 to avoid wasting time on the computation of the often irrelevant deviation EAx(T). The next section discusses how to extract the equations governing the process p(T) from (5.54). The equations that govern EAX(T) are of less importance. Their derivation is, therefore, deferred to appendix D. Note that all analysis is performed up to first order in the perturbation variable E. Furthermore, in what follows, we use C(t,p) (5.55) (5.56) fp(t,p) R N (5.57) as shorthand notations that help to simplify the algebra. 5.5.2 Motion Xs (t,p(t)) over the manifold M The equations that characterize the process p(T) and, therefore, the oscillator's motion over the P-dimensional manifold M, are obtained by projecting (5.54) onto the man- ifold M. More specifically, at each point, we project them onto the tangent space of M. Hence, we focus on the behavior induced by those parts of the perturbation terms that are oriented along the surface M. Fig. 5.14 makes the analogy with a ball that is constrained to move on a curved rod (a 1-dimensional stable manifold). Since the ball cannot leave the rod, only the force components that are oriented along the rod can set the ball in motion (along the rod). At each point of M, i.e. for each couple (t,p) E RxRP, the tangent space is spanned by the columns of (5.58) 5.5 OSCILLATOR PERTURBATION ANALYSIS projected Figure 5.14: When a ball (a system) is constrained to move along a curved rod (a 1-dimensional surface), only the force components that are oriented along the rod can set the ball in motion. It hence makes sense to project the perturbation force on the tangent to the rod. As will be discussed later, this projection is not necessarily orthogonal. Since, for a constant state vector p, xs(t,p) solves (5.52) for E : 0, it is readily shown that all P columns of U (t,p) solve the linear N-dimensional homogeneous system of (5.59) Furthermore, if the system of DAEs (5.59) is index-1 [Dem00b, Marz95], then there exist matrices U2(t,p) E RNXQ and U3(t,p) E RNxR, with P+Q+R=N, such that the column