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We study the problem of existence of response solutions for a real-analyticone-dimensional system, consisting of a rotator subjected to a smallquasi-periodic forcing. We prove that at least one response solution alwaysexists, without any assumption on the forcing besides smallness andanalyticity. This strengthens the results available in the literature, wheregeneric non-degeneracy conditions are assumed. The proof is based on adiagrammatic formalism and relies on renormalisation group techniques, whichexploit the formal analogy with problems of quantum field theory; a crucialrole is played by remarkable identities between classes of diagrams.

Response solutions for arbitrary quasi-periodic perturbations with Bryuno frequency vector