{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:30:26Z","timestamp":1787229026375,"version":"build-2736575974"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100009193","name":"Marsden Fund","doi-asserted-by":"publisher","award":["21-UOA-048"],"award-info":[{"award-number":["21-UOA-048"]}],"id":[{"id":"10.13039\/501100009193","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2026,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>The usual method to analyze the stability of heteroclinic cycles and networks is with transition matrices that are derived from return maps. In this paper, we introduce an extension to this methodology, the projected map, which we define by identifying trajectories that have, in a certain sense, qualitatively the same dynamics. The projected map is a discrete, piecewise-smooth map of one dimension fewer than the order of the transition matrix. We use these maps to describe the dynamics of trajectories near three heteroclinic networks in [Formula: see text] with four equilibria. We find in all three cases that the onset of trajectories that switch between cycles of the network is caused by either a fold bifurcation or a border-collision bifurcation, where fixed points of the map no longer exist in the corresponding function\u2019s domain of definition. We are able to show that a given initial condition near any of these three networks is asymptotic to only one subcycle and cannot switch between subcycles multiple times, resolving a 30-year-old claim by Brannath. We are also able to generalize certain results to all quasi-simple networks, proving that a border-collision bifurcation of the projected map, corresponding to a condition on the eigenvectors of certain transition matrices, causes a cycle to lose stability.<\/jats:p>","DOI":"10.1137\/24m1704944","type":"journal-article","created":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T15:28:02Z","timestamp":1772724482000},"page":"588-628","source":"Crossref","is-referenced-by-count":0,"title":["Analysis of Dynamics near Heteroclinic Networks in \\(\\boldsymbol{\\mathbb{R}^{4}}\\) with a Projected Map"],"prefix":"10.1137","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7031-828X","authenticated-orcid":true,"given":"David C. Groothuizen","family":"Dijkema","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Auckland, Auckland, 1142 New Zealand."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7519-1145","authenticated-orcid":true,"given":"Vivien","family":"Kirk","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Auckland, Auckland, 1142 New Zealand."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0654-6736","authenticated-orcid":true,"given":"Claire M.","family":"Postlethwaite","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Auckland, Auckland, 1142 New Zealand."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,3,5]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/29\/5\/1645"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/18\/1\/019"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-019-09566-z"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2013.09.006"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/7\/5\/006"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/ad03cf"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/21M1435215"},{"key":"ref8","first-page":"1","volume":"39","author":"Castro S. 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