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 \(\varepsilon\).


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