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An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. (English) Zbl 0765.70016
Summary: The measure of the splitting of the separatrices of the rapidly forced pendulum \(\ddot x+\sin x=\mu \sin(t/\varepsilon)\) is considered as a model problem that has been studied by different authors. Here \(\varepsilon\), \(\mu\) are small parameters, \(\varepsilon>0\), but otherwise independent. The following formula for the angle \(\alpha\) between separatrices is established: \(\alpha=\pi\mu[2\varepsilon\;\text{cosh}(\pi/2\varepsilon)]^{-1}[1+O(\mu,\varepsilon^ 2)]\). This formula is also valid for the particular case \(\mu=\varepsilon^ p\), with \(p>0\), \(\varepsilon>0\), and agrees with the one provided by the first order Poincaré-Melnikov theory that cannot be applied directly, due to the exponentially small dependence of \(\alpha\) on the parameter \(\varepsilon\).

MSC:
70K40 Forced motions for nonlinear problems in mechanics
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
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