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Subharmonic oscillations with prescribed minimal period for a class of Hamiltonian systems. (English) Zbl 0872.34022

The authors consider the classical pendulum equation \(\ddot x+A\sin x=f(t)\) and the following second order Hamiltonian system \[ \ddot x+V_x(t,x)=0, \qquad x(0)= x(pT), \qquad \dot x(0)=\dot x(pT). \] Their main goal is to establish conditions for existence of subharmonic solutions with prescribed minimal period. To this end variational methods are employed and solutions are obtained as critical points of certain functionals defined in appropriate Hilbert spaces.
Reviewer: V.Petrov (Plovdiv)

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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[1] Birkhoff, G. D.; Lewis, D. C., On the periodic motions near a given periodic motion of a dynamical system, Ann. Mat. Pura Appl., 12, 117-133 (1933) · Zbl 0007.37104
[2] Moser, J., Proof of a generalized fixed point theorem due to G. D. Birkhoff, (Lecture Notes in Mathematics, 597 (1977), Springer: Springer New York) · Zbl 0358.58009
[3] Rabinowitz, P. H., On subharmonic solutions of Hamiltonian systems, Communs pure appl. Math., 33, 609-633 (1980) · Zbl 0425.34024
[4] Clarke, F.; Ekeland, I., Hamiltonian trajectories having prescribed minimal period, Communs pure appl. Math., 33, 103-116 (1980) · Zbl 0403.70016
[5] Clarke, F.; Ekeland, I., Nonlinear oscillations and boundary value problem for hamiltonian systems, Archs ration. Mech. Analysis, 78, 315-333 (1982) · Zbl 0514.34032
[6] Benci, V.; Fortunato, D., A Birkhoff-Lewis type result for a class of Hamiltonian systems, Manuscripta Math., 59, 441-456 (1987) · Zbl 0616.58014
[7] Benci, V.; Fortunato, D., Subharmonic solutions of prescribed minimal period for nonautonomous differential equations, (Proc. Recent Advances in Hamiltonian Systems (L’Aquila) (1987)), 83-96
[8] Ekeland, I.; Hofer, H., Periodic solutions with prescribed minimal period for convex autonomous Hamil-tonian systems, Invent. Math., 81, 155-188 (1985) · Zbl 0594.58035
[9] Ekeland, I.; Hofer, H., Subharmonics for convex nonautonomous Hamiltonian systems, Communs pure appl. Math., 40, 1-36 (1987) · Zbl 0601.58035
[10] Ekeland, I.; Lasry, J. M., On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. Math., 112, 283-319 (1980) · Zbl 0449.70014
[11] Fonda, A.; Lazer, A. C., Subharmonic solutions of conservative systems with nonconvex potentials, (Proc. Am. math. Soc., 115 (1992)), 183-190 · Zbl 0752.34027
[12] Fonda, A.; Ramos, M.; Willem, M., Subharmonic solutions for second order differential equations, Topo. Meth. Nonlinear Analysis, 1, 49-66 (1993) · Zbl 0803.34029
[13] Fonda, A.; Willem, M., Subharmonic oscillations of forced pendulum type equations, J. diff. Eqns, 81, 215-220 (1989) · Zbl 0708.34028
[14] Hirano N., Subharmonic solutions for second order differential systems, preprint.; Hirano N., Subharmonic solutions for second order differential systems, preprint. · Zbl 0847.34047
[15] Liu, J.-Q.; Wang, Z.-Q., Remarks on subharmonics with minimal periods of Hamiltonian systems, Nonlinear Analysis, 20, 803-821 (1993) · Zbl 0789.58030
[16] Mawhin, J.; Willem, M., Multiple solutions of the periodic boundary value problem for some forced pendulum type equations, J. diff. Eqns, 52, 264-287 (1984) · Zbl 0557.34036
[17] Michalek, R.; Tarantello, G., Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems, J. diff. Eqns, 72, 28-55 (1988) · Zbl 0645.34038
[18] Solimini, S.; Terracini, S., A multiplicity result for periodic solutions of autonomous conservative systems, Nonlinear Analysis, 16, 903-916 (1992) · Zbl 0744.34042
[19] Tarantello, G., Subharmonic solutions for Hamiltonian systems via a \(Z_p\) pseudoindex theory, Ann. Sci. Ec. Norm. Sup., 357-409 (1989) · Zbl 0755.34035
[20] Willem, M., Periodic oscillations of odd second order Hamiltonian systems, Bollettino U.M.I., 3-B, 293-304 (1984) · Zbl 0582.58014
[21] Willem, M., Subharmonic oscillations of convex Hamiltonian systems, Nonlinear Analysis, 9, 1303-1311 (1985) · Zbl 0579.34030
[22] Conley, C.; Zehnder, E., The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold, Invent. Math., 73, 33-49 (1983) · Zbl 0516.58017
[23] Conley, C.; Zehnder, E., Subharmonic solutions and Morse theory, Phisica, 124A, 649-658 (1984) · Zbl 0605.58015
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