Nonlinear Dynamics

, Volume 77, Issue 4, pp 1617–1627 | Cite as

Homoclinic, subharmonic, and superharmonic bifurcations for a pendulum with periodically varying length

  • Anton O. Belyakov
  • Alexander P. Seyranian
Original Paper


Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the simplest model of child’s swing. Melnikov’s analysis is carried out to find bifurcations of homoclinic, subharmonic oscillatory, and subharmonic rotational orbits. For the analysis of superharmonic rotational orbits, the averaging method is used and stability of obtained approximate solution is checked. The analytical results are compared with numerical simulation results.


Homoclinic bifurcation Rotational orbits Averaging method Parametric excitation 



This research was partly supported by the Austrian Science Fund (FWF) under Grant P25979-N25 and by the Russian Foundation for Basic Research (RFBR), Grant No. 13-01-00261.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institute of MechanicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.ORCOS, Institute of Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria

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