Resonant bifurcations

We consider dynamical systems depending on one or more real parameters, and assuming that, for some "critical" value of the parameters, the eigenvalue s of the linear part are resonant, we discuss the existence-under suitable hypotheses-of a general class of bifurcating solutions in correspondence wi th 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 Poincare and Dulac) and to impose that the normalizing transformation is convergent, usi ng the convergence conditions in the form given by A. Bruno. Some specifica lly interesting situations, including the cases of multiple-periodic soluti ons and of degenerate eigenvalues in the presence of symmetry, are also dis cussed in some detail. (C) 2000 Academic Press.