files / problems / hilbert16-limit-cycles / compute / q3 / gg-pt-lyapunov / certs / lyapunov.json

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{
  "L1_formula": {
    "cubic": [
      {
        "coeff": 1,
        "term": "a12"
      },
      {
        "coeff": 3,
        "term": "a30"
      },
      {
        "coeff": 3,
        "term": "b03"
      },
      {
        "coeff": 1,
        "term": "b21"
      }
    ],
    "quad": [
      {
        "coeff": 1,
        "term": "a02*a11"
      },
      {
        "coeff": 2,
        "term": "a02*b02"
      },
      {
        "coeff": 1,
        "term": "a11*a20"
      },
      {
        "coeff": -2,
        "term": "a20*b20"
      },
      {
        "coeff": -1,
        "term": "b02*b11"
      },
      {
        "coeff": -1,
        "term": "b11*b20"
      }
    ]
  },
  "centers": [
    {
      "L1_E": "0",
      "L1_cubic": "0",
      "L1_full": "0",
      "det": 44,
      "jacobian": [
        0,
        4,
        -11,
        0
      ],
      "point": [
        0,
        0
      ],
      "trace": 0
    },
    {
      "L1_E": "9/2",
      "L1_cubic": "-9/2",
      "L1_full": "0",
      "det": 64,
      "jacobian": [
        0,
        -8,
        8,
        0
      ],
      "point": [
        1,
        2
      ],
      "trace": 0
    },
    {
      "L1_E": "-9/2",
      "L1_cubic": "9/2",
      "L1_full": "0",
      "det": 64,
      "jacobian": [
        0,
        -8,
        8,
        0
      ],
      "point": [
        1,
        -2
      ],
      "trace": 0
    }
  ],
  "claim": "unperturbed PT Darboux centers have vanishing first Lyapunov quantity after the q1 normal form plus the cubic jet; not a dent of H(4)",
  "field": {
    "P": [
      {
        "coeff": "4",
        "y": 1
      },
      {
        "coeff": "-3",
        "x": 1,
        "y": 1
      },
      {
        "coeff": "-1",
        "x": 1,
        "y": 3
      },
      {
        "coeff": "2",
        "x": 2,
        "y": 1
      },
      {
        "coeff": "1",
        "x": 3,
        "y": 1
      }
    ],
    "Q": [
      {
        "coeff": "6",
        "y": 2
      },
      {
        "coeff": "-2",
        "y": 4
      },
      {
        "coeff": "-11",
        "x": 1
      },
      {
        "coeff": "7",
        "x": 1,
        "y": 2
      },
      {
        "coeff": "-15",
        "x": 2
      },
      {
        "coeff": "3",
        "x": 2,
        "y": 2
      },
      {
        "coeff": "-21",
        "x": 3
      },
      {
        "coeff": "15",
        "x": 4
      }
    ],
    "deg_P": 4,
    "deg_Q": 4,
    "variables": [
      "x",
      "y"
    ]
  },
  "origin_jet": {
    "P1": "4*y",
    "P2": "-3*x*y",
    "P3": "2*x^2*y",
    "Q1": "-11*x",
    "Q2": "-15*x^2+6*y^2",
    "Q3": "-21*x^3+7*x*y^2"
  },
  "perturbations": {
    "tracefree_xy": {
      "L1_full": [
        "0",
        "-1*mu",
        "1*mu"
      ],
      "delta_P": "mu*x*(x-1)^2*y",
      "delta_Q": "0",
      "traces": [
        "0",
        "0",
        "0"
      ]
    },
    "user_xy": {
      "L1_full": [
        "0",
        "strong",
        "strong"
      ],
      "delta_P": "mu*x*(x-1)*y",
      "delta_Q": "0",
      "traces": [
        "0",
        "2*mu",
        "-2*mu"
      ]
    },
    "x2_shift": {
      "L1_full": [
        "-351/968*sqrt(11)*mu",
        "-1/8*mu",
        "-1/8*mu"
      ],
      "delta_P": "mu*x^2*(x-1)^2",
      "delta_Q": "0",
      "traces": [
        "0",
        "0",
        "0"
      ]
    }
  },
  "schema": "hilbert16-gg-pt-lyapunov/v1"
}