Resonant Bifurcations

We consider dynamical systems depending on one or more real parameters, and assuming that, for some ‘‘critical’’ value of the parameters, the eigenvalues of the linear part are resonant, we discuss the existence}under suitable hypotheses}of a general class of bifurcating solutions in correspondence with this resonance. These bifurcating solutions include, as particular cases, the usual stationary and Hopf bifurcations. The main idea is to transform the given dynamical system into normal form in the sense of Poincar´e and Dulac.and to impose that the normalizing transformation is convergent, using the convergence conditions in the form given by A. Bruno. Some specifically interesting situations, including the cases of multiple-periodic solutions and of degenerate eigenvalues in the presence of symmetry, are also discussed in some detail