{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T00:18:55Z","timestamp":1778372335256,"version":"3.51.4"},"reference-count":26,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2018,1,3]],"date-time":"2018-01-03T00:00:00Z","timestamp":1514937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2018,9,30]]},"abstract":"<jats:p>\n            We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing one-dimensional stable and unstable manifolds. The methods are implemented in M\n            <jats:sc>at<\/jats:sc>\n            C\n            <jats:sc>ont<\/jats:sc>\n            M, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits, and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows us to detect all known codimension 1 and 2 bifurcations in three-dimensional (3D) maps, including tangencies and generalized tangencies. M\n            <jats:sc>at<\/jats:sc>\n            C\n            <jats:sc>ont<\/jats:sc>\n            M provides a graphical user interface, enabling interactive control for all computations. As the prime new feature, we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing us to compute all homoclinic orbits to saddle points in 3D maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis, and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features.\n          <\/jats:p>","DOI":"10.1145\/3134443","type":"journal-article","created":{"date-parts":[[2018,1,4]],"date-time":"2018-01-04T16:27:31Z","timestamp":1515083251000},"page":"1-19","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["Numerical Bifurcation Analysis of Homoclinic Orbits Embedded in One-Dimensional Manifolds of Maps"],"prefix":"10.1145","volume":"44","author":[{"given":"Niels","family":"Neirynck","sequence":"first","affiliation":[{"name":"Ghent University, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Willy","family":"Govaerts","sequence":"additional","affiliation":[{"name":"Ghent University, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuri A.","family":"Kuznetsov","sequence":"additional","affiliation":[{"name":"Utrecht University and University of Twente, Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hil G. E.","family":"Meijer","sequence":"additional","affiliation":[{"name":"University of Twente, Enschede, Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2018,1,3]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2014.10.010"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/20\/5\/012"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142995281693"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1134\/S1560354710520023"},{"key":"e_1_2_1_5_1","volume-title":"Bifurcations de points fixes elliptiques\u2014III. Orbites p\u00e9riodiques de \u00abpetites\u00bb p\u00e9riodes et \u00e9limination r\u00e9sonnante des couples de courbes invariantes","author":"Chenciner A.","year":"1987","unstructured":"A. Chenciner . 1987. Bifurcations de points fixes elliptiques\u2014III. Orbites p\u00e9riodiques de \u00abpetites\u00bb p\u00e9riodes et \u00e9limination r\u00e9sonnante des couples de courbes invariantes . Publications Math\u00e9matiques de l\u2019IHES 66 ( 1987 ), 5--91. A. Chenciner. 1987. Bifurcations de points fixes elliptiques\u2014III. Orbites p\u00e9riodiques de \u00abpetites\u00bb p\u00e9riodes et \u00e9limination r\u00e9sonnante des couples de courbes invariantes. Publications Math\u00e9matiques de l\u2019IHES 66 (1987), 5--91."},{"key":"e_1_2_1_6_1","first-page":"623","article-title":"Une description compl\u00e8te du portrait de phase d\u2019un mod\u00e8le d\u2019\u00e9limination r\u00e9sonante","volume":"305","author":"Chenciner A.","year":"1987","unstructured":"A. Chenciner , A. Gasull , and J. Llibre . 1987 . Une description compl\u00e8te du portrait de phase d\u2019un mod\u00e8le d\u2019\u00e9limination r\u00e9sonante . C. R. Acad. Sci. Paris Ser. I Math. 305 , 13 (1987), 623 -- 626 . A. Chenciner, A. Gasull, and J. Llibre. 1987. Une description compl\u00e8te du portrait de phase d\u2019un mod\u00e8le d\u2019\u00e9limination r\u00e9sonante. C. R. Acad. Sci. Paris Ser. I Math. 305, 13 (1987), 623--626.","journal-title":"C. R. Acad. Sci. Paris Ser. I Math."},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/779359.779362"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/030600131"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000075"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(88)90101-6"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1051\/mmnp\/20138504"},{"key":"e_1_2_1_12_1","unstructured":"S. V. Gonchenko and V. S. Gonchenko. 2000. On andronov-hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies. Preprint WIAS No. 556 (2000).  S. V. Gonchenko and V. S. Gonchenko. 2000. On andronov-hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies. Preprint WIAS No. 556 (2000)."},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1134\/S156035470703001X"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1137\/04060487X"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/060653858"},{"key":"e_1_2_1_16_1","volume-title":"Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC\u201911)","author":"Govaerts W.","unstructured":"W. Govaerts , Yu. A. Kuznetsov , H. G. E. Meijer , and N. Neirynck . 2011. A study of resonance tongues near a chenciner bifurcation using matcontm . In Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC\u201911) , D. Bernardini, G. Rega, and F. Romeo (Eds.). Euromech. W. Govaerts, Yu. A. Kuznetsov, H. G. E. Meijer, and N. Neirynck. 2011. A study of resonance tongues near a chenciner bifurcation using matcontm. In Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC\u201911), D. Bernardini, G. Rega, and F. Romeo (Eds.). Euromech."},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1137\/120902860"},{"key":"e_1_2_1_18_1","first-page":"401","article-title":"Algorithme num\u00e9rique d\u00e9finissant la bifurcation d\u2019un point homocline","volume":"293","author":"Kawakami H.","year":"1981","unstructured":"H. Kawakami . 1981 . Algorithme num\u00e9rique d\u00e9finissant la bifurcation d\u2019un point homocline . C. R. Acad. Sci. Paris 293 , 1 (1981), 401 -- 403 . H. Kawakami. 1981. Algorithme num\u00e9rique d\u00e9finissant la bifurcation d\u2019un point homocline. C. R. Acad. Sci. Paris 293, 1 (1981), 401--403.","journal-title":"C. R. Acad. Sci. Paris"},{"key":"e_1_2_1_20_1","first-page":"8","article-title":"Numerical continuation of connecting orbits of maps in Matlab","volume":"15","author":"Khoshsiar Ghaziani R.","year":"2009","unstructured":"R. Khoshsiar Ghaziani , W. Govaerts , Yu. A. Kuznetsov , and H. G. E. Meijer . 2009 . 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Academic Press , New York. Reprint SIAM 2000 . J. M. Ortega and W. C. Rheinboldt. 1970. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York. Reprint SIAM 2000."},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2012.10.017"},{"key":"e_1_2_1_26_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127498001285"},{"key":"e_1_2_1_27_1","first-page":"1560","article-title":"A method to calculate homoclinic points of a two-dimensional noninvertible map","volume":"80","author":"Yoshinaga T.","year":"1997","unstructured":"T. Yoshinaga , H. Kitajima , H. Kawakami , and C. Mira . 1997 . A method to calculate homoclinic points of a two-dimensional noninvertible map . IEICE Trans. Fund. Electron. Commun. Comput. Sci. 80 , 9 (1997), 1560 -- 1566 . T. Yoshinaga, H. Kitajima, H. Kawakami, and C. Mira. 1997. A method to calculate homoclinic points of a two-dimensional noninvertible map. IEICE Trans. Fund. Electron. Commun. Comput. Sci. 80, 9 (1997), 1560--1566.","journal-title":"IEICE Trans. Fund. Electron. Commun. Comput. 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