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.