{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:10:26Z","timestamp":1760058626476,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T00:00:00Z","timestamp":1745193600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Agencia Estatal de Investigaci\u00f3n of Spain","award":["PID2022-136613NB-100","2021SGR00113"],"award-info":[{"award-number":["PID2022-136613NB-100","2021SGR00113"]}]},{"name":"AGAUR (Generalitat de Catalunya)","award":["PID2022-136613NB-100","2021SGR00113"],"award-info":[{"award-number":["PID2022-136613NB-100","2021SGR00113"]}]},{"name":"Reial Acad\u00e8mia de Ci\u00e8ncies i Arts de Barcelona","award":["PID2022-136613NB-100","2021SGR00113"],"award-info":[{"award-number":["PID2022-136613NB-100","2021SGR00113"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The 3-dimensional modified van der Pol\u2013Duffing oscillator has been studied by several authors. We complete its study, first characterizing its zero-Hopf equilibria and then its zero-Hopf bifurcations\u2014i.e., we provide sufficient conditions for the existence of three, two or one periodic solutions, bifurcating from the zero-Hopf equilibrium localized at the origin of coordinates. Recall that an equilibrium point of a 3-dimensional differential system whose eigenvalues are zero and a pair of purely imaginary eigenvalues is a zero-Hopf equilibrium. Finally, we determine the dynamics of this system near infinity, i.e., we control the orbits that escape to or come from the infinity.<\/jats:p>","DOI":"10.3390\/axioms14040321","type":"journal-article","created":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T20:38:26Z","timestamp":1745267906000},"page":"321","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Dynamics of a Modified van der Pol\u2013Duffing Oscillator"],"prefix":"10.3390","volume":"14","author":[{"given":"Oscar A. R.","family":"Cespedes","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidad Distrital Francisco Jos\u00e9 de Caldas, La Candelaria, Bogot\u00e1 111611, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9511-5999","authenticated-orcid":false,"given":"Jaume","family":"Llibre","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra, Catalonia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.chaos.2004.12.029","article-title":"Adaptive synchronization of two chaotic systems consisting of modified Van der Pol-Duffing and Chua oscillators","volume":"26","author":"Fotsin","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1363","DOI":"10.1016\/j.chaos.2004.09.101","article-title":"Adaptive synchronization of a modified and uncertain chaotic Van der Pol-Duffing oscillator based on parameter identification","volume":"24","author":"Fotsin","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"149563","DOI":"10.1155\/2009\/149563","article-title":"Bifurcation analysis of a Van der Pol-Duffing circuit with parallel resistor","volume":"2009","author":"Braga","year":"2009","journal-title":"Math. 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Soc."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/321\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:18:47Z","timestamp":1760030327000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/321"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,21]]},"references-count":18,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040321"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040321","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,4,21]]}}}