In this paper, we present an averaging method for obtaining quasi-periodic response solutions in perturbed, real analytic, quasi-periodic systems with Diophantine frequency vectors. Under the assumptions that the averaged system possesses a non-degenerate equilibrium and that the eigenvalues of its linearized matrix are pairwise distinct, we show that the original system admits a quasi-periodic response solution for parameters in a Cantorian set. The proof relies on KAM techniques. It is worth mentioning that our results do not require the equilibrium to be hyperbolic, meaning that the eigenvalues of the linearized matrix of the averaged system may be purely imaginary. Furthermore, the proposed averaging method is applicable to second-order systems, and a higher-order averaging framework is also established.
Xing et al. (2026) studied this question.