Cicogna G.
Resonant Bifurcations
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ABSTRACT. 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 to 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-Dulac), and to impose that the normalizing
transformation is convergent, using the convergence conditions in the
form given by A. Bruno. Some specially interesting situations, including
the cases of multiple-periodic solutions, and of degenerate eigenvalues
in the presence of symmetry, are also discussed with some detail.