files / problems / hilbert16-limit-cycles / compute / q3 / ff-two-well / certs / core.json

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{
  "H_saddle": "0",
  "H_wells": "-1/4",
  "I0_left": "4/15*mu",
  "I0_right": "4/15*mu",
  "I_at_well_bottom": "0",
  "L1_cubic_minus": "-sqrt(2)*mu/2",
  "L1_cubic_plus": "-sqrt(2)*mu/2",
  "L1_minus": "sqrt(2)*mu",
  "L1_plus": "sqrt(2)*mu",
  "L1_quad_minus": "3*sqrt(2)*mu/2",
  "L1_quad_plus": "3*sqrt(2)*mu/2",
  "L1_squared_coeff": 2,
  "V1_plus": "sqrt(2)*mu/8",
  "claim": "Unperturbed two-well cubic Hamiltonian classification, dH/dt identities, I(h) formula, I(0)=4 mu/15 per figure-eight lobe, and L1=sqrt(2)*mu at each well. Not 14 zeros of I(h) and not a bound on H(n).",
  "cycles_proved": 0,
  "degree": 3,
  "equilibria": [
    {
      "H": "0",
      "det": "-1",
      "kind": "saddle",
      "trace": "0",
      "x": 0,
      "y": 0
    },
    {
      "H": "-1/4",
      "det": "2",
      "kind": "center",
      "trace": "0",
      "x": 1,
      "y": 0
    },
    {
      "H": "-1/4",
      "det": "2",
      "kind": "center",
      "trace": "0",
      "x": -1,
      "y": 0
    }
  ],
  "fourteen_zeros_produced": false,
  "hn_moved": false,
  "perturbed_equilibria": [
    {
      "det": "-1",
      "kind": "saddle",
      "trace": "mu",
      "x": 0,
      "y": 0
    },
    {
      "det": "2",
      "kind": "center",
      "trace": "0",
      "x": 1,
      "y": 0
    },
    {
      "det": "2",
      "kind": "center",
      "trace": "0",
      "x": -1,
      "y": 0
    }
  ],
  "potential_shift": "1/4",
  "regular_I_zeros_exhibited": 0,
  "saddle_stays_saddle": true,
  "schema": "hilbert16-ff-two-well-core/v1",
  "trace_at_wells": "0",
  "weak_focus_order": 1,
  "what_this_is_not": [
    "not a dent of H(n)",
    "not fourteen zeros of an Abelian integral",
    "not a proved pair of limit cycles",
    "not a new H(3)"
  ]
}