Weak nonlinearity for strong nonnormality

We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of transiently growing and harmonically forced nonnormal systems. This reconciles the non-modal 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 demonstrate that small operator perturbations suffice to let respectively the inverse resolvent and the inverse propagator become singular. The methodology is outlined for a generic nonlinear dynamical system and two application cases are chosen which highlight two common nonnormal mechanisms in hydrodynamic instability: a backward-facing step prone to streamwise convective nonnormality and a plane Poiseuille flow subjected to lift-up nonnormality.