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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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