{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:16:23Z","timestamp":1775052983336,"version":"3.50.1"},"reference-count":48,"publisher":"SAGE Publications","issue":"14","license":[{"start":{"date-parts":[[2011,12,16]],"date-time":"2011-12-16T00:00:00Z","timestamp":1323993600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Vibration and Control"],"published-print":{"date-parts":[[2012,12]]},"abstract":"<jats:p> In this paper we consider a complex-order forced van der Pol oscillator. The complex derivative [Formula: see text], with \u03b1, \u03b2\u2009\u2208\u2009\u211d<jats:sup>+<\/jats:sup>, is a generalization of the concept of an integer derivative, where \u03b1\u2009=\u20091, \u03b2\u2009=\u20090. The Fourier transforms of the periodic solutions of the complex-order forced van der Pol oscillator are computed for various values of parameters such as frequency \u03c9 and amplitude b of the external forcing, the damping \u03bc, and parameters \u03b1 and \u03b2. Moreover, we consider two cases: (i) b\u2009=\u20091, \u03bc\u2009=\u2009{1.0, 5.0, 10.0}, and \u03c9\u2009=\u2009{0.5, 2.46, 5.0, 20.0}; (ii) \u03c9\u2009=\u200920.0, \u03bc\u2009=\u2009{1.0, 5.0, 10.0}, and b\u2009=\u2009{1.0, 5.0, 10.0}. We verified that most of the signal energy is concentrated in the fundamental harmonic \u03c9<jats:sub>0<\/jats:sub>. We also observed that the fundamental frequency of the oscillations \u03c9<jats:sub>0<\/jats:sub> varies with \u03b1 and \u03bc. For the range of tested values, the numerical fitting led to logarithmic approximations for system (7) in the two cases (i) and (ii). In conclusion, we verify that by varying the parameter values \u03b1 and \u03b2 of the complex-order derivative in expression (7), we accomplished a very effective way of perturbing the dynamical behavior of the forced van der Pol oscillator, which is no longer limited to parameters b and \u03c9. <\/jats:p>","DOI":"10.1177\/1077546311429150","type":"journal-article","created":{"date-parts":[[2011,12,18]],"date-time":"2011-12-18T01:35:11Z","timestamp":1324172111000},"page":"2201-2209","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":28,"title":["Complex-order forced van der Pol oscillator"],"prefix":"10.1177","volume":"18","author":[{"given":"Carla MA","family":"Pinto","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica da Universidade do Porto\r        and Department of Mathematics, Institute of Engineering of Porto, Portugal"}]},{"given":"JA 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