Convergence of normalizations for partially integrable differential systems

Abstract

This paper provides some criteria to characterize convergence of normalizations which transform partially integrable analytic differential systems to their Poincaré–Dulac normal forms. For a family of four-dimensional partially integrable differential systems near an equilibrium which has one pair of conjugate imaginary eigenvalues and a pair of resonant nonzero real eigenvalues, we prove convergence of their normalizations. For analytic differential systems with dimension larger than 4, we illustrate that partial integrability may not be sufficient to ensure convergence of the normalizations even though Bruno’s condition $\omega$ holds. This work generalizes in a natural way the classical results by Poincaré and Lyapunov for a monodromic equilibrium, as well as the one by Moser for a hyperbolic saddle of analytic Hamiltonian systems of one degree of freedom.