Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T07:21:35.140Z Has data issue: false hasContentIssue false

Periodic oscillations of forced pendulums with very small length*

Published online by Cambridge University Press:  14 November 2011

Alessandro Fonda
Affiliation:
Dipartimento di Scienze Matematiche, Università di Trieste, P.le Europa 1, 34127 Trieste, Italy
Fabio Zanolin
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze 208, 33100 Udine, Italy

Abstract

We prove the existence of an arbitrarily large number of periodic solutions for a class of nonlinear differential equations generalising the dynamics of a forced pendulum with small length.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Battelli, F. and Palmer, K. J.. Chaos in the Duffing equation. J. Differential Equations 101 (1993), 276301.CrossRefGoogle Scholar
2Ding, T., lannacci, R. and Zanolin, F.. Existence and multiplicity results for periodic solutions of semilinear Duffing equations. J. Differential Equations 105 (1993), 364409.CrossRefGoogle Scholar
3Ding, W.. A generalization of the Poincaré-Birkhoff theorem. Proc. Amer. Math. Soc. 88(1983), 341–6.Google Scholar
4Fonda, A. and Willem, M.. Subharmonic oscillations of forced pendulum-type equations. J. Differential Equations 81 (1989), 215–20.CrossRefGoogle Scholar
5Hammerstein, A.. Eine nichtlineare Randwertaufgabe. Jahresber. Deutsch. Math.-Verein. 39 (1930), 5964.Google Scholar
6Hartman, P.. Ordinary Differtential Equations (New York: Wiley, 1964).Google Scholar
7Iglish, R.. Uber die Losungen des Dufflngschen Schwingungsproblems bei grossen Parameterwerten. Math. Ann. 111 (1936), 568–81.CrossRefGoogle Scholar
8Lazer, A. C.. Small periodic perturbations of a class of conservative systems. J. Differential Equations 13(1973), 438–56.CrossRefGoogle Scholar
9Loud, W. S.. Periodic solutions of x” + cx' + g(x) = tf(t). Mem. Amer. Math. Soc. 31 (Providence, RI: AMS, 1959).Google Scholar
10Martinez-Amores, P., Mawhin, J., Ortega, R. and Willem, M.. Generic results for the existence of nondegenerate periodic solutions of some differential systems with periodic nonlinearities. J. Differential Equations 91 (1991), 138–48.CrossRefGoogle Scholar
11Mawhin, J.. The forced pendulum: a paradigm for nonlinear analysis and dynamical systems. Exposition Math. 6 (1988), 271–87.Google Scholar
12Opial, Z.. Sur les solutions périodiques de l'équation différentielle x)” + g(x)= p(t). Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 151–6.Google Scholar
13Palmer, K. J.. Exponential dichotomies and transversal homoclinic points. J. Differential Equations 55(1984), 225–6.CrossRefGoogle Scholar
14Rebelo, C. and Zanolin, F.. Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities. Trans. Amer. Math. Soc. 348 (1996), 2349–89.CrossRefGoogle Scholar
15Willem, M.. Perturbations of non degenerate periodic orbits of Hamiltonian systems. In Periodic Solutions of Hamiltonian Systems and Related Topics, eds Rabinowitz, P. et al. 261–6 (Boston: Reidel Publishing Company, 1987).CrossRefGoogle Scholar