AN ASYMPTOTIC SYMMETRY OF THE RAPIDLY FORCED PENDULUM

[摘要] The inhomogeneous differential equation x + sin x = delta sin(t + t0)/epsilon describes the motion of a sinusoidally forced pendulum. The orbits that connect the two saddle points of the unforced (delta = 0) pendulum are called separatrices. If epsilon = O(1), then one can use Melinikov's method to show that these separatrices can split for weak forcing (delta << 1), and that the perturbed motion is chaotic. If epsilon << 1, Melinikov's method fails because the perturbation term is not analytic in epsilon at epsilon = 0. In this paper we show that for delta << 1 and epsilon << 1, the solution of the perturbed problem exhibits a symmetry to all orders in an asymptotic expansion. From the asymptotic expansion it follows that the separatrices split by an amount that is at most transcendentally small. This proof differs from that of Holmes, Marsden and Scheurle.

[发布日期] 1991-08-01