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Melnikov-Arnold Integral


\begin{displaymath}
A_m(\lambda) \equiv \int_{-\infty}^\infty \cos\left[{{\textstyle{1\over 2}}m\phi(t)-\lambda t}\right]dt,
\end{displaymath}

where the function

\begin{displaymath}
\phi(t) \equiv 4\tan^{-1}(e^t)-\pi
\end{displaymath}

describes the motion along the pendulum Separatrix. Chirikov (1979) has shown that this integral has the approximate value

\begin{displaymath}
A_m(\lambda)\approx\cases{
{4\pi (2\lambda)^{m-1}\over\Gamm...
...rt l\vert)^{m+1}}\Gamma(m+1)\sin(\pi m) & for $\lambda<0$.\cr}
\end{displaymath}


References

Chirikov, B. V. ``A Universal Instability of Many-Dimensional Oscillator Systems.'' Phys. Rep. 52, 264-379, 1979.




© 1996-9 Eric W. Weisstein
1999-05-26