In this paper, a differential calculus for the non-operator norms |·|₁ and |·|_∞ of m-times continuously differentiable matrix function χ(t), t ≥ t₀, is presented and combined with the study of the asymptotic behavior of the evolution Φ(t, t₀) for periodic linear dynamical systems. The upper bound describing the asymptotic behavior (for short, asymptotic bound or asymptotic estimate) is based on Floquet's theory and on a bound containing the spectral abscissa of a constant matrix; it compares favorably with other asymptotic bounds. The minimal constant in the asymptotic estimate is computed by the differential calculus of norms. As far as we are aware, the achieved result cannot be obtained by other methods.