Averaging theory at any order for computing periodic orbits

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

► Formulas for computing bifurcation functions of differential equations are given.
► An explicit expression for the solution of perturbed differential equations is provided.
► We use averaging theory at any order of the perturbation parameter.
► Limit cycles of planar polynomial differential systems are studied.