Regular and Chaotic Dynamics

, Volume 18, Issue 6, pp 686–696 | Cite as

Capture into resonance and escape from it in a forced nonlinear pendulum

  • Anatoly I. NeishtadtEmail author
  • Alexey A. Vasiliev
  • Anton V. Artemyev


We study the dynamics of a nonlinear pendulum under a periodic force with small amplitude and slowly decreasing frequency. It is well known that when the frequency of the external force passes through the value of the frequency of the unperturbed pendulum’s oscillations, the pendulum can be captured into resonance. The captured pendulum oscillates in such a way that the resonance is preserved, and the amplitude of the oscillations accordingly grows. We consider this problem in the frames of a standard Hamiltonian approach to resonant phenomena in slow-fast Hamiltonian systems developed earlier, and evaluate the probability of capture into resonance. If the system passes through resonance at small enough initial amplitudes of the pendulum, the capture occurs with necessity (so-called autoresonance). In general, the probability of capture varies between one and zero, depending on the initial amplitude. We demonstrate that a pendulum captured at small values of its amplitude escapes from resonance in the domain of oscillations close to the separatrix of the pendulum, and evaluate the amplitude of the oscillations at the escape.


autoresonance capture into resonance adiabatic invariant pendulum 

MSC2010 numbers

34E13 70H11 70K30 70K65 


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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • Anatoly I. Neishtadt
    • 1
    • 2
    Email author
  • Alexey A. Vasiliev
    • 1
  • Anton V. Artemyev
    • 1
  1. 1.Space Research InstituteMoscowRussia
  2. 2.Dept. of Math. SciencesLoughborough UniversityLoughborough, LeicestershireUK

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