We investigate the notion of complex rotation number which was introduced by V.

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We study global bifurcations in generic three-parameter families of vector fields on S 2 . In the recent article by Ilyashenko et al (2018 Invent. Math. 213 461-506), the authors show that three-parameter unfoldings of vector fields with the polycycle 'tears of the heart' are structurally unstable. We consider three-parameter unfoldings of vector fields with separatrix graphs 'ears' and 'glasses', and prove that these families are structurally unstable as well. We also study in more details the classical bifurcation of a saddle loop, and use it as a building block in our main example.

In this article we prove in a new way that a generic polynomial vector field in C 2 possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.