14) as an expression for the first component of the oscillator's deviation from the manifold. Boundedness of Ax2 (T) In order to prove that Ax2(t) remains bounded, it is sufficient to show that the homo- geneous part in (D.14) (the first term) is uniformly exponentially stable [Cor91], i.e. At, to and for an arbitrary Ax2(To) with ||Ax2(To)| = 1, it holds (D.15) with K, B>0. Since the initial condition AX2(TO) is arbitrary, (D.15) is equivalent with < Ke-B(t-to) ,To (D.16) Using (D.16), it is readily shown that (D.17) L = max |f, (t,p)|| (D.18) is an upper bound for ||Ax2(T)||. For this yields (D.19) Hence, the deviation grows larger when the strength of the perturbing force fp grows large especially the strength of the components oriented along the space spanned by U2- and when the time constant 1/ increases. This time constant is a measure for how quick the manifold attracts solutions in its neighborhood. It remains of course to be answered whether it is possible to determine suitable values for the constants K,B In what follows, we outline an argument for (D.7) being a system of DAEs that is periodic in T. If p is considered constant and assuming the system of linear DAEs (D.7) to be T- periodic in T and stable, Floquet theory [Lam97] shows that we can write U2(t,p) = Up,2(t,p)e^(P) (D.20) (D.21) D.3 THE BEHAVIOR OF AX3(T) Here, both Up,2(t,p) and VT p,2 (t,p) are periodic in T with their norm bounded At. Furthermore (D.22) "g(p) is a diagonal matrix containing the nonzero Floquet exponents corresponding to (D.7). Since the system is assumed to be stable, we can order the exponents Mk such that (D.23) Using (D.20)-(D.23), it is readily shown that < K (p)e-B(p)(t-to) (D.24) B(p) = |Re{u1(p)} (D.25) (D.26) Substituting these results in (D.19), we then obtain (D.27) as an upper bound on the deviation. The reasoning above, however, does not take into account that the state vector p also varies with time. Fortunately, (D.27) holds as long as the time constant 1/B(p), which characterizes the time it takes to attract