Periodic Orbits in Trigonometric Series

Abstract

A method is given for the study of families of periodic motion using trigonometric series to represent the individual solutions. The continuous deformations of a periodic orbit along a family are represented by the variations of the trigonometric coefficients with respect to a parameter.For the applications the series are truncated, while the coefficients and their variations are determined numerically. In this form, the continuation with respect to the parameter is given by a mapping f: R" R" of the space of coefficients into itself. The values of the coefficients are then improved by another mapping which is a contraction operator in some neighborhood of the fixed point representing the solution.The method is applied to the natural families defined by Wintner and to the families of the first and second kinds of Poincare in the restricted problem of three bodies.