Resonant Bifurcations

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 existenceᎏunder suitable hypothesesᎏof 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 Ž . normal form in the sense of Poincare and Dulac and to impose that thé 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. ᮊ

Convergence of normal form transformations: The role of symmetries

“…A nontrivial example in dimension n > 2, and corresponding to the case of coupled oscillators with multiple frequencies, is given by the following corollary, which immediately follows from the theorem. For an explicit example, see [12]. Then there is a multiple-periodic bifurcating solution preserving the frequency resonance 1 : m.…”

“…We then show the existence -under suitable hypotheses -of a general class of bifurcating solutions in correspondence to this resonance. Details and complete proofs can be found in [12].…”

“…However, the presence of degenerate eigenvalues is typically connected to the existence of some symmetry property of the problem (indeed, in the absence of symmetries, the degeneration is "non-generic", being possibly removed by arbitrarily small perturbations); in this situation, the arguments used above are still applicable, with the same result [12]. An example, given by coupled oscillators with degenerate frequencies and in the presence of a rotation symmetry, is described in [12].…”