×

On periodic motions of lattices of Toda type via critical point theory. (English) Zbl 0809.34056

The ordinary differential systems of Toda type are studied, in which the potential of the interaction between the \(n\)th and the \((n+1)\)st particle is superquadratic at \(-\infty\) and has weaker growth at \(+\infty\). Under fixed end points condition as well as the identified end points condition, the authors prove that there exist infinitely many \(T\)- periodic solutions for every \(T>0\). The global variational method and the relative \(S^ 1\)-index theory are used.

MSC:

34C25 Periodic solutions to ordinary differential equations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arnold, V. I., Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russ. Math. Surv. 18, 1963, pp. 9-36. · Zbl 0129.16606
[2] Ambrosetti, A. & Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14, 1973, pp. 349-381. · Zbl 0273.49063
[3] Benci, V., A geometrical index for the group S 1 and some applications to the study of periodic solutions, Comm. Pure Appl. Math. 34, 1981, pp. 393-432. · Zbl 0447.34040
[4] Benci, V., On the critical point theory for indefinite functions in the presence of symmetries, Trans. Amer. Math. Soc. 274, 1982, pp. 533-572. · Zbl 0504.58014
[5] Berestycki, H., Lasry, J.-M., Mancini, G. & Ruf, B., Existence of multiple periodic orbits on star-shaped hamiltonian systems, Comm. Pure Appl. Math. 38, 1985, pp. 253-289. · Zbl 0569.58027
[6] Fadell, E. & Husseini, S., Relative cohomological index theories, Advances Math. 64, 1987, pp. 1-31. · Zbl 0619.58012
[7] Fadell, E., Husseini, S. & Rabinowitz, P. H., Borsuk-Ulam theorems for arbitrary S 1 actions and applications, Trans. 274, 1982, 345-361. · Zbl 0506.58010
[8] Fadell, E. & Rabinowitz, P. H., Generalized cohomological index theories for Lie group actions with applications to bifurcation questions for Hamiltonian systems, Invent. Math. 45, 1978, pp. 139-174. · Zbl 0403.57001
[9] Krasnoselskii, M. A., Topological methods in the theory of nonlinear integral equations, Pergamon Press, 1964.
[10] Moser, J., On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, Kl. 2, 1962, p. 1.
[11] Mawhin, J. & Willem, M., Critical point theory and Hamiltonian systems, Springer, 1989. · Zbl 0676.58017
[12] Nirenberg, L., Variational methods in nonlinear problems, Lecture Notes in Mathematics 1365, M. Giaquinta (ed.), Springer-Verlag, 1987. · Zbl 0679.58021
[13] Rabinowitz, P. H., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31, 1978, pp. 157-184. · Zbl 0369.70017
[14] Toda, M., Theory of Nonlinear Lattices, Springer-Verlag, 1989. · Zbl 0694.70001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.