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The Hopf-van der Pol System: Failure of the Homotopy Method
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The purpose of this article to provide an explicit example where continuation based on the homotopy method fails. The example is a one-parameter homotopy for periodic orbits between two well-known nonlinear systems, the normal form of the Hopf bifurcation and the van der Pol system. Our analysis shows that various types of obstructions can make approximation over the whole range of the homotopy parameter impossible. The Hopf-van der Pol system demonstrates that homotopy methods may fail even for seemingly
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<div class="JournalArticle Publication short-list"> <div class="authors"> <span class="author-name" > Meijer, H. G. E. </span> <span class="author-type"> </span> ; <span class="author-name" mtid="10048329"> <a href="/gui2/?type=authors&mode=browse&sel=10048329" target="_blank">Kalmár-Nagy, T.</a> </span> <span class="author-type"> </span> </div ><div class="title"><a href="/gui2/?mode=browse¶ms=publication;2827587" mtid="2827587" target="_blank">The Hopf-van der Pol System: Failure of the Homotopy Method</a></div> <div class="pub-info"> <span class="journal-title">DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS</span> <span class="journal-volume">20</span> : <span class="journal-issue">3</span> <span class="page"> pp. 323-328. , 6 p. </span> <span class="year">(2012)</span> </div> <div class="pub-end"><div class="identifier-list"> <span class="identifiers"> <span class="id identifier oa_none" title="none"> <a style="color:black" title="10.1007/s12591-011-0091-5" target="_blank" href="https://doi.org/10.1007/s12591-011-0091-5"> DOI </a> </span> <span class="id identifier oa_none" title="none"> <a style="color:black" title="84858700715" target="_blank" href="http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84858700715"> Scopus </a> </span> <span class="id identifier oa_none" title="none"> <a style="color:black" title="10765152280831072530" target="_blank" href="http://scholar.Google.com/scholar?hl=en&lr=&cluster=10765152280831072530"> Google scholar </a> </span> </span> </div> <div class="short-pub-prop-list"> <span class="short-pub-mtid"> Közlemény:2827587 </span> <span class="status-holder"><span class="status-data status-ADMIN_APPROVED"> Admin láttamozott </span></span> <span class="pub-core">Forrás </span> <span class="pub-type">Folyóiratcikk (Szakcikk ) </span> <!-- && !record.category.scientific --> <span class="pub-category">Tudományos</span> <div class="publication-citation" style="margin-left: 0.5cm;"> <span title="Nyilvános idézőközlemények összesen, említések nélkül" class="citingPub-count">Nyilvános idéző összesen: 8</span> | Független: 7 | Függő: 1 | Nem jelölt: 0 | WoS jelölt: 7 | Scopus jelölt: 6 | WoS/Scopus jelölt: 7 | DOI jelölt: 7 </div> </div> </div> </div><div class="JournalArticle Publication long-list"> <div class="authors"> <img title="Forrásközlemény" style="float: left" src="/frontend/resources/grid/publication-core-icon.png"> <div class="autype autype0"> <span class="author-name" >Meijer H. G. E. </span> ; <span class="author-name" mtid="10048329"><a href="/gui2/?type=authors&mode=browse&sel=10048329" target="_blank">Kalmár-Nagy T. (<span class="authorship-author-name">Kalmár-Nagy Tamás</span> <span class="authorAux-mtmt"> Áramlástan, Elméleti és alkalmazott mechanika</span>) </a> </span> <span class="author-affil"><span title="Budapesti Műszaki és Gazdaságtudományi Egyetem">BME</span>/<span title="Gépészmérnöki Kar">GPK</span>/Áramlástan Tanszék</span> </div> </div> <div class="title"><a href="/gui2/?mode=browse¶ms=publication;2827587" target="_blank">The Hopf-van der Pol System: Failure of the Homotopy Method</a></div> <div> <span class="journal-title">DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS</span> <span class="journal-issn">(<a target="_blank" href="https://portal.issn.org/resource/ISSN/0971-3514">0971-3514</a> <a target="_blank" href="https://portal.issn.org/resource/ISSN/0974-6870">0974-6870</a>)</span>: <span class="journal-volume">20</span> <span class="journal-issue">3</span> <span class="page"> pp 323-328 </span> <span class="year">(2012)</span> </div> <div class="pub-footer"> <span class="language" xmlns="http://www.w3.org/1999/html">Nyelv: Angol | </span> <span class="identifiers"> <span class="id identifier oa_none" title="none"> <a style="color:black" title="10.1007/s12591-011-0091-5" target="_blank" href="https://doi.org/10.1007/s12591-011-0091-5"> DOI </a> </span> <span class="id identifier oa_none" title="none"> <a style="color:black" title="84858700715" target="_blank" href="http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84858700715"> Scopus </a> </span> <span class="id identifier oa_none" title="none"> <a style="color:black" title="10765152280831072530" target="_blank" href="http://scholar.Google.com/scholar?hl=en&lr=&cluster=10765152280831072530"> Google scholar </a> </span> </span> <div class="publication-citation" style="margin-left: 0.5cm;"> <span title="Nyilvános idézőközlemények összesen, említések nélkül" class="citingPub-count">Nyilvános idéző összesen: 8</span> | Független: 7 | Függő: 1 | Nem jelölt: 0 | WoS jelölt: 7 | Scopus jelölt: 6 | WoS/Scopus jelölt: 7 | DOI jelölt: 7 </div> <div class="mtid"><span class="long-pub-mtid">Közlemény: 2827587</span> | <span class="status-data status-ADMIN_APPROVED"> Admin láttamozott </span> <span class="oldId">Régi azonosító: 2827587</span> | Forrás | <span class="type-subtype">Folyóiratcikk ( Szakcikk ) </span> | <span class="pub-category">Tudományos</span> | <span class="publication-sourceOfData">kézi felvitel</span> </div> <div class="lastModified">Utolsó módosítás: 2017.01.12. 15:29 Nagy László (BME GPK Áramlástan admin 5) </div> </div></div>