{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T22:09:52Z","timestamp":1777932592445,"version":"3.51.4"},"reference-count":16,"publisher":"SAGE Publications","issue":"9-10","license":[{"start":{"date-parts":[[2007,9,1]],"date-time":"2007-09-01T00:00:00Z","timestamp":1188604800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Vibration and Control"],"published-print":{"date-parts":[[2007,9]]},"abstract":"<jats:p>In this paper a modified version of the classical Van der Pol oscillator is proposed, introducing fractional-order time derivatives into the state-space model. The resulting fractional-order Van der Pol oscillator is analyzed in the time and frequency domains, using phase portraits, spectral analysis and bifurcation diagrams. The fractional-order dynamics is illustrated through numerical simulations of the proposed schemes using approximations to fractional-order operators. Finally, the analysis is extended to the forced Van der Pol oscillator.<\/jats:p>","DOI":"10.1177\/1077546307077463","type":"journal-article","created":{"date-parts":[[2007,9,27]],"date-time":"2007-09-27T08:03:13Z","timestamp":1190880193000},"page":"1291-1301","source":"Crossref","is-referenced-by-count":114,"title":["Analysis of the Van der Pol Oscillator Containing Derivatives of Fractional Order"],"prefix":"10.1177","volume":"13","author":[{"given":"Ramiro S.","family":"Barbosa","sequence":"first","affiliation":[{"name":"Institute of Engineering of Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4200-072 Porto, Portugal"}]},{"given":"J.A. Tenreiro","family":"Machado","sequence":"additional","affiliation":[{"name":"Institute of Engineering of Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4200-072 Porto, Portugal,"}]},{"given":"B.M.","family":"Vinagre","sequence":"additional","affiliation":[{"name":"Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, Badajoz, Spain"}]},{"given":"A.J.","family":"Calder\u00f3n","sequence":"additional","affiliation":[{"name":"Escuela de Ingenier\u00edas Industriales, Universidad de Extremadura, Badajoz, Spain"}]}],"member":"179","published-online":{"date-parts":[[2007,9,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(02)00438-1"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1142\/4507"},{"key":"atypb3","volume-title":"Proceedings of the IEEE International Conference on Computational Cybernetics(ICCC'04)","author":"Barbosa, R.S."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/9.159595"},{"issue":"8","key":"atypb5","first-page":"485","volume":"42","author":"Hartley, T.T.","year":"1995","journal-title":"Theory and Applications"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1142\/3779"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1109\/9.995039"},{"issue":"1","key":"atypb8","first-page":"47","volume":"4","author":"Machado, J.A.T.","year":"2001","journal-title":"Fractional Calculus & Applied Analysis"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2003.12.039"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1006\/jsvi.2001.4172"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1006\/jsvi.2002.5170"},{"key":"atypb12","volume-title":"The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order","author":"Oldham, K.B.","year":"1974"},{"key":"atypb13","volume-title":"La D\u00e9rivation Non Enti\u00e8re: Th\u00e9orie, Synth\u00e8se et Applications","author":"Oustaloup, A.","year":"1995"},{"key":"atypb14","volume-title":"Proceedings of the First IFAC Workshop on Fractional Differentiation and Its Applications (FDA'04)","author":"Pereira, E."},{"key":"atypb15","volume-title":"Fractional Differential Equations","author":"Podlubny, I.","year":"1999"},{"issue":"3","key":"atypb16","first-page":"231","volume":"3","author":"Vinagre, B.M.","year":"2000","journal-title":"Fractional Calculus & Applied Analysis"}],"container-title":["Journal of Vibration and Control"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/1077546307077463","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/1077546307077463","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T01:24:19Z","timestamp":1777685059000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/1077546307077463"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9]]},"references-count":16,"journal-issue":{"issue":"9-10","published-print":{"date-parts":[[2007,9]]}},"alternative-id":["10.1177\/1077546307077463"],"URL":"https:\/\/doi.org\/10.1177\/1077546307077463","relation":{},"ISSN":["1077-5463","1741-2986"],"issn-type":[{"value":"1077-5463","type":"print"},{"value":"1741-2986","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,9]]}}}