Non–continuation of the periodic oscillations of a forced pendulum in the presence of friction
HTML articles powered by AMS MathViewer
- by Rafael Ortega, Enrico Serra and Massimo Tarallo PDF
- Proc. Amer. Math. Soc. 128 (2000), 2659-2665 Request permission
Abstract:
A well known theorem says that the forced pendulum equation has periodic solutions if there is no friction and the external force has mean value zero. In this paper we show that this result cannot be extended to the case of linear friction.References
- José Miguel Alonso, Nonexistence of periodic solutions for a damped pendulum equation, Differential Integral Equations 10 (1997), no. 6, 1141–1148. MR 1608049
- Peter W. Bates, Reduction theorems for a class of semilinear equations at resonance, Proc. Amer. Math. Soc. 84 (1982), no. 1, 73–78. MR 633281, DOI 10.1090/S0002-9939-1982-0633281-9
- E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl. (4) 131 (1982), 167–185. MR 681562, DOI 10.1007/BF01765151
- Lucien Guillou, Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff, Topology 33 (1994), no. 2, 331–351 (French). MR 1273787, DOI 10.1016/0040-9383(94)90016-7
- G. Hamel, Über erzwungene Schwingungen bei endlichen Amplituden, Math. Ann., 86 (1922), 1-13.
- Jean Mawhin, Periodic oscillations of forced pendulum-like equations, Ordinary and partial differential equations (Dundee, 1982) Lecture Notes in Math., vol. 964, Springer, Berlin-New York, 1982, pp. 458–476. MR 693131
- J. Mawhin, Recent results on periodic solutions of the forced pendulum equation, Rend. Istit. Mat. Univ. Trieste 19 (1987), no. 2, 119–129 (English, with Italian summary). MR 988376
- Jean Mawhin, The forced pendulum: a paradigm for nonlinear analysis and dynamical systems, Exposition. Math. 6 (1988), no. 3, 271–287. MR 949785
- P. Murthy, Periodic solutions of two-dimensional forced systems: the Massera Theorem and its extensions, J. Dynamics and Diff. Eq., 10 (1998), 275-302.
- R. Ortega, A counterexample for the damped pendulum equation, Acad. Roy. Belg. Bull. Cl. Sci. (5) 73 (1987), no. 10, 405–409. MR 1026970
- Rafael Ortega, A forced pendulum equation with many periodic solutions, Rocky Mountain J. Math. 27 (1997), no. 3, 861–876. MR 1490280, DOI 10.1216/rmjm/1181071898
- E. Serra, M. Tarallo, S. Terracini, Subharmonic solutions to second order differential equations with periodic nonlinearities, Nonlin. Anal. TMA, to appear.
Additional Information
- Rafael Ortega
- Affiliation: Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad deGranada, 18071 Granada, Spain
- Email: rortega@goliat.ugr.es
- Enrico Serra
- Affiliation: Dipartimento di Matematica del Politecnico, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
- Email: serra@polito.it
- Massimo Tarallo
- Affiliation: Dipartimento di Matematica dell’Università, Via Saldini 50, 20133 Milano, Italy
- Email: tarallo@vmimat.mat.unimi.it
- Received by editor(s): October 22, 1998
- Published electronically: February 28, 2000
- Communicated by: Hal L. Smith
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2659-2665
- MSC (1991): Primary 34C25
- DOI: https://doi.org/10.1090/S0002-9939-00-05389-2
- MathSciNet review: 1670407