s of (5.60) span the entire space RN. Here, the columns of U2(t,p) correspond to the stable modes of (5.59). They solve (5.59) and satisfy lim 1|U2(t,p)||=0,VpER' (5.61) T->00 a property that expresses the stability of the manifold M. The columns of U3(t,p) satisfy C(t,p)U3(t,p)= (5.62) Hence, they span the null space of C(t,p). In order to project (5.54) onto the tangent space of M, we need to determine a projector V[(t,p) ERP^" that satisfies (5.63) Here, Ip E RPxP is the P-dimensional unity matrix. This projector is a generalization the perturbation projection vector introduced in [Dem98] (see also section 5.7). The condition (5.63) is automatically satisfied for a projector V(t,p) that meets (5.64) (5.65) 5.5.2 Motion Xs (t,p(t)) over the manifold M Indeed, combining (5.64) with the fact that U1 (t,p) solves (5.59) implies that the par- tial derivative with respect to T of the left-hand side of (5.63) equals zero. The linear homogeneous system of DAEs (5.64) is called the system adjoint to (5.59). It is impor- tant to note that (5.64) and (5.65) do not uniquely determine V(t,p). There are still Q degrees of freedom left with Q the number of columns in U2(t,p). As is discussed below, the extra conditions needed to fix V1 (t,p) stem from the requirement that the deviation EAX(T) must remain bounded. By multiplying (5.54) with VT(t,p) we obtain9 dp du (( = = (()((()(566 -E Here, we used (5.55)-(5.57) and (5.64)-(5.65). The ordinary differential equation (5.66) governs the behavior of the state vector p(t). Its right-hand side, however, depends on both p(t) and Ax(t). The dependency on Ax(t) can be removed by properly fixing the projector's remaining degrees of freedom. The extra conditions build on the re- quirement that Ax(t) must remain bounded. It can be shown that this requires Ax(t) to belong to the space spanned by the columns of [ U2 (t,p(t)) U3 (t,p(t)) (see appendix D), i.e. Ax(T) = (5.67) = U2(t,p(t))a2(t)+U3(t,p(t))a3(T) (5.68) Using (5.68), Ax(T) is now removed from (5.66) by forcing the projector V(