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Melnikov-Arnold Integral

A_m(lambda)=int_(-infty)^inftycos[1/2mphi(t)-lambdat]dt,
(1)

where the function

phi(t)=4tan^(-1)(e^t)-pi
(2)

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

A_m(lambda) approx {(4pi(2lambda)^(m-1))/(Gamma(m))e^(-pilambda/2) for lambda>0; -(4e^(-pi|lambda|/2))/((2|l|)^(m+1))Gamma(m+1)sin(pim) for lambda<0.
(3)

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References

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

Referenced on Wolfram|Alpha

Melnikov-Arnold Integral

Cite this as:

Weisstein, Eric W. "Melnikov-Arnold Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Melnikov-ArnoldIntegral.html

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