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On the Rapidly Forced Pendulum

  • Prem Kumar N. Swami
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 63)

Abstract

The rapidly forced pendulum equation \( \mathop x\limits^{..} + \sin x = \delta \sin t/\varepsilon \), with δ = δ 0 ε p , p ≥ −2 and δ 0, ε sufficiently small, is considered. We sketch our proof that the term proportional to δ 2 in the splitting distance d(t 0) in the (\( \mathop x\limits^. \),t) plane has the form
$$ \delta ( - \frac{\pi }{4}\varepsilon \sec h\frac{\pi }{{2\varepsilon }} + O({\varepsilon ^2}\exp ( - \pi /2\varepsilon )))\sin \frac{{2{t_0}}}{\varepsilon. }$$

From this it follows that for −2 ≤ p < −1, the Melnikov term πδ sin \( \frac{1}{e}\) t 0 sech \( \frac{1}{{2e}}\pi \) is dominated by the O(δ 2)term as ε ↓0.

Keywords

Periodic Solution Half Plane Fourier Coefficient Unstable Manifold Stable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    V.G. Gelfreich, Separatrices splitting for the rapidly forced pendulum, preprint.Google Scholar
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    A paper with the same title and result but with the sharper condition p > 3 (instead of p > 5) is available in preprint form from the authors of [7].Google Scholar
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Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • Prem Kumar N. Swami
    • 1
  1. 1.Department of MathematicsUniversity of ToledoToledoUSA

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