Generalized stability theory part II: non-autonomous operators

An extension of classical stability theory to address the stability of perturbations to time-dependent systems is described. Nonormality is found to play a central role in determining the stability of systems governed by nonautonomous operators associated with time-dependent systems. This pivotal role of nonnormality provides a conceptual bridge by which the generalized stability theory developed for analysis of autonomous operators can be extended naturally to nonautonomous operators. It has been shown that nonormality leads to transient growth in nonautonomous systems, and this result can be extended to show further that time-dependent nonormality or nonautonomous operators is capable of sustaining this transient growth leading to asympotic instability. This general destabilizing effect associated with the time-dependent operators. Simple dynamical systems are used as examples including the parametrically destabilized harmonic oscillator, growth of errors in the Lorenz system, and the asymptotic destabilization of the quasigeostrophic three-layer model b stochastic vacillation of the zonal wind.