{"id":"W3162840096","doi":"10.1016/j.jde.2021.04.032","title":"On the Melnikov functions and limit cycles near a double homoclinic loop with a nilpotent saddle of order <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" altimg=\"si1.svg\"> <mml:mover accent=\"true\"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy=\"false\">ˆ</mml:mo> </mml:mrow> </mml:mover> </mml:math>","year":2021,"lang":"lv","type":"article","venue":"Journal of Differential Equations","topic":"Advanced Differential Equations and Dynamical Systems","field":"Mathematics","cited_by":12,"is_retracted":false,"has_abstract":false,"ca_institutions":"Western University","funders":"China Scholarship Council; Western University; Natural Science Foundation of Hebei Province; Natural Sciences and Engineering Research Council of Canada; National Natural Science Foundation of China","keywords":"Homoclinic orbit; Mathematics; Nilpotent; Saddle; Loop (graph theory); Limit (mathematics); Order (exchange); Quotient; Hamiltonian (control theory); Combinatorics; Mathematical analysis; Mathematical physics; Bifurcation; Physics; Quantum mechanics","routes":{"ca_aff":true,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0004609085,0.0007503465,0.0005475477,0.002143696,0.001564794,0.001833891,0.00100193,0.0008793125,0.004704671],"category_scores_gemma":[0.001965185,0.0003059536,0.0005085471,0.0005359887,0.002128952,0.001750433,0.001453914,0.0009384991,0.0006667515],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001339858,"about_ca_system_score_gemma":0.0007395113,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001997397,"about_ca_topic_score_gemma":0.001331019,"domain_scores_codex":[0.9998796,0.00003592423,0.000005215981,0.00001985497,0.00003029586,0.00002912119],"domain_scores_gemma":[0.9995253,0.0001172481,0.00009083758,0.00002704942,0.00007470365,0.0001648754],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.00008072091,0.00003886841,0.0005028108,0.00005796592,0.00001791848,0.0003676241,0.0002815441,0.01582102,0.002347217,0.9737453,0.001793584,0.004945374],"study_design_scores_gemma":[0.00003088502,0.00005013546,0.0006063912,0.0000356524,0.000008842162,0.0001946292,0.0002310079,0.1570775,0.0007252266,0.8390827,0.001926799,0.00003023798],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.7941595,0.001205701,0.1010557,0.002005245,0.0003181201,0.00007989139,0.000145397,0.0003073764,0.1007231],"genre_scores_gemma":[0.9820017,0.0003634485,0.005941271,0.0001150106,0.0001177905,0.00005505703,0.00009166922,0.0001010064,0.01121302],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.004704671,"threshold_uncertainty_score":0.01573873,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03068827632530236,"score_gpt":0.2641730029308158,"score_spread":0.2334847266055134,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}