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This two level decomposition plays an important role in this monograph and w i l l be dealt with in more detail in Chapters 6, 7 and 8. We shall postpone the i l l u s t r a t i o n of the weak connection form to after we have discussed dynamic networks and aggregations. 6 Conclusions We have used a manifold approach to investigate the time-scale characteristics of nonexplicit singularly perturbed models. These mode|s include systems whose storage elements are interconnected with weak and strong springs, etc.

3,15) E dz/dt = ~ Afs(E) y + Af(c) z , where As(C) = PAI(¢)V , Asf(C) = PAI(¢)W , Afs(¢) = QAI(¢)V , Af(c) = QAoW+ ~QAI(C)W . 15) is an explicit model, we need to show that Af(O) is nonsingular. 17) which implies that Af(O) = QAoW= QWG= G is nonsingular. 15) by setting ~ = O. 19) y and z are the representations of x with respect to the bases V and W of N(AO) 44 and R(Ao), respectively. In other words, Vy is the projection of x on N(Ao) along R(A0) and hence, y is the representation of this projection with respect to the basis P.

18) mldXl/dT + m2dx2/d% + m3dx3/dT = (mldXl/d~ + m2dx2/dl + m3dx3/dT)tT=0 = c0. 18) we obtain mlxI + m2x2 + m3x3 = COT + (mlXl(O) + m2x2(O) + m3x3(0)) = COT + c I . 19) defines a moving plane F. 19). 19) When k3=O, the fast When the i n i t i a l motion is momentum c o is small, the plane F slowly d r i f t s remaining orthogonal to the vector [mI m2 m3]T. W i t h the weak connection k3 included, the plane F, instead of d r i f t i n g with constant speed, oscillates slowly about the origin while maintaining orthogonality to the vector [mI m2 m3]T.

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