ude changes, i.e. changes in the value of the state vector p, are slow as com- pared to the oscillations. to describe the oscillator's dynamic behavior has now been reduced to three. Hence, in a certain sense, the procedure above amounts to nonlinear model order reduction. Furthermore, as will be discussed below, the equations (5.79)-(5.81), when properly processed, can be solved using a time step that is much larger that the one needed to solve the original system of equations. 5.6 Averaging The ordinary differential equations (ODEs) (5.71) that govern the behavior of the state vector p belongs to the class of ODEs structured as (5.82) where, in this case, h(p,t) -V[ (t,p)f(x,(t,p),t) (5.83) Since the perturbation term ef is small, SO is the right-hand side of (5.82). Hence, the time derivative of the process p(t) is small in magnitude. This in turn implies that it takes a long time for the state vector p(T) to substantially change value. Apart from some small fluctuations, p(T) is therefore inherently a slow-varying process. It only changes significantly over a (normalized) time interval that is at least 1/e in order of magnitude. Example (Harmonic Oscillator): Fig. 5.16 shows the startup behavior of a harmonic oscillator. For the partitioning indicated in the figure, the state vector p = [ A 0 is 2-dimensional, containing amplitude and phase (since there is no control voltage, there is no common-mode component in p). It is observed that amplitude changes, i.e. changes in the value of the state vector p, occur on a time scale that is slow compared to that of the oscillations. Hence, with respect to normalized time we find that dA/dt 1. This is in correspondence with (5.82). Equation (5.82) can be solved efficiently using averaging [Bogo61, Verhu96]11. Aver- aging exploits the fact that solutions of (5.82) only vary substantially on a time scale proportional to 1/e. Hence, these solutions are dominated by a slow-varying com- ponent. Averaging techniques allow us to extract those parts of (5.82) that produce this