 account for this slow-varying common-mode behavior. They treat a third-order voltage-controlled harmonic oscillator as a second-order non-controlled harmonic oscillator, e.g. the one in the previous example. Here, the order of an oscil- lator refers to the dimension of its core system's state vector. 5.5 Oscillator perturbation analysis The perturbation term ef (x, T) drives the unperturbed core from one state to another. In order to unravel the details of this state-changing behavior, we study the structure of the waveforms XE(T) that solve the perturbed equations (5.52) Here, we use the notation XE(T) to indicate the fact that solutions of (5.52) in principle also depend on the perturbation variable E. As a note aside, we mention that the theory that follows embeds the methods in [Dem00a, Dem00b, Kae90] as a special case. 5.5.1 Components of an oscillator's perturbed behavior While all waveforms that solve the unperturbed equations (5.34) converge to the P- dimensional manifold M, the action of ef(x,t) causes the XE(T) to deviate from the manifold. However, the attracting nature of M ensures that, for small perturbations, these deviations remain small. Hence, after some initial transients, any solution X (T) will be located near M. They can therefore be written as (5.53) 5.5 OSCILLATOR PERTURBATION ANALYSIS EAx(T) XE(T) xs(t,p(T)) -2 -2 Figure 5.13: Perturbed harmonic oscillator behavior: (a) the solutions X (T) of the perturbed harmonic oscillator in Fig. 5.11(a) are attracted to the neigh- borhood of the stable, 2-dimensional manifold M. These solutions contain two components: a motion XS (t,p(t)) over the attracting manifold M and a small deviation EAx(t) from the manifold. (b) The motion over the manifold M can be represented as the resonant tank slipping from one state to another. As indicated, a solution X (T) contains two components: a motion Xs (t,p(t,e)) over the manifold M and a deviation EAX(T,E) from the manifold. The first component, XS (t, p (t,e)), corresponds to the state-changing behavior disc