By O. Axelsson, L.S. Frank and A. Van Der Sluis (Eds.)
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Additional resources for Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems
Aa and Sanders [l]. Remark 4 . The case of an eigenvalue on the positive imaginary axis with multiplicity 2, but with dim N = 2, is very interesting because it describes the transition from an elliptic to a hyperbolic equilibrium point. In this case ' A has a non-zero nilpotent part on the corresponding 4-dimensional generalized eigenspace, so the theorem of Moser [ 7 1 does not apply directly. However, we are convinced that a suitable variant also works in this case. An example is given by the Lagrange equilibria in the restricted 3-body problem for a special value of the mass-ratio.
F . Verhulst, Lecture Notes in Plath. 711, Sprinqer-Verlag, Heidelberg 1979. , Integral analytic sets, Dokl. Akad. Nauk SSSR 220 (1975), 1255-1258 - Soviet Math. Dokl. 16 (1975), 224-258. See also his Preprints 97, 98 of the Inst. Appl. Math. Acad. Sci. USSR, Ploscow, 1974. [ 31 Duistermaat, J . J . , On periodic solutions near equilibrium points of conservative systems, Arch. Rat. Mech. Anal. 160. 5 (19721, 143- [ 41 Henrard, J . , Lyapounov's center theorem for resonant equi- librium, J. Diff.
DUISTERMAAT 30 such that for 1-1 close to '1the set of periodic orbits of ( 1 ) near the origin and with period near w o is equal to the Q'-image of the set of periodic orbits of the Hamiltonian system is given a s the set of w E IR4 where: Gul u = U ( U ) ~which 0 (7) grad Gu (w) is a multiple of grad G2(w). Here G: denotes the quadratic part of the Taylor expansion of Go at the origin. (111) The standard Hamiltonian function Gu can be taken as follows (see (11) I (13), (16) below). By a linear canonical change of coordinates and a suitable time scaling (including time reversal) the quadratic part of F o on N can be brouqht into the form (8) 0 G 2 = ilpl + kp2 , k,R Zwithout common factors, 0 E il < lkl.
Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems by O. Axelsson, L.S. Frank and A. Van Der Sluis (Eds.)