The Near-Resonance Theorem: Analysis of the Vicinity of Periodic Solutions of Analytic Differential Systems

Abstract

The near-resonance theorem allows us to simplify the systems of differential equations and to study the vicinity and the stability of their periodic solutions. It leads to suitable transformations that cancel all non-near-resonant terms, even those of very large orders.

The demonstration of the theorem is difficult but short and various applications are presented.

For Hamiltonian systems the near-resonance theorem leads to a strict definition of the notion of “quasi-integral”, to the “generalized Birkhoff differential rotation” and to the “all-order stability” that is not necessarily a true stability but is at worse an extremely slow “Arnold diffusion”. It also discloses many high-order instabilities related to resonances.