For analytic differential systems x˙=Ax+f(x) in Fn with F=R or C, we study the varieties of their partial integrability in a neighborhood of the origin. We also prove the existence of analytic ...normalizations of partially integrable differential systems.
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.
► 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.
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