  This paper is concerned with the theory of generic non-normal nonlinear
evolutionary equations, with potential applications in Fluid Dynamics and
Optics. Two theoretical models are presented. The first is a model two-level
non-normal nonlinear system that not only highlights the phenomena of linear
transient growth, subcritical transition and global modes, but is also of
potential interest in its own right in the field of nonlinear optics. The
second is the fairly familiar inhomogeneous nonlinear complex Ginzburg--Landau
(CGL) equation. The two-level model is exactly solvable for the nonlinear
global mode and its stability, while for the spatially-extended CGL equation,
perturbative solutions for the global mode and its stability are presented,
valid for inhomogeneities with arbitrary scales of spatial variation and global
modes of small amplitude, corresponding to a scenario near criticality. For
other scenarios, a numerical iterative nonlinear eigenvalue technique is
preferred. Two global modes of different amplitudes are revealed in the
numerical approach. For both the two-level system and the nonlinear CGL
equation, the analytical calculations are supplemented with direct numerical
simulation, thus showing the fate of unstable global modes. For the two-level
model this results in unbounded growth of the full nonlinear equations. For the
spatially-extended CGL model in the subcritical regime, the global mode of
larger amplitude exhibits a `one-sided' instability leading to a chaotic
dynamics, while the global mode of smaller amplitude is always unstable (theory
confirms this). However, advection can stabilize the mode of larger amplitude.
