Integrability and Pseudo‐Linearizable Conditions in a Quasi‐Analytic System

Feng Li,Yusen Wu

doi:10.1155/2012/783546

Abstract

This paper deals with the problems of integrability and linearizable conditions at degenerate singular point in a class of quasianalytic septic polynomial differential system. We solve the problems by an indirect method, that is, we transform the quasianalytic system into an analytic system firstly, and the degenerate singular point into an elementary singular point. Then we calculate the singular values at the origin of the analytic system by the known classical methods. We obtain the center conditions and isochronous center conditions. Accordingly, integrability and pseudolinearizable conditions at degenerate singular point in the quasianalytic system are obtained. Especially, when λ = 1, the system has been studied in Wu and Zhang (2010).

Highlights

In the qualitative theory of planar polynomial differential equations, one of open problems for planar polynomial differential systems dx dtP x, y, dt Q x, y, is how to characterize their centers and isochronous centers
We investigate integrability and linearizable conditions at degenerate singular point for a class of quasanalytic polynomial differential system dx dt δx − y x2
√ −1, 2.3 where r, θ are complex numbers, system 2.1 can be transformed into dρ dt iρ ∞ 2k 1 α β k aα,β−1 − bβ,α−1 ei α−β θρk, 2.4 dθ dt 1∞ 2k 1 α β k aα,β−1 bβ,α−1 ei α−β θρk

Summary

Introduction

In the qualitative theory of planar polynomial differential equations, one of open problems for planar polynomial differential systems dx dtP x, y , dt Q x, y , is how to characterize their centers and isochronous centers. We compute the singular point quantities and derive the center conditions of the origin for the transformed system.

Results

Conclusion