From the point of view of perturbation theory, (perturbations of) near-resonant systems are plagued by small denominators. These do not affect (perturbations of) fully resonant systems; so it is in many ways convenient to approximate near resonant systems as fully resonant ones, which correspond to considering the "detuning" as a perturbation itself. This approach has proven very fruitful in Classical Mechanics, but it is also standard in (perturbations of) Quantum Mechanical systems. Actually, its origin may be traced back (at least) to the Rayleigh-Ritz method for computing eigenvalues and eigenvectors of perturbed matrix problems. We will discuss relations between these approaches, and consider some case study models in the different contexts.

We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the deterministic one, such as conditional, partial and asymptotic symmetries. A number of explicit examples are presented.