{"id":"https://openalex.org/W7224098038","doi":"https://doi.org/10.48550/arxiv.2610.36398","title":"Asymptotic Analysis of the Biased van der Pol Equation","display_name":"Asymptotic Analysis of the Biased van der Pol Equation","publication_year":2026,"publication_date":"2026-10-26","ids":{"openalex":"https://openalex.org/W7224098038","doi":"https://doi.org/10.48550/arxiv.2610.36398"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.36398","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.36398","pdf_url":"https://arxiv.org/pdf/2610.36398","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.36398","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082183533","display_name":"Alain J. Brizard","orcid":"https://orcid.org/0000-0002-0192-6273","observed_orcids":["https://orcid.org/0000-0002-0192-6273"]},"institutions":[{"id":"https://openalex.org/I55801708","display_name":"Saint Michael's College","ror":"https://ror.org/03b353k89","country_code":"US","type":"education","lineage":["https://openalex.org/I55801708"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Brizard, Alain J.","raw_affiliation_strings":["Department of Physics , Saint Michael's College , Colchester , VT 05439 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , Saint Michael's College , Colchester , VT 05439 , USA","institution_ids":["https://openalex.org/I55801708"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5160010526","display_name":"Desmond L. Carrico","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I55801708","display_name":"Saint Michael's College","ror":"https://ror.org/03b353k89","country_code":"US","type":"education","lineage":["https://openalex.org/I55801708"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Carrico, Desmond L.","raw_affiliation_strings":["Department of Physics , Saint Michael's College , Colchester , VT 05439 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , Saint Michael's College , Colchester , VT 05439 , USA","institution_ids":["https://openalex.org/I55801708"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I55801708"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Singular Perturbation, Asymptotics and Difference Schemes","score":0.9580000042915344,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12727","display_name":"Singular Perturbation, Asymptotics and Difference Schemes","score":0.9580000042915344,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.035100001841783524,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.0007999999797903001,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/relaxation-oscillations","display_name":"relaxation oscillations","score":0.8859999775886536},{"id":"https://openalex.org/keywords/large-amplitude-oscillations","display_name":"large-amplitude oscillations","score":0.734000027179718},{"id":"https://openalex.org/keywords/stable-limit-cycle","display_name":"stable limit cycle","score":0.8190000057220459},{"id":"https://openalex.org/keywords/singular-perturbation-analysis","display_name":"singular perturbation analysis","score":0.6570000052452087},{"id":"https://openalex.org/keywords/oscillation-period","display_name":"oscillation period","score":0.7860000133514404},{"id":"https://openalex.org/keywords/nonlinear-oscillators","display_name":"nonlinear oscillators","score":0.4189999997615814}],"concepts":[{"id":"https://openalex.org/C98010927","wikidata":"https://www.wikidata.org/wiki/Q2081515","display_name":"Van der Pol oscillator","level":3,"score":0.8748999834060669},{"id":"https://openalex.org/C165160513","wikidata":"https://www.wikidata.org/wiki/Q7524249","display_name":"Singular perturbation","level":2,"score":0.6675999760627747},{"id":"https://openalex.org/C205147927","wikidata":"https://www.wikidata.org/wiki/Q752718","display_name":"Asymptotic analysis","level":2,"score":0.5648000240325928},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5227000117301941},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4763999879360199},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.46560001373291016},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4433000087738037},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4426000118255615},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.43380001187324524},{"id":"https://openalex.org/C174256460","wikidata":"https://www.wikidata.org/wiki/Q911364","display_name":"Perturbation theory (quantum mechanics)","level":2,"score":0.4327000081539154},{"id":"https://openalex.org/C135402231","wikidata":"https://www.wikidata.org/wiki/Q898440","display_name":"Dissipation","level":2,"score":0.39739999175071716},{"id":"https://openalex.org/C2778439541","wikidata":"https://www.wikidata.org/wiki/Q7106412","display_name":"Oscillation (cell signaling)","level":2,"score":0.3716000020503998},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3538999855518341},{"id":"https://openalex.org/C119047807","wikidata":"https://www.wikidata.org/wiki/Q752718","display_name":"Asymptotic analysis","level":2,"score":0.31349998712539673},{"id":"https://openalex.org/C93357160","wikidata":"https://www.wikidata.org/wiki/Q1545916","display_name":"Limit cycle","level":3,"score":0.2946000099182129},{"id":"https://openalex.org/C135854075","wikidata":"https://www.wikidata.org/wiki/Q1421688","display_name":"Relaxation oscillator","level":4,"score":0.2879999876022339},{"id":"https://openalex.org/C11683690","wikidata":"https://www.wikidata.org/wiki/Q752726","display_name":"Asymptotic expansion","level":2,"score":0.28189998865127563},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.26669999957084656},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.26409998536109924},{"id":"https://openalex.org/C97118885","wikidata":"https://www.wikidata.org/wiki/Q565636","display_name":"Frequency analysis","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.25839999318122864},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.25690001249313354}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.36398","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.36398","pdf_url":"https://arxiv.org/pdf/2610.36398","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.36398","doi":"https://doi.org/10.48550/arxiv.2610.36398","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.36398","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.36398","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.36398","pdf_url":"https://arxiv.org/pdf/2610.36398","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7224098038.pdf","grobid_xml":"https://content.openalex.org/works/W7224098038.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2570866504","https://openalex.org/W2040340891","https://openalex.org/W4253490608","https://openalex.org/W2331740684","https://openalex.org/W2130698148","https://openalex.org/W2790784512","https://openalex.org/W2210456795","https://openalex.org/W1979749276","https://openalex.org/W4252313065","https://openalex.org/W1582177716"],"abstract_inverted_index":{"We":[0],"perform":[1],"an":[2,53],"asymptotic":[3],"analysis":[4],"of":[5,60,71],"the":[6,22,30,58,72],"biased":[7],"van":[8],"der":[9],"Pol":[10],"equation":[11],"$\\ddot{x}":[12],"-":[13,15],"\u03bd\\,(1":[14],"x^{2})\\,\\dot{x}":[16],"+":[17],"x":[18],"=":[19],"a$,":[20],"where":[21],"dissipation":[23],"parameter":[24,32],"is":[25,34,64,67],"large":[26],"$(\u03bd\\gg":[27],"1)$":[28],"and":[29,40],"bias":[31],"$a$":[33],"a":[35,48],"real":[36],"number":[37],"between":[38],"$-1$":[39],"$1$,":[41],"corresponding":[42],"to":[43],"large-amplitude":[44],"relaxation":[45,61],"oscillations":[46,62],"along":[47],"stable":[49],"limit":[50],"cycle.":[51],"Using":[52],"abridged":[54],"singular":[55],"perturbation":[56],"analysis,":[57],"period":[59],"$T(a,\u03bd)$":[63],"obtained":[65],"that":[66],"accurate":[68],"within":[69],"$0.1\\%$":[70],"numerical":[73],"results":[74],"for":[75],"$\u03bd\\gg":[76],"1$.":[77]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-29T00:00:00"}