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Separatrix Splitting of a quasi periodically forced pendulum

Contributed by: G. Gallavotti (Roma 1), Nov. 6, 1998.

Abstract:   The degeneracy of the pendulum separatrix is generically removed by a perturbation. The splitting may be hard to compute: this is exemplified by the fact that, in certain simple cases in which the force is quasiperiodic and with a fast frequency (say \h-1/2 times that of the pendulum small oscillations with \h small), it can be determined by a series (in the expansion in powers of the perturbation strength \e) convergent in a domain which may be much larger than the domain of perturbation strengths in which one can derive an expression asymptotic as the rapidity of the forcing frequency tends to $\infty$ (i.e. \h \to 0 ). This is so because the splitting is determined by taking the asymptotic form as \h \to 0 of each perturbation order in \e and the power series of the leading terms is only known to be convergent in a domain smaller than the estimate on the radius of convergence itself. The problem discussed here deals with finding an aymptotic expression for the splitting in a region of values of the coupling \e in which the series in \e for the splitting still converges, but no asymptotics as \h\to 0 can be immediately derived from it, because the series of the terms leading at each order does not (seem to) converge.

Further details are provided in the following TeX document.

    TeX file        postscript (PS)        PDF file

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giovanni.gallavotti@roma1.infn.it (contributor),
aizenman@princeton.edu (editor)