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.