An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. (English) Zbl 0765.70016

Delshams, Amadeu; Seara, Teresa M.

Commun. Math. Phys. 150, No. 3, 433-463 (1992).

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

Keywords:

angle between separatrices; first order Poincaré-Melnikov theory

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