sense, the term steady-state solution can be defined as a solution to (5.34) that is void of fast-varying transient behavior. For typical oscillators, these solutions tend to be periodic or quasi- periodic. Furthermore, the methods here presented can be extended to handle steady-state solutions with varying periods of oscillation (see for example [Vana02e]). However, the assumption of a constant period is often satisfied and helps to simplify analysis. Rloss Figure 5.11: Characterizing a harmonic oscillator's core system: (a) The harmonic oscillator is partitioned into a core system and a set of perturbation components (enclosed by the dashed line). (b) The unperturbed core system is solved for all possible steady-state solutions (states). (c) Each solution (state) is identified by a particular value of the state vector p = [ A 0 ]T. Hence, it can be represented as a single point in a 2-dimensional space. (5.34)8. Roughly, a manifold is a smooth surface [Abra88]. Solutions x(t) of (5.34) are attracted to the manifold if, for T -> 8, x(t) approaches M infinitesimally closely. Example (Harmonic oscillator): Fig. 5.11(b) and (c) illustrate the characterization of the core system associated with the harmonic oscillator in Fig. 5.11(a) The behavior of this oscillator is captured by the system of (unnormalized) DAEs 0(5.38) (5.40) where Vd=V1-V2 - and Vcm =v1+v2 In (5.38)-(5.40), the terms in the square brack- ets denote the perturbation terms. This corresponds with the partitioning indicated in Fig. 5.11(a). As shown in Fig. 5.11(b), the oscillator's core system corresponds to a lossless resonant tank. This core system is described by the equations obtained by neglecting the perturbation terms in (5.38)-(5.40). These (unperturbed) equations are solved for vd(t) = Acos(wor + 0) and id(t) = sin(wot +0), i.e. sine waves with an 8 See appendix D on how to quantify the notion of "a manifold that quickly attracts all other solutions". Note that if M does not attract the other solutions in its neighborhood, this typically in