Bifurcation of limit cycles at degenerate singular point and infinity in a septic system

Yusen Wu,Feng Li

doi:10.1186/1687-1847-2013-5

Abstract

In this paper, the problem of bifurcation of limit cycles from degenerate singular point and infinity in a class of septic polynomial differential systems is investigated. Using the computer algebra system Mathematica, the limit cycle configurations of {(8),3} and {(3),6} are obtained under synchronous perturbation at degenerate singular point and infinity. To our knowledge, up to now, this is the first time that the problem of limit cycles bifurcated from degenerate singular point and infinity under synchronous perturbed conditions in a septic system has been investigated.MSC:34C05, 34C07.

Highlights

In the qualitative theory of planar dynamical systems, bifurcation of limit cycles for planar polynomial differential system dx dy = P(x, y), = Q(x, y), ( . )dt dt which belongs to the second part of the Hilbert th problem, is known as a hot but intractable issue
As far as the number of limit cycles bifurcated from infinity is concerned, representative results are as follows: cubic systems, [ ] got I ≥, [ ] got I ≥, [ ] got I ≥, [, ] got I ≥ ; quintic systems, [ ] got I ≥, [ ] got I ≥, [ ] got I ≥, [ ] got I ≥ ; septic systems, [ ] got I ≥, [ ] got I ≥, [ ] got I ≥
Few papers are concerned with bifurcation of limit cycles from the origin and infinity under synchronous perturbation: [ ] obtained the limit cycle configurations of {( ), } and {( ), } in a cubic polynomial differential system; [ ] and [ ] respectively got the limit cycle configurations of {( ), }, {( ), } and {( ), }, {( ), } in two certain quintic systems

Summary

Introduction

We deal with limit cycles bifurcated from degenerate singular point and infinity under synchronous perturbation in the following real septic polynomial differential system: dx dt ) is either a center or a focus, and it is a degenerate singular point. Singular point quantities and center conditions at degenerate singular point From Theorem . F k(z, w) = α+β= k cαβ zαwβ , and for any positive integer m, the mth singular point quantity at the origin μ(m ) can be determined by the following recursion formulas: c = , when (α = β > ) or α < or β < , cαβ = ; else cαβ = λ(β – α) ( α – β)c– +α,– +β + b (– + α – β)c– +α,– +β

Results

Conclusion