files / problems / hilbert16-limit-cycles / compute / q1 / a-quadratic-five / cert.json
{
"line": "a-quadratic-five",
"imagined_claim": "An explicit quadratic, a perturbation of Shi Songling 1980 with an extra \u03bc y\u00b2 term, has five isolated periodic orbits, hence H(2) \u2265 5.",
"status": "dropped",
"fork": "Unperturbed Shi (\u03bb = \u03b5 = \u03b4 = \u03bc = 0) has a weak focus of order 3 at the origin, V1 = V2 = 0, V3 = 35625/8, and a strong unstable focus at (0, 1) with charpoly t^2 - 5 t + 24.",
"published_Hn_moved": false,
"dropped_claim": "H(2) >= 5",
"proved": {
"system": {
"P": "-y - 10 x^2 + 5 x y + y^2",
"Q": "x + x^2 - 25 x y",
"parameters": {
"lambda": 0,
"epsilon": 0,
"delta": 0,
"mu": 0
}
},
"normalization": {
"F2": "(x^2 + y^2)/2",
"identity": "dF/dt = V1 r^4 + V2 r^6 + V3 r^8 + O(9)",
"gauge": "y^n coefficient of even F_n is 0"
},
"V1": {
"num": 0,
"den": 1,
"str": "0"
},
"V2": {
"num": 0,
"den": 1,
"str": "0"
},
"V3": {
"num": 35625,
"den": 8,
"str": "35625/8"
},
"weak_focus_order": 3,
"li_chengzhi_crosscheck": {
"L1": 0,
"L2": 0,
"L3": 57000,
"L3_over_V3": "64/5",
"source": "Li Chengzhi, as quoted by Llibre\u2013Schlomiuk, Canad. J. Math. 56 (2004), after Proposition 2."
},
"origin": {
"point": [
0,
0
],
"jacobian": [
[
0,
-1
],
[
1,
0
]
],
"trace": 0,
"det": 1,
"disc": -4,
"charpoly": "t^2 + 1",
"type": "linear_center"
},
"second_focus": {
"point": [
0,
1
],
"jacobian": [
[
5,
1
],
[
-24,
0
]
],
"trace": 5,
"det": 24,
"disc": -71,
"charpoly": "t^2 - 5 t + 24",
"eigenvalues": "5/2 \u00b1 i sqrt(71)/2",
"type": "strong_unstable_focus",
"note": "Trace 5 > 0, det 24 > 0, disc \u221271 < 0. Not a weak focus: no Hopf unfolding at (0, 1) inside a small Shi perturbation."
},
"equilibria": {
"real_count": 2,
"complex_count": 2,
"bezout_max": 4,
"y_axis_branch": [
"(0, 0)",
"(0, 1)"
],
"second_branch_quadratic": "6124 x^2 - 102 x + 24",
"second_branch_discriminant": -577500,
"real": [
{
"x": "0",
"y": "0"
},
{
"x": "0",
"y": "1"
}
],
"complex": [
{
"x": "51/6124 - 25*sqrt(231)*I/6124",
"y": "247/6124 - sqrt(231)*I/6124"
},
{
"x": "51/6124 + 25*sqrt(231)*I/6124",
"y": "247/6124 + sqrt(231)*I/6124"
}
]
},
"F": {
"2": [
{
"i": 0,
"j": 2,
"num": 1,
"den": 2
},
{
"i": 1,
"j": 1,
"num": 0,
"den": 1
},
{
"i": 2,
"j": 0,
"num": 1,
"den": 2
}
],
"3": [
{
"i": 0,
"j": 3,
"num": 44,
"den": 3
},
{
"i": 1,
"j": 2,
"num": 0,
"den": 1
},
{
"i": 2,
"j": 1,
"num": 10,
"den": 1
},
{
"i": 3,
"j": 0,
"num": 2,
"den": 1
}
],
"4": [
{
"i": 0,
"j": 4,
"num": 0,
"den": 1
},
{
"i": 1,
"j": 3,
"num": 0,
"den": 1
},
{
"i": 2,
"j": 2,
"num": -540,
"den": 1
},
{
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"j": 1,
"num": 50,
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},
{
"i": 4,
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"den": 1
}
],
"5": [
{
"i": 0,
"j": 5,
"num": -38252,
"den": 3
},
{
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"num": 0,
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},
{
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{
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},
{
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},
{
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"num": -2966,
"den": 1
}
],
"6": [
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"num": 0,
"den": 1
},
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"num": 0,
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{
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"num": 764500,
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{
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},
{
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"den": 1
},
{
"i": 6,
"j": 0,
"num": 5889250,
"den": 9
}
],
"7": [
{
"i": 0,
"j": 7,
"num": 2088923596,
"den": 63
},
{
"i": 1,
"j": 6,
"num": 0,
"den": 1
},
{
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"num": 1051342298,
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},
{
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{
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"num": 33646484,
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{
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],
"8": [
{
"i": 0,
"j": 8,
"num": 0,
"den": 1
},
{
"i": 1,
"j": 7,
"num": -35625,
"den": 8
},
{
"i": 2,
"j": 6,
"num": -8353400884,
"den": 3
},
{
"i": 3,
"j": 5,
"num": 33779125831,
"den": 72
},
{
"i": 4,
"j": 4,
"num": -74502924433,
"den": 9
},
{
"i": 5,
"j": 3,
"num": 74214405481,
"den": 72
},
{
"i": 6,
"j": 2,
"num": -210861830686,
"den": 27
},
{
"i": 7,
"j": 1,
"num": 12274102315,
"den": 24
},
{
"i": 8,
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"num": -63011675575,
"den": 27
}
]
}
},
"five_cycle": {
"mu_moves_second_equilibrium": {
"Q_at_0_1": "mu",
"L1_primitive_at_origin": "27 mu",
"V1_at_origin": "27 mu / 8",
"order_3_requires": "mu = 0"
},
"trace_at_0_1": {
"formula": "lambda + 5 + delta",
"det_formula_mu_0": "24 - 8 epsilon + 9 delta"
},
"delta_minus_5_lambda_eps_0": {
"origin_Li": [
0,
0,
0
],
"origin_type": "center",
"second_point": "(0, 1)",
"second_trace": 0,
"second_det": -21,
"second_type": "saddle"
},
"both_weak_in_family": {
"conditions": "lambda = 0, delta = -5, mu = 0, epsilon < -21/8",
"origin_Li_L1": "-8 epsilon != 0",
"origin_order": 1,
"bautin_budget": "1 + 3 = 4, not 5"
},
"circles_about_0_1_linear": {
"V": "x^2 + (y-1)^2",
"Vdot_r2": "5 + 5 cos(2 theta) - 23 sin(2 theta)",
"amplitude_squared": 554,
"mean_squared": 25,
"sign_change": true,
"note": "Concentric Euclidean circles about (0, 1) already fail at linear order. Axis-aligned ellipses sampled later also change sign (residue, not a trapping pair)."
},
"coppel_unique_singularity": "A quadratic limit cycle surrounds a unique singular point, and that point is a focus (Coppel 1966, as stated in Llibre\u2013Schlomiuk, Canad. J. Math. 56 (2004), \u00a77.1(iv)). A fifth cycle surrounding both foci is excluded for any quadratic, not only this family.",
"bautin": "A quadratic focus has cyclicity at most 3 (Bautin). Four small cycles at the origin are impossible.",
"galias_this_system": "Galias\u2013Tucker\u2013Wilczak (AMC 2022) prove the Songling field with \u03bc = 0 and the classical tiny (\u03b4, \u03b5, \u03bb) has exactly four cycles. That is one system, not H(2) = 4, and it is not a fifth cycle."
},
"annulus": {
"status": "residue",
"tried": "Euclidean circles and axis-aligned ellipses about (0, 1); linear-order circles already change sign (amp^2 = 554 > 25).",
"certificate": null
}
}