Limit cycles near vector fields of center type

Abstract

In this thesis, we focus on three problems, that briefly can be described as follows. The first problem deals with stable bifurcation diagrams of limit cycles near centers, where attention is focused on uniform results as well in phase plane as well as in parameter space. The second problem is the investigation of how 1-parameter techniques, such as the computation of Melnikov functions, can be used in multi-parameter families, to compute its cyclicity near centers. The third problem deals with families (X(v,ε)) of planar vector fields that unfold a Hamiltonian vector field for ε = 0, where ε is a 1-parameter; it is the investigation whether results on linear approximations Iv of the displacement map θ(v,ε), with respect to ε (such as the first order Melnikov function), can be transferred to valuable results on the bifurcation diagram of limit cycles and the cyclicity.