Weak nonlinearity for strong nonnormality.

We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of nonnormal dynamical systems, when they experience transient growth or respond to harmonic forcing. This approach reconciles the nonmodal nature of these growth mechanisms and the need for a center manifold to project the leading-order dynamics. Under the hypothesis of strong nonnormality, we take advantage of the fact that small operator perturbations suffice to make the inverse resolvent and the inverse propagator singular, which we encompass in a multiple-scale asymptotic expansion. The methodology is outlined for a generic nonlinear dynamical system, and two application cases highlight two common nonnormal mechanisms in hydrodynamics: the flow past a backward-facing step, subject to streamwise convective nonnormal amplification, and the plane Poiseuille flow, subject to lift-up nonnormality.