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The number of limit cycles of a quintic polynomial system with center. (English) Zbl 1181.34040

The bifurcation of limit cycles in the system

x ˙=y(1+x 4 ),y ˙=-x(1+x 4 )+εP(x)y 2m-1

is studied for small ε. It is assumed that P(x) is a real polynomial of degree 2n+2 or 2n+3 and m,n are natural numbers. The main result states that an upper bound for the number of limit cycles bifurcating from the periodic orbits of the initial system (ε=0) is given by min{N 1 ,N 2 ,N 3 } where

N 1 =4m+2n-2+sin 2 (nπ/2)+[1 4(m+n-3+sin 2 (nπ/2))],N 2 =3m+2n-2-sin 2 (nπ/2)+[1 2(m+n-1)],N 3 =5m+2n-2+sin 2 (nπ/2)·

Moreover, there are systems with at least 3m+n-2 limit cycles.

The proof follows from an estimation of the number of positive zeros of the integral Φ(h)= x 2 +y 2 =h P(x)y 2m dx/(1+x 4 ) which is elementary and is calculated explicitly in the paper.

(Reviewer’s remark). The integral Φ(h), as taken by the authors, is identically zero. One should consider a similar integral with y 2m-1 instead of y 2m in order to obtain information about the limit cycles in the perturbed system.

MSC:
34C07Theory of limit cycles of polynomial and analytic vector fields
34C05Location of integral curves, singular points, limit cycles (ODE)
34C08Connections of ODE with real algebraic geometry