files / problems / hilbert16-limit-cycles / compute / q1 / b-cubic-fourteen / certificate.json

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{
  "abelian_integral": {
    "I_h": "pi*mu*h*(2 - h)",
    "I_over_pi_mu_coeffs_h_high_to_low": [
      "-1",
      "2",
      "0"
    ],
    "I_prime_at_2": "-2*pi*mu",
    "methods_agree": [
      "trig_parametrization",
      "green_disk_flux"
    ],
    "note": "A simple zero of the first-order Abelian integral gives a unique hyperbolic cycle near the circle of radius 2 for sufficiently small \u03bc>0 (Poincar\u00e9\u2013Pontryagin). Uniqueness for every \u03bc>0 is the Li\u00e9nard theorem, not this integral.",
    "perturbation_P1": "0",
    "perturbation_Q1": "-(x**2 - 1)*y",
    "positive_zeros": [
      "2"
    ],
    "simple_at_h2": true,
    "unperturbed_H": "(x**2 + y**2)/2"
  },
  "energy": {
    "cartesian_dE_dt": "-mu*(x**2 - 1)*y**2",
    "closed_orbit_must_meet_both_strips": true,
    "lienard_dW_dt": "-x*F(x)",
    "lienard_dW_dt_expanded": "-mu*x**2*(x**2/3 - 1)"
  },
  "equilibrium": {
    "charpoly_lambda": "lambda**2 - mu*lambda + 1",
    "det_at_0": "1",
    "jacobian_at_0": [
      [
        "0",
        "1"
      ],
      [
        "-1",
        "mu"
      ]
    ],
    "origin_is_source": true,
    "trace_at_0": "mu",
    "unique_equilibrium": [
      0,
      0
    ]
  },
  "exact_statement": "For every real mu>0 the cubic van der Pol field dx/dt=y, dy/dt=-x-mu*(x**2-1)*y satisfies the hypotheses of Lienard's uniqueness theorem, hence has exactly one periodic orbit, and that orbit is asymptotically stable. Separately, its first-order Abelian integral on the ovals of H=(x**2+y**2)/2 is I(h)=pi*mu*h*(2-h), with exactly one positive simple zero, at h=2.",
  "field": {
    "P": "y",
    "Q": "-x - mu*(x**2 - 1)*y",
    "deg_P": 1,
    "deg_Q": 3,
    "degree": 3
  },
  "h3_lower_bound_claimed": false,
  "hn_moved": false,
  "imagined_claim": "An explicit real cubic planar field with 14 isolated periodic orbits, beating Li\u2013Liu\u2013Yang H(3)>=13.",
  "isolated_periodic_orbits": 1,
  "lienard_hypotheses": {
    "F": "mu*(x**3/3 - x)",
    "F_factored": "(mu/3)*x*(x**2 - 3)",
    "F_negative_on": "(0, sqrt(3))",
    "F_odd": true,
    "F_positive_on": "(sqrt(3), oo)",
    "F_prime_at_0": "-mu",
    "F_prime_at_sqrt3": "2*mu",
    "F_prime_positive_on_tail": true,
    "F_tends_to_plus_infinity": true,
    "G": "x**2/2",
    "G_tends_to_plus_infinity": true,
    "f": "mu*(x**2 - 1)",
    "g": "x",
    "g_odd": true,
    "positive_zero_of_F": "sqrt(3)",
    "theorem": "Li\u00e9nard uniqueness for \u1e8b=y-F(x), \u1e8f=-x with F odd, one positive zero a, F<0 on (0,a), F>0 and F'>0 on [a,\u221e), F\u2192\u221e. Conclusion: exactly one periodic orbit, asymptotically stable.",
    "xg_positive_off_zero": true
  },
  "line": "B",
  "not_the_radial_family": {
    "distinct_from_radial_family": true,
    "radial_P": "y - x*(x**2 + y**2 - rho**2)",
    "radial_Q": "-x - y*(x**2 + y**2 - rho**2)",
    "this_P": "y",
    "this_Q": "-x - mu*(x**2 - 1)*y"
  },
  "parameter": "mu>0",
  "published_record": "H(3)>=13, Li\u2013Liu\u2013Yang, J. Differ. Equations 246 (2009). Not replayed (paywalled). Not beaten.",
  "status": "fork",
  "what_this_is_not": [
    "not a dent of H(3)",
    "not a claim that H(3)>=14",
    "not a replay of Li\u2013Liu\u2013Yang's 13 cycles",
    "not a lower bound from numerics"
  ]
}