By H. W. Broer, G. Vegter (auth.), C. K. R. T. Jones, U. Kirchgraber, H. O. Walther (eds.)

**Dynamics Reported** is a sequence of books devoted to the exposition of the math of dynamcial structures. Its target is to make the new learn available to complicated scholars and more youthful researchers. The sequence is additionally a medium for mathematicians to exploit to maintain up to date with the paintings being performed in neighboring fields. the fashion is better defined as expository, yet whole. hence, there's an emphasis on examples and motives, but in addition theorems regularly happen with their proofs. the focal point is at the analytic method of dynamical platforms, emphasizing the origins of the topic within the thought of differential equations. **Dynamics Reported** presents a very good starting place for seminars on dynamical systems.

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We treat the following examples in some detail: the quadratic Zeeman effect, orbiting dust, a three dimensional lunar problem, and the main problem of artificial satellite theory. 1. Introduction This paper discusses the use of normal form for analyzing Hamiltonian systems which are small perturbations of the Kepler Hamiltonian. Recall that the Kepler Hamiltonian describes the motion of two bodies in three space under mutual gravitational attraction. The basic idea of normal form theory is to construct a change of coordinates, which preserves the Hamiltonian form of the equations of motion, so that up to some finite order in the perturbation parameter the Hamiltonian in the new coordinates commutes with the flow of the Kepler Hamiltonian.

W. Broer and G. Vegter Given these local solutions we take a partition of unity in order to patch them together to a global solution Y on a full neighbourhood of the set DT. Notice that here we use the compactness of DT. 2 we now show that the time-evolution of the vector field Y can be used to construct a right-equivalence between HI and HO. Let yt be the time t map of Y, then Y1i == implies that 1i 0 yt = 1i. Obviously yt is of the form ° yt(x, y, J1" to) = (Y(x, y, J1" t), t + to) for some mapping Y : ~2 x ~3 ---+ ~2 X ~3.

L and t using the Malgrange-Mather Preparation Theorem, cf. [Maj. We begin the construction of the family Yp. in a neighbourhood of the most degenerate singularities, subsequently determining the solutions in neighbourhoods of the Morse-singularities respectively the regular points. Due to the linearity of the Infinitesimal Stability Condition these local solutions can be glued together to obtain a more global infinitesimal solution, also to be denoted by Yw The time 1 flow lffp. ,l of Yp. then is a right-equivalence between HZ and H~.