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: 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. © 2013 Elsevier B.V. All rights reserved.
- Averaging theory
- First-order analytic differential equations
- Limit cycles
- Periodic orbits
- Polynomial differential equations