In this thesis, we propose to formally derive amplitude equations governing the weakly nonlinear evolution of non-normal dynamical systems, when they respond to harmonic or stochastic forcing, or to an initial condition. This approach reconciles the non-modal nature of these growth mechanisms and the need for a centre manifold to project the leading-order dynamics. Under the hypothesis of strong non-normality, small operator perturbations suffice to make singular the inverse of the operator which is relevant to the considered problem. The adjective "small" is relative to the choice of an induced norm, under which the operator induces a large input-output amplification. Such operator perturbation can be encompassed in a multiple-scale asymptotic expansion, closed by a standard compatibility condition. The resulting amplitude equations are tested in parallel and non-parallel two-dimensional flows, where they bring insight into the weakly nonlinear mechanisms that modify the gains as we increase the amplitude of the harmonic or stochastic forcing, or that of the initial condition.