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Limit cycles appearing from the perturbation of a system with a multiple line of critical points. (English) Zbl 1241.34037

Consider the family of planar systems

x ˙=-yC(x,y)+ϵP(x,y);y ˙=xC(x,y)+ϵQ(x,y)

where P,Q and C are real polynomials, C(0,0)0, and ϵ is a small real parameter. The number of zeros of the abelian integral M(r) on x 2 +y 2 =r 2 controls the number of limit cycles that bifurcate from the periodic orbits of the unperturbed system with ϵ=0. The authors consider the case C(x,y)=(1-y) m and P,Q are of degree n. They prove that M(r)=[m+n 2]-1 when n<m-1 and n when nm-1.

MSC:
34C08Connections of ODE with real algebraic geometry
34C07Theory of limit cycles of polynomial and analytic vector fields
34C23Bifurcation (ODE)
34C05Location of integral curves, singular points, limit cycles (ODE)
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