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Agraïments: The first author is partially supported by NSFC key program of China (no.  11231001). The MINECO/FEDER grant UNAB13-4E-1604. And the third is supported by NSFC for Young Scientists of China (no. 11001047) and NSF of Jiangsu, China (no.  BK20131285).For almost periodic differential systems ˙x = εf(x, t, ε) with x ∈ Cn, t ∈ R and ε > 0 small enough, we get a polynomial normal form in a neighborhood of a hyperbolic singular point of the system ˙x = ε limT→∞1T∫ T0f(x,t, 0) dt, if its eigenvalues are in the Poincaré domain. The normal form linearizes if the real part of the eigenvalues are non–resonant

Polynomial and linearized normal forms for almost periodic differential systems