Rotating solutions and stability of parametric pendulum by perturbation method

S. Lenci, E. Pavlovskaia, G. Rega, M. Wiercigroch

Research output: Contribution to journalArticlepeer-review

85 Citations (Scopus)


Rotating solutions of a parametrically driven pendulum are studied via a perturbation method by assuming the undamped unforced Hamiltonian case as basic solution, and damping and excitation as small perturbations. The existence and stability of the harmonic solution are determined analytically. First-order terms are mainly considered, but the extensions to higher-order terms, as well as to subliarmonic rotations, are straightforward. Setting up the analysis in the phase space instead of in the physical space has facilitated development of simple but comprehensive formulas. Comparison with numerical simulations shows that the analytical results are very effective in accurate predictions, even for fairly large amplitudes, and in detecting the saddle-node bifurcation where the rotating solution is born. A limited accuracy is observed, to first order, in detecting the period-doubling bifurcation where it loses stability, and an explanation is proposed. (c) 2007 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)243-259
Number of pages17
JournalJournal of Sound and Vibration
Issue number1-2
Early online date21 Sept 2007
Publication statusPublished - 5 Feb 2008

Bibliographical note

The support of the British Council, Italian MIUR and Italian CRUI, and by the Royal Society of London is gratefully acknowledged.


  • excited pendulum
  • forced pendulum
  • driven pendulum
  • chaos
  • bifurcations
  • orbits
  • dynamics
  • systems
  • motion


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