h the perturbation variable E. D.1 Components of the deviation Ax(t) By substituting the solution (D.2) into (D.1), it is obtained that (D.3) Here, C(t,p) E RNXN,being fp (t,p) E RN and U1 (t,p) E RNxP are defined by (5.55)-(5.58) in section 5.5 of chapter 5. Furthermore, the process p(T) is solved from 5.66. D.2 BEHAVIOR OF AX2 As stated in that same section, in order for Ax(t) to remain bounded, it must belong to the space spanned by the columns of the [U2 (t,p) U3 (t,p)] RNx(Q+R). or Ax(t) (D.4) = U2(t,p(t))a2(t)+U3(t,p(t))a3(T) (D.5) (D.6) In the above, U2 (t,p) E RNXQ spans the stable modes of the N-dimensional homoge- neous system of DAEs (C(t,p)y(t))+G(t,p)y(t)=0 (D.7) Moreover, U3 (t,p) E RNxR is defined by C(t,p)U3(t,p) = 0, i.e. the directions stored in the columns of U3 (t,p) capture the deviation from the manifold defined by the constraints that are implicit to the DAEs (5.34) that model the core system. Note that U3 (t,p) can always be chosen to be normalized, i.e. U3 (t,p) Il = 1. Having obtained these results, we continue with a discussion on both the behavior of Ax2( (T) and Ax3 (T). D.2 Behavior of Ax2 (T) The first component of the (normalized) deviation is oriented along space spanned by the stable modes of the attracting manifold M. Since the manifold is stable along these directions, it is expected that the corresponding "attracting force" keeps the component Ax2 (T) bounded. In what follows, we first derive a zeroth-order approximation for Ax2 (T). Next, we address the issue of its boundedness. An expression for Ax2 (T) By left multiplication of both sides of (D.3) with the projector V2 (t,p) satisfying (D.8) ).C(0,p).U2(0,p) = (D.9) V$(0.p).C(0,p)-U((0,p) = 0 (D.10) with I the unity matrix, we obtain = -V2(tp)f,(t,p)+(e) (D.11) Since U3 (t,p, satisfies ((t,p)U3(t,p) = 0 and with (D.5) and (D.8)-(D.10) it is then readily shown that (D.12) is equivalent to dot((p((tp())+() + (D.12) Boundedness of Ax2 (T) Hence, up to zeroth order in E, we find AX2(T) = U2(t,p(T))A2(T) (D.13) = C(To,p(To)) (D.