Averaging theory at any order for computing periodic orbits

We provide a recurrence formula for the coefficients of the powers of a in the series expansion of the solutions around epsilon = 0 of the perturbed first-order differential equations. Using it, we give an averaging theory at any order in epsilon for the following two kinds of analytic differential equation: dx/d theta = Sigma(k >= 1) epsilon F-k(k)(theta, x), dx/d theta = Sigma(k >= 0) epsilon F-k(k)(theta,x). A planar polynomial differential system with a singular point at the origin can be transformed, using polar coordinates, to an equation of the previous form. Thus, we apply our results for studying the limit cycles of a planar polynomial differential systems. (C) 2013 Elsevier B.V. All rights reserved.