Averaging methods of arbitrary order, periodic solutions and integrability. (English) Zbl 1336.34062
Authors’ abstract: 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 \(C^1\) 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.
Reviewer: Adriana Buică (Cluj-Napoca)
MSC:
| 34C29 | Averaging method for ordinary differential equations |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
Keywords:
periodic differential systems; averaging methods; limit cycles; integrability; polynomial differential systemsReferences:
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