t,p) to satisfy VT(t,p)C(t,p)U2(t,p)=0(t,p) E RxR (5.69) With (5.62), it also automatically holds that '{(t,p)C(t,p)U3(t,p) = 0. Furthermore, since T(p, solves (5.64), it is sufficient that 7/(0,p)C(0,p)U2(0,p) = 0 in order for (5.69) to hold. Using (5.57) and (5.66)-(5.69), we finally obtain =-VT(t,p)f(xs(t,p),t)+(e2) = (5.70) as the P-dimensional system of equations that governs the behavior of the oscillator's state vector p and, therefore, the motion over the manifold M. Hence, changes in the state vector are driven the projected perturbation term ef (xs (t,p), T). Hereby, the con- dition (5.63) states that this projection is onto the hyperplane tangent to the manifold M. Moreover, (5.69) expresses that it runs, in a certain sense, in parallel to the hyperplane spanned by the stable modes of (5.59) augmented with the null space of C(t,p)¹0. 9In deriving (5.66), we interchanged the partial derivative with the full derivative dV|/dt = This only introduces an error that is O(e2) and, therefore, makes no difference for results up to first order in E. Note that this projection is not necessarily done in a direction that is orthogonal to the hyperplane tangent to M. This actually only holds for some special cases. 5.5 OSCILLATOR PERTURBATION ANALYSIS Figure 5.15: The motion over the manifold M is computed by projecting the perturbation terms onto the tangent space to M. This tangent space is spanned by the columns of U1. The projection runs, in a certain sense, paral- lel to the space spanned by the columns of U2 and U3. This projection is not necessarily orthogonal to the tangent space. 5.5.3 In summary At this point, we recapitulate the main results SO far. Up to first order in E, the equation that governs the behavior of the oscillator's state vector p -and as such its motion over M- is (5.71) Hence, changes in the state vector are driven by the projected perturbation term. As illustrated in Fig. 5.15, the perturbation term (xx(t) is projected onto the tangent space of the manifold M. In most cases, the projecti