Oscillatory Differential Equations with Varying High Frequencies

Abstract

New aspects come into play when the high frequencies in an oscillatory system and their associated eigenspaces do not remain nearly constant, as in the previous chapter, but change with time or depend on the solution. We begin by studying linear differential equations with a time-dependent skew-hermitian matrix and then turn to nonlinear oscillatory mechanical systems with time- or solution-dependent frequencies. Our analysis uses canonical coordinate transforms that separate slow and fast motions and relate the fast oscillations to the skew-hermitian linear case. For the numerical treatment we consider suitably constructed long-time-step methods (“adiabatic integrators”) and multiple time-stepping methods.