{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:39:45Z","timestamp":1787236785829,"version":"build-2736575974"},"reference-count":69,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2020M670965"],"award-info":[{"award-number":["2020M670965"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007567","name":"City University of Hong Kong","doi-asserted-by":"publisher","award":["7004848"],"award-info":[{"award-number":["7004848"]}],"id":[{"id":"10.13039\/100007567","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002878","name":"Consejer\u00eda de Econom\u00eda, Innovaci\u00f3n, Ciencia y Empleo, Junta de Andaluc\u00eda","doi-asserted-by":"publisher","award":["FQM-276"],"award-info":[{"award-number":["FQM-276"]}],"id":[{"id":"10.13039\/501100002878","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002878","name":"Consejer\u00eda de Econom\u00eda, Innovaci\u00f3n, Ciencia y Empleo, Junta de Andaluc\u00eda","doi-asserted-by":"publisher","award":["TIC-0130"],"award-info":[{"award-number":["TIC-0130"]}],"id":[{"id":"10.13039\/501100002878","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002878","name":"Consejer\u00eda de Econom\u00eda, Innovaci\u00f3n, Ciencia y Empleo, Junta de Andaluc\u00eda","doi-asserted-by":"publisher","award":["UHU-1260150"],"award-info":[{"award-number":["UHU-1260150"]}],"id":[{"id":"10.13039\/501100002878","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002878","name":"Consejer\u00eda de Econom\u00eda, Innovaci\u00f3n, Ciencia y Empleo, Junta de Andaluc\u00eda","doi-asserted-by":"publisher","award":["P12-FQM-1658"],"award-info":[{"award-number":["P12-FQM-1658"]}],"id":[{"id":"10.13039\/501100002878","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC2018-096265-B-I00"],"award-info":[{"award-number":["PGC2018-096265-B-I00"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["MTM2017-87915-C2-1-P"],"award-info":[{"award-number":["MTM2017-87915-C2-1-P"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["Nos 12001110"],"award-info":[{"award-number":["Nos 12001110"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009018","name":"Shanghai Municipal Human Resources and Social Security Bureau","doi-asserted-by":"publisher","award":["2019173"],"award-info":[{"award-number":["2019173"]}],"id":[{"id":"10.13039\/501100009018","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Based on the nonlinear time transformation method, in this paper we propose a recursive algorithm for arbitrary order approximation of heteroclinic orbits. This approach works fine for a wide class of systems that are perturbations of non-Hamiltonian integrable planar vector fields. Specifically, our method can provide an approximation up to any desired order, both for the locus where the heteroclinic bifurcation occurs in the parameter space and for the orbit in the phase space. We also give proofs that guarantee the existence and uniqueness of the solution. As illustrations, we apply it to the generic Hopf-zero and nonresonant double Hopf singularities which give rise to an intricate bifurcation scenario in the cases where a heteroclinic cycle appears in the corresponding unfolding of normal forms. The obtained high-order approximations are expressed in terms of symbolic nonzero normal form coefficients which improve the existing first-order approximations in the literature. They are also in excellent agreement with numerical continuations. Note that the derived formulas can be practically important in many control engineering applications in which certain constants of the problem may not be known in advance.<\/jats:p>","DOI":"10.1137\/20m1325101","type":"journal-article","created":{"date-parts":[[2021,3,11]],"date-time":"2021-03-11T10:06:11Z","timestamp":1615457171000},"page":"403-437","source":"Crossref","is-referenced-by-count":9,"title":["High-Order Approximation of Heteroclinic Bifurcations in Truncated 2D-Normal Forms for the Generic Cases of Hopf-Zero and Nonresonant Double Hopf Singularities"],"prefix":"10.1137","volume":"20","author":[{"given":"B. W.","family":"Qin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0153-8801","authenticated-orcid":true,"given":"K. W.","family":"Chung","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Algaba","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9959-0789","authenticated-orcid":true,"given":"A. J.","family":"Rodr\u00edguez-Luis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,3,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-019-05025-2"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-019-04941-7"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2020.132384"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2019.105012"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498001583"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127499000936"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/12\/4\/324"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008328027179"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127407018178"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2015.02.017"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-006-9051-y"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127400000190"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2019.01.066"},{"key":"atypb14","unstructured":"A. Algaba, M. Merino, and A. J. Rodr\u00edguez-Luis,\n                      Evolution of Arnold's tongues in a $\\mathbb{Z}_2$-symmetric electronic circuit\n                      , IEICE T. Fund. Electr., E82-A (1999), pp. 1714-1721."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127401002286"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2019.105162"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-005-0736-z"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008316010341"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008353609572"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700002613"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/14\/3\/310"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-011-9990-9"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2004.03.004"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127400000360"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/S1631-073X(03)00043-8"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsv.2008.11.015"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-009-9489-9"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7462(91)90066-3"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"S. N. Chow, C. Li, and D. Wang,\n                      Normal Forms and Bifurcation of Planar Vector Fields\n                      , Cambridge University Press, Cambridge, UK, 1994,https:\/\/doi.org\/10.1017\/CBO9780511665639.","DOI":"10.1017\/CBO9780511665639"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-96-01501-2"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1080\/13873950701742754"},{"key":"atypb32","unstructured":"E. J. Doedel, A. R. Champneys, F. Dercole, T. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, R. Paffenroth, B. Sandstede, X. Wang, and C. Zhang,\n                      AUTO-$07$P: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont)\n                      , Technical report, Concordia University, Montreal, QC, Canada, 2012."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2010.12.013"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/11\/4\/015"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2013.33.4435"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127402006175"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(93)90276-7"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/26\/10\/2809"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/28\/2\/311"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2012.11.004"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1137\/141000403"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1137\/18M1210769"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1002\/mma.4569"},{"key":"atypb44","doi-asserted-by":"crossref","unstructured":"J. Guckenheimer and P. Holmes,\n                      Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields\n                      , Springer-Verlag, New York, 1983,https:\/\/doi.org\/10.1007\/978-1-4612-1140-2.","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012900372714"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7462(00)00116-5"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7462(00)00117-7"},{"key":"atypb48","first-page":"379","author":"Homburg A. J.","year":"2010","journal-title":"New York"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1111\/j.1749-6632.1980.tb29712.x"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012995290288"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1137\/0141010"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1137\/110839461"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/10\/5\/007"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2004.832199"},{"key":"atypb55","doi-asserted-by":"crossref","unstructured":"Y. A. Kuznetsov,\n                      Elements of Applied Bifurcation Theory\n                      , Springer-Verlag, New York, 2004,https:\/\/doi.org\/10.1007\/978-1-4757-3978-7.","DOI":"10.1007\/978-1-4757-3978-7"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2005.02.019"},{"key":"atypb57","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-018-1597-8"},{"key":"atypb58","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2018.08.003"},{"key":"atypb59","doi-asserted-by":"crossref","unstructured":"L. M. Perko,\n                      Differential Equations and Dynamical Systems\n                      , Springer-Verlag, New York, 2001,https:\/\/doi.org\/10.1007\/978-1-4613-0003-8.","DOI":"10.1007\/978-1-4613-0003-8"},{"key":"atypb60","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2020.125042"},{"key":"atypb61","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127420500789"},{"key":"atypb62","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127420500170"},{"key":"atypb63","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-020-05575-w"},{"key":"atypb64","doi-asserted-by":"publisher","DOI":"10.1063\/1.5030692"},{"key":"atypb65","doi-asserted-by":"publisher","DOI":"10.1137\/140968197"},{"key":"atypb66","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/16\/6\/316"},{"key":"atypb67","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-010-9844-x"},{"key":"atypb68","doi-asserted-by":"crossref","unstructured":"S. Wiggins,\n                      Introduction to Applied Nonlinear Dynamical Systems and Chaos\n                      , Springer-Verlag, New York, 2003,https:\/\/doi.org\/10.1007\/b97481.","DOI":"10.1007\/b97481"},{"key":"atypb69","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijnonlinmec.2004.07.015"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1325101","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:52:43Z","timestamp":1787233963000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1325101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":69,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1325101"],"URL":"https:\/\/doi.org\/10.1137\/20m1325101","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}