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Applications numériques de la dualité en mécanique Hamiltonienne. (Numerical applications of duality in Hamiltonian mechanics). (French) Zbl 0624.70015
This paper is concerned with the numerical search of periodic solutions of Hamiltonian systems. It has been proved by F. H. Clarke and I. Ĕkeland [Commun. Pure Appl. Math. 33, 103-116 (1980; Zbl 0403.70016)] that under certain assumptions a Hamiltonian system of n degrees of freedom has at least n distinct periodic solutions contained in each variety of positive constant energy. Later the same result was also proved by A. Ambrosetti and G. Mancini [J. Differ. Equations. 43, 249-256 (1982; Zbl 0492.70018)]. Here the author, following the ideas of the Ambrosetti and Mancini proof, has designed a constructive method to compute this periodic solutions. The convergence of the proposed algorithms is rigorously proved.
Finally this method is applied to a Hamiltonian system of two degrees of freedom whose behavior has been studied extensively by M. Henon [Numerical exploration of Hamiltonian systems (1983; Zbl 0578.70019)].
Reviewer: M.Calvo
MSC:
70H05 Hamilton’s equations
65L05 Numerical methods for initial value problems
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C25 Periodic solutions to ordinary differential equations
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References:
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