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On the nonexistence, existence and uniqueness of limit cycles. (English) Zbl 0886.58087
Summary: We present two new criteria for studying the nonexistence, existence and uniqueness of limit cycles of planar vector fields. We apply these criteria to some families of quadratic and cubic polynomial vector fields, and to compute an explicit formula for the number of limit cycles which bifurcate out of the linear centre x ˙=-y, y ˙=x, when we deal with the system x ˙=-y+ε i+j=1 n a ij x i y j , y ˙=x+ε i+j=1 n b ij x i y j . Moreover, by using the second criterion we present a method to derive the shape of the bifurcated limit cycles from a centre.

MSC:
37G99Local and nonlocal bifurcation theory
37G15Bifurcations of limit cycles and periodic orbits
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion