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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

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

Citations:

Zbl 0514.34032
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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.
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