Hamiltonian Dynamical Systems pp 363-372 | Cite as

# On the Rapidly Forced Pendulum

Conference paper

## 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

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### References

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*Separatrices splitting for the rapidly forced pendulum*, preprint.Google Scholar - [2]P. Holmes, J. Marsden and J. Scheurle,
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## Copyright information

© Springer-Verlag New York, Inc. 1995