A forced pendulum equation with many periodic solutions. (English) Zbl 0899.34031

The author shows that if \(a\) is a positive real number and \(n\) an integer greater than or equal to \(1\), then there exists \(p(t)\) satisfying \(p\in L^1(\mathbb{R}/T\mathbb{Z})\), and \(\int^T_0 p(t)dt= 0\), such that the forced pendulum equation \(x''+ a\sin x= p(t)\) has at least \(2n\) \(T\)-periodic solutions which are geometrically different.
Reviewer: O.Akinyele (Bowie)


34C25 Periodic solutions to ordinary differential equations
Full Text: DOI Link


[1] P.W. Bates, Reduction theorems for a class of semilinear equations at resonance , Proc. Amer. Math. Soc. 84 (1982), 73-78. · Zbl 0496.47052
[2] H.T. Davis, Introduction to nonlinear differential and integral equations , Dover, New York, 1962. · Zbl 0106.28904
[3] F. Donati, Sur l’existence de quatre solutions périodiques pour l’équation du pendule forcé , C.R. Acad. Sci. Paris, Ser. I. Math, 317 (1993), 667-672. · Zbl 0783.34033
[4] J. Hale, Ordinary differential equations , R.E. Krieger, Malabar, 1980.
[5] W.S. Loud, Periodic solutions of \(x^\pp+cx^\p+g(x)= \e f(t)\) , Mem. Amer. Math. Soc. 31 (1959), 1-58. · Zbl 0085.30701
[6] J. Mawhin, The forced pendulum equation : A paradigm for nonlinear analysis and dynamical systems , Expo. Math. 6 (1988), 271-287. · Zbl 0668.70028
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.