Computation of periodic solutions to pendulum type systems with a small parameter

Periodic solutions of pendulum-type ODE systems are considered. Finding such solutions is a classical problem in mechanics. Numerous methods are available for computing periodic solutions, and these methods have existed as long as the problems themselves. However, they were designed for manual calculation, and attempts to program them in computer algebra systems (CAS) are sometimes ineffective. For computing such solutions, we propose a method intended for CAS. The method is based on the use of high-order variational equations and symbolic differentiation. It is shown on a number of examples that all computations are reduced to operations with polynomials.