Global Instability in Fully Nonlinear Systems

Existence of a saturated steady solution of a nonlinear evolution equation subject to a boundary condition at $x=0\mathrm{}$, called a nonlinear global mode, is illustrated on the real subcritical Ginzburg-Landau model. Such a nonlinear global mode is shown to exist whereas the flow is linearly stable, convectively unstable, or absolutely unstable. If the linearized evolution operator is absolutely unstable, then a global mode exists but the converse is false. This result relies only on the existence of a structurally unstable heteroclinic orbit in the phase space and is likely to be generic as demonstrated by the supercritical Ginzburg-Landau and the van der Pol-Duffing equations.