In this thesis we work with Cinfinity or analytic families of vector fields or diffeomorphisms. We are interested in local equivalences and conjugacies between such families and families in a "simple" form, sometimes called a normal form.;As in this thesis we are working with families of vector fields or diffeomorphisms, we will encounter the same problems concerning hyperbolicity and resonance as in the case of individual systems. An additional problem can be caused by the parameters: as the parameter perturbs the eigenvalues, it can cause resonances which are absent for the unperturbed system. This phenomenon also has its impact on the smoothness of the equivalence or conjugacy.;This thesis is structured as follows.;In Chapter 1 we introduce the most important objects used in this thesis: vector fields, flows, fixed points, singular points, conjugacies and equivalences. We give a brief introduction on analytic functions in several variables. After this we give a profound discussion on normal forms. The chapter ends with a short discussion on transition maps of planar vector fields.;In Chapter 2 the aim is to give an explicit construction for equivalences and conjugacies between nearly-resonant planar saddles and their linear parts. We start by proving a lower bound on the degree of the resonant terms that appear as the parameter varies. After this we discuss the explicit form for a C1 equivalence or conjugacy between nearly-resonant planar saddles and their linear parts. Introducing two new variables we prove that this conjugacy is Cinfinity with respect to the two original and the two new variables. Next the conjugacies between nearly-resonant planar saddle diffeomorphisms and their linear parts are studied. To conclude we try to repeat the calculations that were made in the saddle case for a deformation of planar singularity of center type.;In Chapter 3 we consider the Poincare map of a deformation of a planar singularity of center type. (Abstract shortened by UMI.).
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