Averaging theory at any order for computing periodic orbits

We provide a recurrence formula for the coefficients of the powers of ε in the series expansion of the solutions around ε = 0 of the perturbed first-order differential equations. Using it, we give an averaging theory at any order in ε for the following two kinds of analytic differential equation: d x d θ = ∑ k ≥ 1 ε k F k ( θ , x ) , d x d θ = ∑ k ≥ 0 ε k F k ( θ , 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.