Averaging methods of arbitrary order, periodic solutions and integrability
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In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods. Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability. For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number $N$ such that if its first $N$ averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for $n=2$. Furthermore, in a punctured neighborhood of the origin, the system is $C^\infty$ completely integrable for $n>2$ provided that each periodic orbit has a trivial holonomy. Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities.