Resonant Bifurcations

TL;DR

This paper investigates bifurcations in dynamical systems with resonant eigenvalues, using normal form theory and convergence conditions to establish the existence of various bifurcating solutions, including stationary and periodic ones.

Contribution

It extends bifurcation analysis to resonant cases by applying convergent normal form transformations under Bruno's conditions, covering complex scenarios like multiple-periodic solutions and symmetric degeneracies.

Findings

Normal form transformations can be convergent under Bruno's conditions.

Resonant bifurcations include stationary and Hopf types.

Analysis of degenerate eigenvalues with symmetry.

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.