Archiv der Mathematik

, Volume 102, Issue 5, pp 459–468 | Cite as

Subharmonic solutions of the forced pendulum equation: a symplectic approach



Using the Poincaré–Birkhoff fixed point theorem, we prove that for every β > 0 and for a large (both in the sense of prevalence and of category) set of continuous and T-periodic functions \({f: \mathbb{R} \to \mathbb{R}}\) with \({\int_0^T f(t)\,dt = 0}\), the forced pendulum equation
$$x'' + \beta \sin x = f(t) $$
has a subharmonic solution of order k for every large integer number k. This improves the well known result obtained with variational methods, where the existence when k is a (large) prime number is ensured.

Mathematics Subject Classification (2010)

Primary 34C25 


Subharmonic solutions Pendulum equation Poincaré–Birkhoff theorem 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanoItaly
  2. 2.Departamento de Matemática AplicadaUniversidad de GranadaGranadaSpain
  3. 3.Dipartimento di Matematica e InformaticaUniversità di UdineUdineItaly

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