -1 0  |  | -1 0  1  0  |  | 1 
     0  -1 2  -1 |  | 0  1  0  -1 |  | 0  -1 2  -1 |  | 0  1  -2 1  |  | 0 
     1  0  -1 0  |  | 1  0  -1 0  |  | 1  0  -1 0  |  | -1 0  1  0  |  | 1 
     ------------------------------------------------------------------------
     1  0  1  |, | 0  1  0  -1 |, | 0  -1 0  1  |, | 0  -1 0  1  |, | 0  1 
     -2 1  0  |  | 1  0  -1 0  |  | -1 2  -1 0  |  | -1 0  1  0  |  | 1  -2
     1  -2 1  |  | 0  -1 0  1  |  | 0  -1 0  1  |  | 0  1  0  -1 |  | 0  1 
     0  1  -2 |  | -1 0  1  0  |  | 1  0  1  -2 |  | 1  0  -1 0  |  | -1 0 
     ------------------------------------------------------------------------
     0  -1 |, | -2 1  0  1  |, | 0  -1 0  1  |, | 2  -1 0  -1 |, | 0  1  0 
     1  0  |  | 1  0  -1 0  |  | -1 2  -1 0  |  | -1 0  1  0  |  | 1  -2 1 
     0  -1 |  | 0  -1 2  -1 |  | 0  -1 0  1  |  | 0  1  -2 1  |  | 0  1  0 
     -1 2  |  | 1  0  -1 0  |  | 1  0  1  -2 |  | -1 0  1  0  |  | -1 0  -1
     ------------------------------------------------------------------------
     -1 |, | 0  -1 0  1  |, | 0  1  0  -1 |, | -2 1  0  1  |, | 2  -1 0  -1
     0  |  | -1 2  -1 0  |  | 1  -2 1  0  |  | 1  0  -1 0  |  | -1 0  1  0 
     -1 |  | 0  -1 0  1  |  | 0  1  0  -1 |  | 0  -1 2  -1 |  | 0  1  -2 1 
     2  |  | 1  0  1  -2 |  | -1 0  -1 2  |  | 1  0  -1 0  |  | -1 0  1  0 
     ------------------------------------------------------------------------
     |}
     |
     |
     |

o9 : List

i10 : jacs/eigenvalues

o10 = {{-2.82843    }, {-2.82843    }, {2.82843     }, {2.82843    },
       {-1.2484e-16 }  {2.82843     }  {-2.82843    }  {1.2484e-16 } 
       {2.82843     }  {-1.17387e-16}  {1.17387e-16 }  {-2.82843   } 
       {-1.59293e-33}  {4.34285e-17 }  {-4.34285e-17}  {1.59293e-33} 
      -----------------------------------------------------------------------
      {2                          }, {2                          },
      {-2                         }  {-2                         } 
      {-2.66445e-25+1.79147e-26*ii}  {5.05671e-26+4.09034e-25*ii } 
      {2.66445e-25-1.79147e-26*ii }  {-5.05672e-26-4.09034e-25*ii} 
      -----------------------------------------------------------------------
      {-4.71028e-16}, {-4.71028e-16}, {2              }, {2.82843    },
      {4           }  {4           }  {-2             }  {1.2484e-16 } 
      {2           }  {2           }  {7.88837e-25*ii }  {-2.82843   } 
      {2           }  {2           }  {-7.88837e-25*ii}  {1.59293e-33} 
      -----------------------------------------------------------------------
      {-2.82843    }, {-2.82843    }, {2.82843     }, {-2.82843   },
      {2.82843     }  {-1.2484e-16 }  {-2.82843    }  {2.94928e-16} 
      {-1.17387e-16}  {2.82843     }  {-1.44685e-16}  {2.82843    } 
      {4.34285e-17 }  {-1.59293e-33}  {9.06977e-18 }  {4.81379e-34} 
      -----------------------------------------------------------------------
      {2.82843     }, {2.82843    }, {2.82843     }, {2.82843     },
      {-2.02439e-16}  {-2.82843   }  {-5.45246e-18}  {-2.82843    } 
      {-2.82843    }  {1.00102e-16}  {-2.82843    }  {-8.85661e-17} 
      {-6.91373e-34}  {5.50481e-18}  {1.00194e-34 }  {9.28259e-17 } 
      -----------------------------------------------------------------------
      {-2.82843    }, {-2.82843    }, {2                          },
      {2.82843     }  {5.45246e-18 }  {-2                         } 
      {8.85661e-17 }  {2.82843     }  {5.05671e-26+4.09034e-25*ii } 
      {-9.28259e-17}  {-1.00194e-34}  {-5.05672e-26-4.09034e-25*ii} 
      -----------------------------------------------------------------------
      {2.82843     }, {3.14018e-16}, {-2.82843    }, {2.82843     },
      {-2.94143e-17}  {-4         }  {2.82843     }  {-2.82843    } 
      {-2.82843    }  {-2         }  {-9.07823e-17}  {7.58339e-17 } 
      {1.22371e-32 }  {-2         }  {1.21707e-16 }  {-3.05993e-17} 
      -----------------------------------------------------------------------
      {-2.82843    }, {2                          }, {-2.82843    },
      {9.14415e-17 }  {-2                         }  {-6.57495e-17} 
      {2.82843     }  {5.05671e-26+4.09034e-25*ii }  {2.82843     } 
      {-8.15529e-34}  {-5.05672e-26-4.09034e-25*ii}  {-4.14771e-33} 
      -----------------------------------------------------------------------
      {2.82843     }, {3.14018e-16}, {2                          },
      {-9.71557e-17}  {-4         }  {-2                         } 
      {-2.82843    }  {-2         }  {5.05671e-26+4.09034e-25*ii } 
      {7.32065e-33 }  {-2         }  {-5.05672e-26-4.09034e-25*ii} 
      -----------------------------------------------------------------------
      {2.82843     }, {2                          }, {2.82843    },
      {-2.82843    }  {-2                         }  {-2.82843   } 
      {-9.24027e-17}  {5.05671e-26+4.09034e-25*ii }  {1.00102e-16} 
      {4.85289e-17 }  {-5.05672e-26-4.09034e-25*ii}  {5.50481e-18} 
      -----------------------------------------------------------------------
      {-2.82843    }, {2.82843     }, {2.82843    }, {-2.82843    },
      {5.45246e-18 }  {-2.82843    }  {8.07983e-17}  {2.82843     } 
      {2.82843     }  {-8.85661e-17}  {-2.82843   }  {8.85661e-17 } 
      {-1.00194e-34}  {9.28259e-17 }  {1.43002e-32}  {-9.28259e-17} 
      -----------------------------------------------------------------------
      {-2.82843    }, {-2.82843   }, {-2.82843    }, {2.82843    }}
      {2.82843     }  {2.82843    }  {-6.57495e-17}  {1.13935e-16}
      {-1.00102e-16}  {2.17323e-16}  {2.82843     }  {-2.82843   }
      {-5.50481e-18}  {1.4608e-17 }  {-4.14771e-33}  {1.34215e-32}

o10 : List

i11 : jacs/eigenvalues/identifyStability

o11 = {Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, Unstable,
      -----------------------------------------------------------------------
      Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, Unstable,
      -----------------------------------------------------------------------
      Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, Unstable,
      -----------------------------------------------------------------------
      Unstable, Stable, Unstable, Unstable, Unstable, Unstable, Unstable,
      -----------------------------------------------------------------------
      Unstable, Stable, Unstable, Unstable, Unstable, Unstable, Unstable,
      -----------------------------------------------------------------------
      Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, Unstable}

o11 : List

Ways to use oscJacobian:

oscJacobian(Graph)
oscJacobian(Graph,Ring)
oscJacobian(Ideal)

For the programmer

The object oscJacobian is a method function with options.

The source of this document is in Oscillators/Documentation.m2:318:0.

oscJacobian -- create the Jacobian for the oscillator system associated to a graph

Description

See also

Ways to use oscJacobian:

For the programmer