ussed in section 5.1.3: the perturbation term drives the core system from one state (steady-state solution) to another. Hence, the state vector p does no longer remain constant but changes with time. This time-varying behavior of the state vector p entirely characterizes the motion over M. The second component in (5.53), EAx (t,e), describes a deviation from the manifold. Here, the manifold's stability guarantees that there exists a process p(t,E) such that (5.53) is satisfied for a bounded (small) deviation EAx (T,E). Note that the processes p(t,E) and Ax (t,e) in principle depend on both T and E. In what follows, for notational convenience, the dependence on E will no longer be mentioned explicitly unless when relevant. Example (Harmonic oscillator): Fig. 5.13 illustrates the components of the perturbed behavior of the harmonic oscillator from Fig. 5.11(a). Solutions XE(T) are attracted to the neighborhood of the stable, 2-dimensional manifold M obtained from the character- ization of the oscillator's unperturbed core. These solutions contain two components: a motion XS (t,p(t)) over the attracting manifold M and a small deviation EAX(T) from 5.5.2 Motion XS (t,p(t)) over the manifold M the manifold. Moreover, the motion over the manifold M can be represented as the oscillator's resonant tank (core system) slipping from one state to another. Hence, this implies that the state vector p(T) = [ A(T) ((t) ] (slowly) varies with time. Substituting the signal model (5.53) into (5.52) yields ( dc dx (5.54) Here we used the fact that, for an arbitrary but fixed state vector p, XS (T,p) solves (5.52) for E = 0. By projecting (5.54) on appropriate subspaces of RN it is possible to compute the motion over M, i.e. the behavior of the state vector p(t), without needing the details of EAX(T). This is quite convenient. In most cases, the motion over M is of greatest interest as it captures the slow-varying components of the oscillator's dynamics, e.g. long-term amplitude and phase variations. Hence, proper projection helps us