files / problems / hilbert16-limit-cycles / compute / q2 / i-lienard / certs / family.json
{
"F": {
"F": [
{
"alpha": 1,
"coeff": "1",
"x": 1
},
{
"beta": 1,
"coeff": "1",
"x": 3
}
],
"Fprime": [
{
"alpha": 1,
"coeff": "1"
},
{
"beta": 1,
"coeff": "3",
"x": 2
}
],
"factored": [
{
"alpha": 1,
"coeff": "1",
"x": 1
},
{
"beta": 1,
"coeff": "1",
"x": 3
}
],
"fg_num": [
{
"alpha": 1,
"coeff": "-1"
},
{
"beta": 1,
"coeff": "3",
"x": 2
}
],
"odd_sum": [],
"variables": [
"x",
"alpha",
"beta"
]
},
"H31_full_proved": false,
"claim": "Energy identity dE/dt = -alpha x^2 - beta x^4 for dx/dt = y-(alpha x + beta x^3), dy/dt = -x. Odd cubic F meets the Li\u00e9nard algebraic hypotheses when beta>0 and alpha<0. Not a bound on planar H(n) or on all deg-F<=3 fields.",
"cousin_of_van_der_Pol": {
"classical_F": "mu*(x**3/3 - x)",
"new_H3": false,
"slice": "alpha = -mu, beta = mu/3, equivalently alpha + 3*beta = 0 with beta>0"
},
"energy": {
"P": [
{
"coeff": "1",
"y": 1
},
{
"alpha": 1,
"coeff": "-1",
"x": 1
},
{
"beta": 1,
"coeff": "-1",
"x": 3
}
],
"Q": [
{
"coeff": "-1",
"x": 1
}
],
"claimed": [
{
"alpha": 1,
"coeff": "-1",
"x": 2
},
{
"beta": 1,
"coeff": "-1",
"x": 4
}
],
"dEdt": [
{
"alpha": 1,
"coeff": "-1",
"x": 2
},
{
"beta": 1,
"coeff": "-1",
"x": 4
}
],
"factored": [
{
"alpha": 1,
"coeff": "-1",
"x": 2
},
{
"beta": 1,
"coeff": "-1",
"x": 4
}
],
"variables": [
"x",
"y",
"alpha",
"beta"
]
},
"even_term": {
"claimed": [
{
"coeff": "2",
"gamma": 1,
"x": 2
}
],
"note": "F(x)+F(-x)=2 gamma x^2. A quadratic term kills oddness; full deg-F<=3 is dropped.",
"odd_sum": [
{
"coeff": "2",
"gamma": 1,
"x": 2
}
],
"variables": [
"x",
"alpha",
"beta",
"gamma"
]
},
"field": {
"P": "y - (alpha*x + beta*x**3)",
"Q": "-x"
},
"hn_moved": false,
"isolated_periodic_orbits_when_alpha_beta_nonneg_not_both_zero": 0,
"isolated_periodic_orbits_when_beta_pos_alpha_neg": 1,
"linear": {
"P": [
{
"coeff": "1",
"y": 1
},
{
"alpha": 1,
"coeff": "-1",
"x": 1
}
],
"Q": [
{
"coeff": "-1",
"x": 1
}
],
"charpoly": "lambda**2 + alpha*lambda + 1",
"d2Edt": [
{
"alpha": 1,
"coeff": "-2",
"x": 2
}
],
"det": "1",
"trace": "-alpha",
"variables": [
"x",
"y",
"alpha"
]
},
"schema": "hilbert16-i-lienard-family/v1",
"vdp_chart": {
"F": [
{
"a": 2,
"beta": 1,
"coeff": "-1",
"x": 1
},
{
"beta": 1,
"coeff": "1",
"x": 3
}
],
"Fprime": [
{
"a": 2,
"beta": 1,
"coeff": "-1"
},
{
"beta": 1,
"coeff": "3",
"x": 2
}
],
"factored": [
{
"a": 2,
"beta": 1,
"coeff": "-1",
"x": 1
},
{
"beta": 1,
"coeff": "1",
"x": 3
}
],
"minus_F": [
{
"a": 2,
"beta": 1,
"coeff": "1",
"x": 1
},
{
"beta": 1,
"coeff": "-1",
"x": 3
}
],
"sign_pos": [
{
"a": 2,
"beta": 1,
"coeff": "1",
"x": 1
},
{
"beta": 1,
"coeff": "-1",
"x": 3
}
],
"substitution": "alpha = -beta * a**2",
"tail": [
{
"a": 2,
"beta": 1,
"coeff": "-1"
},
{
"beta": 1,
"coeff": "3",
"x": 2
}
],
"two_beta_a2": [
{
"a": 2,
"beta": 1,
"coeff": "2"
}
],
"variables": [
"x",
"a",
"beta"
]
},
"what_this_is_not": [
"not a dent of planar H(n)",
"not a field that beats B(n)",
"not a proof that every deg-F<=3 Li\u00e9nard has at most one cycle",
"not a new H(3)"
]
}