The Parametrically Forced Pendulum: A Case Study in 1½ Degree of Freedom

H.W. Broer, I. Hoveijn, M. van Noort, C. Simó, G. Vegter

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This paper is concerned with the global coherent (i.e., non-chaotic) dynamics of the parametrically forced pendulum. The system is studied in a 1½ degree of freedom Hamiltonian setting with two parameters, where a spatio-temporal symmetry is taken into account. Our explorations are restricted to large regions of coherent dynamics in phase space and parameter plane. At any given parameter point we restrict to a bounded subset of phase space, using KAM theory to exclude an infinitely large region with rather trivial dynamics. In the absence of forcing the system is integrable. Analytical and numerical methods are used to study the dynamics in a parameter region away from integrability, where the analytic results of a perturbation analysis of the nearly integrable case are used as a starting point. We organize the dynamics by dividing the parameter plane in fundamental domains, guided by the linearized system at the upper and lower equilibria. Away from integrability some features of the nearly integrable coherent dynamics persist, while new bifurcations arise. On the other hand, the chaotic region increases.
Original languageEnglish
Pages (from-to)897-947
Number of pages51
JournalJournal of dynamics and differential equations
Issue number4
Publication statusPublished - 2004


  • KAM theory
  • Numerical methods
  • Bifurcations
  • Hamiltonian dynamics

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