Dynamical systems and stability: The improvements given by the near-resonance theorem

Abstract

The near-resonance theorem represents a major advance in the analysis of analytic dynamical systems. It leads to great simplifications in the study of the vicinity and the stability of periodic solutions, it leads also to the following. 1. (i) The asymptotic solutions lying in the stable and unstable subsets are integrable and simple. 2. (ii) Almost all periodic, “quasi-periodic” and “almost-periodic” solutions of conservative systems (such as the quasi-periodic solutions of the “Arnold tori”) are without neighbouring asymptotic solutions and have the first-order stability: all their Liapunov characteristic exponents are zero. Conversely almost all bounded solutions with one or several positive Liapunov characteristic exponents are chaotic. 3. (iii) In analytic Hamiltonian problems the near-resonance theorem leads to the notion of quasi-integral (i.e. state functions without secular variations to any order) and gives a sign to the eigenfrequencies. In most critical cases the quasi-integrals lead to the “all-order stability”, i.e. either the true stability or the Arnold diffusion. However there exist destabilizing resonances, the “positive resonances” related to the signs of the eigenfrequencies.