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Journal of Dynamics and Differential Equations

, Volume 25, Issue 3, pp 627–652 | Cite as

Complex Dynamics in Pendulum-Type Equations with Variable Length

  • Alessandro Margheri
  • Carlota Rebelo
  • Fabio Zanolin
Article

Abstract

We prove the existence of complex dynamics for a generalized pendulum type equation with variable length. The solutions we find switch from an oscillatory behavior around the stable vertical position to a rotational type behavior crossing the unstable position with positive or negative velocity following any prescribed two-sided sequence of symbols. Moreover, to any periodic sequence of symbols corresponds a periodic solution of the equation. The proof is based on a topological approach and the results are robust with respect to small perturbations. In particular a small friction term can be added to the equation.

Keywords

Pendulum type equations Periodic solutions Complex dynamics 

Notes

Acknowledgments

Alessandro Margheri—Supported by FCT, Financiamento Base 2010 ISFL-1-209 and project PTDC/MAT/113383/2009. Carlota Rebelo—Supported by FCT, Financiamento Base 2010 ISFL-1-209 and project PTDC/MAT/113383/2009. Fabio Zanolin—The author acknowledges the support of the PRIN project “Equazioni Differenziali Ordinarie e Applicazioni”.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alessandro Margheri
    • 1
  • Carlota Rebelo
    • 1
  • Fabio Zanolin
    • 2
  1. 1.Departamento de MatemáticaFac. Ciências de Lisboa, Campo GrandeLisboaPortugal
  2. 2.Department of Mathematics and Computer ScienceUniversity of UdineUdineItaly

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