Complex Dynamics in Pendulum-Type Equations with Variable Length

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.

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Fig. 1
Fig. 2

Notes

  1. 1.

    Recall that \(T_B\) is the time length for which the points are subjected to the effects of system \((Eq_B)\) in the period interval \([0,T].\)

  2. 2.

    Recall that \(T_A\) is the time length for which the points are subjected to the effects of system \((Eq_A)\) in the period \([0,T].\)

  3. 3.

    Here, by “crossing”, we mean go through the set and cross two opposite sides of its boundary.

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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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Margheri, A., Rebelo, C. & Zanolin, F. Complex Dynamics in Pendulum-Type Equations with Variable Length. J Dyn Diff Equat 25, 627–652 (2013). https://doi.org/10.1007/s10884-013-9295-4

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Keywords

  • Pendulum type equations
  • Periodic solutions
  • Complex dynamics