{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T08:04:17Z","timestamp":1649059457113},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2009,2]]},"abstract":"<jats:p> The study of the local behavior of nonlinear systems in the neighborhood of a periodic orbit is a classical problem in nonlinear dynamics. Most of our knowledge stems from simulations or the numerical integration of the variational equation. Only in the case of planar oscillators, the solution of the variational equation can be found analytically, provided that an explicit expression for the periodic trajectory is available. The aim of this paper is to extend a classical theorem due to S. P. Diliberto to higher dimensional systems. In doing so, we show how the fundamental matrix solution to the variational equation of higher order differential equations can be obtained in a closed analytical form. To obtain this result, the knowledge of the periodic trajectory is not sufficient anymore, and a specific set of orthogonal vectors has to be determined. The analysis of some examples reveals that finding these vectors may be easier than solving the variational equations. <\/jats:p>","DOI":"10.1142\/s0218127409023251","type":"journal-article","created":{"date-parts":[[2009,5,8]],"date-time":"2009-05-08T11:52:17Z","timestamp":1241783537000},"page":"629-637","source":"Crossref","is-referenced-by-count":2,"title":["DILIBERTO'S THEOREM IN HIGHER DIMENSION"],"prefix":"10.1142","volume":"19","author":[{"given":"MICHELE","family":"BONNIN","sequence":"first","affiliation":[{"name":"Department of Electronics, Politecnico di Torino, Turin, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"FERNANDO","family":"CORINTO","sequence":"additional","affiliation":[{"name":"Department of Electronics, Politecnico di Torino, Turin, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MARCO","family":"GILLI","sequence":"additional","affiliation":[{"name":"Department of Electronics, Politecnico di Torino, Turin, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","volume-title":"Handbook of Mathematical Functions","author":"Abramowitz M.","year":"1964"},{"key":"rf2","series-title":"Translations of Mathematical Monographs","doi-asserted-by":"crossref","DOI":"10.1090\/mmono\/146","volume-title":"Introduction to Linear Systems of Differential Equations","volume":"146","author":"Adrianova L. 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P.\u00a0Diliberto, Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies\u00a020 (Princeton University Press, 1950)\u00a0pp. 1\u201338.","DOI":"10.1515\/9781400882632-002"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4211-4"},{"key":"rf8","volume-title":"Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields","author":"Guckenheimer J.","year":"1982"},{"key":"rf9","volume-title":"Nonlinear Oscillations in Physical Systems","author":"Hayashi C.","year":"1964"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1137\/1026077"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-69689-3"},{"key":"rf12","volume-title":"Nonlinear Oscillations","author":"Minorsky N.","year":"1962"},{"key":"rf13","volume-title":"Nonlinear Oscillations","author":"Nayfeh A.","year":"1979"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01246445"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22492-2"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127409023251","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:02:42Z","timestamp":1565136162000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127409023251"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,2]]},"references-count":14,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2009,2]]}},"alternative-id":["10.1142\/S0218127409023251"],"URL":"https:\/\/doi.org\/10.1142\/s0218127409023251","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,2]]}}}