Romano, M. Étude numérique des solutions périodiques d’une équation du second ordre. (Numerical study of the periodic solutions of a second order equation). (French. English summary) Zbl 0761.65053 RAIRO, Modélisation Math. Anal. Numér. 26, No. 4, 493-506 (1992). Dans cet article, l’auteur propose une méthode de Newton pour déterminer numériquement un grand nombre de sous-harmoniques (ou solutions \(2k\pi\) périodiques avec \(k\in \mathbb{N}\), \(k\geq 2\)) de l’équation \(\ddot X+V(X)=\cos t\) avec \(V(X)=\alpha^{-1}| X|^ \alpha\).Il utilise pour cela la méthode d’action duale introduite par F. H. Clarke et I. Ekeland [Arch. Ration. Mech. Anal. 78, 315-333 (1982; Zbl 0514.34032)] dont la méthode est rappelée. Suivant l’index des sous-harmoniques et de nombreux résultats numériques. Reviewer: M.Sibony (Tours) MSC: 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 37-XX Dynamical systems and ergodic theory 34C25 Periodic solutions to ordinary differential equations Keywords:periodic solutions; second order equation; subquadratic Hamiltonian system; dual action principle; subharmonics; solutions index Citations:Zbl 0514.34032 PDFBibTeX XMLCite \textit{M. Romano}, RAIRO, Modélisation Math. Anal. Numér. 26, No. 4, 493--506 (1992; Zbl 0761.65053) Full Text: DOI EuDML References: [1] F. CLARKE et I. EKELAND, Nonlinear oscillation and boundary-value problems for Hamiltonian Systems, Arch. Rational Mech. Anal., An 78, 1982, p. 315-333. Zbl0514.34032 MR653545 · Zbl 0514.34032 · doi:10.1007/BF00249584 [2] I. EKELAND, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag. Zbl0707.70003 MR1051888 · Zbl 0707.70003 [3] I. EKELAND et H. HOFER, Subharmonics for convex Nonautonomous Hamiltonian Systems, Comm. Pure Appl. Math. 40 (1987) 419-467. Zbl0601.58035 MR865356 · Zbl 0601.58035 · doi:10.1002/cpa.3160400102 [4] J. MAWHIN et M. WHILEM, Critical point and Hamiltonian mechanics, Springer-Verlag. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.