Abstract:
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.
Keywords:
autoresonance, capture into resonance, adiabatic invariant, pendulum.
The work was supported in part by the Russian Foundation for Basic Research (project no. 13-01-00251) and Russian Federation Presidential Program for the State Support of Leading Scientific Schools (project NSh-2519.2012.1). The work of A.V.A. and V.A.A. was also partially supported by the Russian Academy of Science (OFN-15).
Citation:
Anatoly I. Neishtadt, Alexey A. Vasiliev, Anton V. Artemyev, “Capture into Resonance and Escape from it in a Forced Nonlinear Pendulum”, Regul. Chaotic Dyn., 18:6 (2013), 686–696
\Bibitem{NeiVasArt13}
\by Anatoly I. Neishtadt, Alexey A. Vasiliev, Anton V. Artemyev
\paper Capture into Resonance and Escape from it in a Forced Nonlinear Pendulum
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 6
\pages 686--696
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\crossref{https://doi.org/10.1134/S1560354713060087}
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Linking options:
https://www.mathnet.ru/eng/rcd159
https://www.mathnet.ru/eng/rcd/v18/i6/p686
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