{ "status": "Numerical corroboration; not a validated computation or a proof.", "model": "x'=x(alpha(x)-y), y'=y(x-1)", "local_growth": "alpha(x)=1-s(x)^2/2+delta*s(x)^3*P(s(x)^2)", "coordinate": "s=sign(x-1)*sqrt(2*(x-1-log(x))); q=y-1; p=s*sqrt(y)", "moment": "J_j=pi*2^(-j-1)*binomial(2j+4,j+2)*h^(j+2)*2F1((2j+3)/4,(2j+5)/4;j+3;2h)", "parameter_delta": 0.1, "mpmath_decimal_digits": 80, "solver": "scipy.solve_ivp DOP853, positive-direction q=0 return section", "solver_controls": { "standard": { "rtol": 2e-12, "atol": 2e-14, "max_step": 0.075 }, "refined": { "rtol": 5e-14, "atol": 2e-15, "max_step": 0.02 } }, "software_versions": { "numpy": "1.26.4", "scipy": "1.13.1", "mpmath": "1.2.1", "matplotlib": "3.5.1" }, "cases": { "3": { "number_of_selected_cycles": 3, "delta": 0.1, "interpolation": { "coefficients": [ "0.00462584243209378141896988778179405308170216039803811623598483361222148538681498", "-0.097551638990176428865306517348822759142131971300039947155724740573960883880035435", "0.57579363490571894653606472697653648295605984754182171815670067039215301171026256", "-1.0" ], "target_energies": [ "0.05", "0.10", "0.15" ], "melnikov_derivatives": [ "-0.00081217393062896312207246760001090070291497337542116370007220505255006723451164794", "0.0018323582924361989120896606564293042342003353261874092063369762480692367585135075", "-0.0094054132042072024686201736018943601013167393165185803371808355947009822402927221" ], "interpolation_residual_max": "1.0295115e-84", "quadrature_relative_error_max": "2.6139085e-81", "quadrature_check_energy": "0.117" }, "return_maps": { "standard": [ { "displacement": -3.469446951953614e-17, "multiplier": 0.999918785904978, "log_multiplier": -8.121739306519811e-05, "period_sigma": 6.28318530722577, "return_function_evaluations": 1304, "cycle_index": 1, "target_energy": 0.05, "actual_section_energy": 0.0499999109050964, "energy_shift": -8.909490360059502e-08, "stability": "stable", "bracket": [ 0.02500000000000001, 0.075 ], "bracket_displacements": [ 8.993082755391624e-07, -1.8172729700782764e-06 ] }, { "displacement": -1.3877787807814457e-17, "multiplier": 1.0001832526179668, "log_multiplier": 0.00018323582925691473, "period_sigma": 6.283185307243242, "return_function_evaluations": 1304, "cycle_index": 2, "target_energy": 0.1, "actual_section_energy": 0.10000028155356193, "energy_shift": 2.8155356192327563e-07, "stability": "unstable", "bracket": [ 0.07500000000000001, 0.125 ], "bracket_displacements": [ -1.8172729700921542e-06, 5.7249657550162425e-06 ] }, { "displacement": 2.220446049250313e-16, "multiplier": 0.9990599008489506, "log_multiplier": -0.0009405413214006424, "period_sigma": 6.283185307459679, "return_function_evaluations": 1376, "cycle_index": 3, "target_energy": 0.15, "actual_section_energy": 0.14999882002789444, "energy_shift": -1.1799721055527623e-06, "stability": "stable", "bracket": [ 0.125, 0.175 ], "bracket_displacements": [ 5.7249657550162425e-06, -6.425984050595357e-05 ] } ], "refined": [ { "displacement": 5.551115123125783e-17, "multiplier": 0.999918785904979, "log_multiplier": -8.121739306426408e-05, "period_sigma": 6.283185307225769, "return_function_evaluations": 6017, "period_original_t": 6.435440616565398, "independent_log_multiplier_original_coordinates": -8.121739306427012e-05, "coordinate_inversion_error_max": 3.918030179808629e-13, "min_x": 0.7128935460172515, "max_x": 1.3554506586949773, "min_y": 0.683772233981349, "max_y": 1.3162277660226298, "max_abs_s": 0.3203645425315451, "cycle_index": 1, "target_energy": 0.05, "actual_section_energy": 0.049999910904630934, "energy_shift": -8.909536906853699e-08, "stability": "stable", "bracket": [ 0.02500000000000001, 0.075 ], "bracket_displacements": [ 8.993082755148762e-07, -1.8172729702031765e-06 ] }, { "displacement": 1.1102230246251565e-16, "multiplier": 1.0001832526179737, "log_multiplier": 0.0001832358292636248, "period_sigma": 6.283185307243243, "return_function_evaluations": 6017, "period_original_t": 6.6069154857487336, "independent_log_multiplier_original_coordinates": 0.00018323582926316195, "coordinate_inversion_error_max": 3.238805165901859e-13, "min_x": 0.6080118748311936, "max_x": 1.532422937384849, "min_y": 0.5527864044907478, "max_y": 1.4472135955225056, "max_abs_s": 0.45950589849828877, "cycle_index": 2, "target_energy": 0.1, "actual_section_energy": 0.10000028155438578, "energy_shift": 2.8155438577814884e-07, "stability": "unstable", "bracket": [ 0.07500000000000001, 0.125 ], "bracket_displacements": [ -1.8172729702170543e-06, 5.724965754849709e-06 ] }, { "displacement": 2.220446049250313e-16, "multiplier": 0.999059900848934, "log_multiplier": -0.0009405413214172094, "period_sigma": 6.283185307459679, "return_function_evaluations": 6017, "period_original_t": 6.802880502129139, "independent_log_multiplier_original_coordinates": -0.0009405413214158553, "coordinate_inversion_error_max": 3.3431119201079995e-13, "min_x": 0.5317404962318957, "max_x": 1.6852590073672775, "min_y": 0.45227744242617385, "max_y": 1.5477225577815985, "max_abs_s": 0.5715596100459746, "cycle_index": 3, "target_energy": 0.15, "actual_section_energy": 0.1499988200280812, "energy_shift": -1.179971918785494e-06, "stability": "stable", "bracket": [ 0.125, 0.175 ], "bracket_displacements": [ 5.724965754849709e-06, -6.425984050648093e-05 ] } ] }, "solver_comparisons": [ { "cycle_index": 1, "energy_difference": 4.654679419680008e-13, "multiplier_difference": 9.992007221626409e-16 }, { "cycle_index": 2, "energy_difference": 8.238548732109052e-13, "multiplier_difference": 6.8833827526759706e-15 }, { "cycle_index": 3, "energy_difference": 1.8676726831756696e-13, "multiplier_difference": 1.6542323066914832e-14 } ], "concavity_check": { "method": "50001-point grid plus local scalar refinement; not interval arithmetic", "cycle_projection_x": [ 0.5317404962318957, 1.6852590073672775 ], "checked_interval_x": [ 0.508670126009188, 1.708329377589985 ], "sample_count": 50002, "estimated_max_alpha_second": -0.36360650925762533, "estimated_min_alpha_second": -3.7024797865515686, "min_alpha_on_checked_interval": 0.8154468811174469, "alpha_at_peak": 1.0, "alpha_prime_at_peak": 0.0, "alpha_second_at_peak": -1.0 } }, "4": { "number_of_selected_cycles": 4, "delta": 0.1, "interpolation": { "coefficients": [ "-0.00073482072391701243163932586967882398517522342399029506355707080162221410140904602", "0.022063943726098817354776749351536391595061715934417546240123077822253509863742263", "-0.20968599892322634928663454200154469836880154449914141485846529326019304888973129", "0.78396493974049915208417517639362868779949532330534757014180058739527224642892921", "-1.0" ], "target_energies": [ "0.04", "0.08", "0.12", "0.16" ], "melnikov_derivatives": [ "0.000077294429728410458401433628709358315019533169409701249064805574168061185934048544", "-0.00011584406854584101392459897717241084293931044808200113096827244837713135636226924", "0.00029540898823835746441002510824316768212293886512119930466556469776428872709351663", "-0.0018029976428871051272113861719423396871105290023817421924744480263642190412412334" ], "interpolation_residual_max": "4.1180461e-84", "quadrature_relative_error_max": "3.1887899e-81", "quadrature_check_energy": "0.117" }, "return_maps": { "standard": [ { "displacement": -1.3877787807814457e-17, "multiplier": 1.0000077294728449, "log_multiplier": 7.729442972554532e-06, "period_sigma": 6.283185307180133, "return_function_evaluations": 1304, "cycle_index": 1, "target_energy": 0.04, "actual_section_energy": 0.04000000656337952, "energy_shift": 6.563379520430068e-09, "stability": "unstable", "bracket": [ 0.020000000000000004, 0.06 ], "bracket_displacements": [ -7.995730813559199e-08, 1.1513035290400264e-07 ] }, { "displacement": 1.3877787807814457e-17, "multiplier": 0.9999884156602402, "log_multiplier": -1.1584406858766781e-05, "period_sigma": 6.283185307180047, "return_function_evaluations": 1304, "cycle_index": 2, "target_energy": 0.08, "actual_section_energy": 0.07999998571874233, "energy_shift": -1.4281257676440617e-08, "stability": "stable", "bracket": [ 0.060000000000000005, 0.1 ], "bracket_displacements": [ 1.1513035289706375e-07, -2.1654425254313825e-07 ] }, { "displacement": -1.3877787807814457e-17, "multiplier": 1.0000295413351574, "log_multiplier": 2.9540898820855812e-05, "period_sigma": 6.283185307180333, "return_function_evaluations": 1304, "cycle_index": 3, "target_energy": 0.12, "actual_section_energy": 0.12000003854074703, "energy_shift": 3.85407470382404e-08, "stability": "unstable", "bracket": [ 0.1, 0.13999999999999999 ], "bracket_displacements": [ -2.1654425254313825e-07, 8.054515476529911e-07 ] }, { "displacement": -2.7755575615628914e-17, "multiplier": 0.9998197164887792, "log_multiplier": -0.0001802997642464184, "period_sigma": 6.283185307184757, "return_function_evaluations": 1328, "cycle_index": 4, "target_energy": 0.16, "actual_section_energy": 0.1599998568667637, "energy_shift": -1.431332362911153e-07, "stability": "stable", "bracket": [ 0.14, 0.18 ], "bracket_displacements": [ 8.05451547791769e-07, -1.071934440266542e-05 ] } ], "refined": [ { "displacement": 0.0, "multiplier": 1.0000077294728449, "log_multiplier": 7.729442972629201e-06, "period_sigma": 6.283185307180132, "return_function_evaluations": 6017, "period_original_t": 6.403589538427969, "independent_log_multiplier_original_coordinates": 7.729442972642664e-06, "coordinate_inversion_error_max": 5.973702435491113e-13, "min_x": 0.7407740153735952, "max_x": 1.313621578380874, "min_y": 0.7171572875323373, "max_y": 1.282842712467696, "max_abs_s": 0.28577505802968645, "cycle_index": 1, "target_energy": 0.04, "actual_section_energy": 0.04000000656406758, "energy_shift": 6.56406758114958e-09, "stability": "unstable", "bracket": [ 0.020000000000000004, 0.06 ], "bracket_displacements": [ -7.995730813559199e-08, 1.1513035286930817e-07 ] }, { "displacement": 1.3877787807814457e-17, "multiplier": 0.9999884156602418, "log_multiplier": -1.1584406857182488e-05, "period_sigma": 6.283185307180047, "return_function_evaluations": 6017, "period_original_t": 6.535724495633134, "independent_log_multiplier_original_coordinates": -1.1584406857501062e-05, "coordinate_inversion_error_max": 2.810199447277062e-13, "min_x": 0.6450365287613713, "max_x": 1.466070647307902, "min_y": 0.599999999981924, "max_y": 1.4000000000181285, "max_abs_s": 0.4086192820041784, "cycle_index": 2, "target_energy": 0.08, "actual_section_energy": 0.07999998571435295, "energy_shift": -1.4285647054190775e-08, "stability": "stable", "bracket": [ 0.060000000000000005, 0.1 ], "bracket_displacements": [ 1.1513035286236928e-07, -2.1654425266803834e-07 ] }, { "displacement": 5.551115123125783e-17, "multiplier": 1.0000295413351667, "log_multiplier": 2.9540898830114774e-05, "period_sigma": 6.283185307180331, "return_function_evaluations": 6017, "period_original_t": 6.681998259594113, "independent_log_multiplier_original_coordinates": 2.9540898829616413e-05, "coordinate_inversion_error_max": 1.7187043888783338e-13, "min_x": 0.5752233847654779, "max_x": 1.5952518196426482, "min_y": 0.5101020514365939, "max_y": 1.4898979485636388, "max_abs_s": 0.5063994771205457, "cycle_index": 3, "target_energy": 0.12, "actual_section_energy": 0.12000003854557544, "energy_shift": 3.8545575439807855e-08, "stability": "unstable", "bracket": [ 0.1, 0.13999999999999999 ], "bracket_displacements": [ -2.1654425266803834e-07, 8.054515472921686e-07 ] }, { "displacement": 1.942890293094024e-16, "multiplier": 0.9998197164887173, "log_multiplier": -0.00018029976430840896, "period_sigma": 6.283185307184757, "return_function_evaluations": 6017, "period_original_t": 6.845634550589294, "independent_log_multiplier_original_coordinates": -0.00018029976430888883, "coordinate_inversion_error_max": 4.136525106820943e-13, "min_x": 0.518431464430689, "max_x": 1.7145008101609756, "min_y": 0.4343145750477292, "max_y": 1.5656854249573, "max_abs_s": 0.5922480982517307, "cycle_index": 4, "target_energy": 0.16, "actual_section_energy": 0.15999985686955298, "energy_shift": -1.4313044702229938e-07, "stability": "stable", "bracket": [ 0.14, 0.18 ], "bracket_displacements": [ 8.054515476252355e-07, -1.0719344402471132e-05 ] } ] }, "solver_comparisons": [ { "cycle_index": 1, "energy_difference": 6.880607195114408e-13, "multiplier_difference": 0.0 }, { "cycle_index": 2, "energy_difference": 4.389377750158019e-12, "multiplier_difference": 1.5543122344752192e-15 }, { "cycle_index": 3, "energy_difference": 4.828401567458229e-12, "multiplier_difference": 9.325873406851315e-15 }, { "cycle_index": 4, "energy_difference": 2.789268815917012e-12, "multiplier_difference": 6.195044477408373e-14 } ], "concavity_check": { "method": "50001-point grid plus local scalar refinement; not interval arithmetic", "cycle_projection_x": [ 0.518431464430689, 1.7145008101609756 ], "checked_interval_x": [ 0.4945100775160833, 1.7384221970755813 ], "sample_count": 50002, "estimated_max_alpha_second": -0.3383078144673508, "estimated_min_alpha_second": -4.017558093306436, "min_alpha_on_checked_interval": 0.8013227329337567, "alpha_at_peak": 1.0, "alpha_prime_at_peak": 0.0, "alpha_second_at_peak": -1.0 } } } }