{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,18]],"date-time":"2024-06-18T17:56:26Z","timestamp":1718733386506},"reference-count":25,"publisher":"World Scientific Pub Co Pte Lt","issue":"10","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2005,10]]},"abstract":"<jats:p> This paper is devoted to the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, submitted to an external periodical excitation of period \u03c4 = 2\u03c0\/\u03c9 , having an amplitude E. In absence of this excitation the equation of Duffing type does not give rise to self oscillations. It may be a model of a series R-L-C electrical circuit with validity conditions in the parameter space. The purpose is essentially an analysis of the harmonics behavior of the period \u03c4 solutions according to points of the parameter plane (\u03c9, E). Let r be the place occupied by a rank-m harmonic from an ordering based on line amplitudes of a frequency spectrum in descending order. Domains of the (\u03c9, E) plane, for which the amplitude of the rank-m harmonic has the place r in the ordering mentioned above, are defined from properties of their boundaries. When r = 2 they are regions of predominance for the rank-m harmonic. When r = 1 they are regions of full predominance for the harmonic m, m = 2,3,4,\u2026, which contain a set of points leading to a resonance for this harmonic. These regions fulfil the following important property: each of them is directly related to a fold bifurcation structure, called rank-mlip (two folds curves joining at two cusps), associated with a well-defined rank-m harmonic. The set of such structures, with m = 2,3,4,\u2026, constitutes an isoordinal lips cascade. As an opening to Part II (to be published) period k\u03c4 solutions related to fractional harmonics behavior are also considered for k = 2,3. <\/jats:p>","DOI":"10.1142\/s0218127405014076","type":"journal-article","created":{"date-parts":[[2005,12,22]],"date-time":"2005-12-22T05:26:18Z","timestamp":1135229178000},"page":"3181-3221","source":"Crossref","is-referenced-by-count":5,"title":["BEHAVIOR OF HARMONICS GENERATED BY A DUFFING TYPE EQUATION WITH A NONLINEAR DAMPING: PART I"],"prefix":"10.1142","volume":"15","author":[{"given":"H\u00c9DI","family":"KHAMMARI","sequence":"first","affiliation":[{"name":"LARA-Ecole Nationale d'Ing\u00e9nieurs de Tunis, BP 37, le Belv\u00e9d\u00e8re, 1002 Tunis, Tunisia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CHRISTIAN","family":"MIRA","sequence":"additional","affiliation":[{"name":"CESNLA, 19 rue d'Occitanie, 31130 Quint, France"},{"name":"Istituto di Scienze Economiche, University of Urbino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JEAN-PIERRE","family":"CARCASS\u00c9S","sequence":"additional","affiliation":[{"name":"11 rue Fouque, 31400 Toulouse, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","first-page":"183","author":"Carcass\u00e8s J. 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