×

zbMATH — the first resource for mathematics

On forced oscillations in groups of interacting nonlinear systems. (English) Zbl 1353.70050
Summary: Consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them. Each subsystem is assumed to be a massive point moving with friction on a compact surface, possibly with a boundary, in an external periodic field. We present sufficient conditions for the existence of a periodic solution for the whole system. The result is illustrated by a series of examples including a chain of strongly coupled pendulums in a periodic field.

MSC:
70K40 Forced motions for nonlinear problems in mechanics
70K42 Equilibria and periodic trajectories for nonlinear problems in mechanics
34C25 Periodic solutions to ordinary differential equations
37K60 Lattice dynamics; integrable lattice equations
70F40 Problems involving a system of particles with friction
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alonso, J. M.; Ortega, R., Global asymptotic stability of a forced Newtonian system with dissipation, J. Math. Anal. Appl., 196, 3, 965-986, (1995) · Zbl 0844.34047
[2] Benci, V.; Degiovanni, M., Periodic solutions of dissipative dynamical systems, (Variational Methods, (1990), Springer), 395-411
[3] Drábek, P.; Invernizzi, S., Periodic solutions for systems of forced coupled pendulum-like equations, J. Differential Equations, 70, 3, 390-402, (1987) · Zbl 0652.34049
[4] Flach, S.; Willis, C. R., Discrete breathers, Phys. Rep., 295, 5, 181-264, (1998)
[5] Furi, M., Second order differential equations on manifolds and forced oscillations, (Topological Methods in Differential Equations and Inclusions, (1995), Springer), 89-127 · Zbl 0843.58008
[6] Furi, M.; Pera, M., On the existence of forced oscillations for the spherical pendulum, Boll. Unione Mat. Ital., 4, 2, 381-390, (1990) · Zbl 0711.70025
[7] Furi, M.; Pera, M. P., The forced spherical pendulum does have forced oscillations, (Delay Differential Equations and Dynamical Systems, (1991), Springer), 176-182 · Zbl 0736.34031
[8] Hamel, G., Über erzwungene schwingungen bei endlichen amplituden, (Festschrift David Hilbert zu Seinem Sechzigsten Geburtstag am 23, Januar 1922, (1922), Springer), 326-338 · JFM 48.0519.03
[9] Kartashov, Y. V.; Malomed, B. A.; Torner, L., Solitons in nonlinear lattices, Rev. Modern Phys., 83, 1, 247, (2011)
[10] Marin, J.; Aubry, S., Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit, Nonlinearity, 9, 6, 1501, (1996) · Zbl 0926.70028
[11] Marlin, J., Periodic motions of coupled simple pendulums with periodic disturbances, Int. J. Non-Linear Mech., 3, 4, 439-447, (1968) · Zbl 0169.55605
[12] Mawhin, J., Une généralisation de théorèmes de J.A. marlin, Int. J. Non-Linear Mech., 5, 2, 335-339, (1970) · Zbl 0202.09501
[13] Mawhin, J., Global results for the forced pendulum equation, (Handbook of Differential Equations: Ordinary Differential Equations 1, (2000)), 533-589 · Zbl 1091.34019
[14] Polekhin, I., Forced oscillations of a massive point on a compact surface with a boundary, Nonlinear Anal. TMA, 128, 100-105, (2015) · Zbl 1372.70023
[15] Srzednicki, R.; Wójcik, K.; Zgliczyński, P., Fixed point results based on the ważewski method, (Handbook of Topological Fixed Point Theory, (2005), Springer), 905-943 · Zbl 1079.37012
[16] Toda, M., Theory of nonlinear lattices, vol. 20, (2012), Springer Science & Business Media
[17] Wazewski, T., Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math., 20, 279-313, (1947) · Zbl 0032.35001
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.