Averaging methods of arbitrary order, periodic solutions and integrability

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.The first author is partially supported by a MICINN/FEDER grant number MTM2014-53703-P and by a Generalitat de Catalunya grant number 2014SGR 1204. The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014SGR-568, the grants FP7-PEOPLE-2012-IRSES 318999 and 316338, from the recruitment program of high–end foreign experts of China. The third and fourth authors are partially supported by NNSF of China grant number 11271252, by RFDP of Higher Education of China grant number 20110073110054, and by FP7-PEOPLE-2012-IRSES-316338 of Europe. The fourth author also is supported by the innovation program of Shanghai Municipal Education Commission grant 15ZZ01