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By O. Axelsson, L.S. Frank and A. Van Der Sluis (Eds.)

ISBN-10: 0080871585

ISBN-13: 9780080871585

ISBN-10: 0444861319

ISBN-13: 9780444861313

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IIb) A smooth mapping p a local diffeomorphism H 0 0 (defined near p = p , ~ ( )p = 0) and from W4 to N ( u ) depending smoothly on p , u(p) H. 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.

160. 5 (19721, 143- [ 41 Henrard, J . , Lyapounov's center theorem for resonant equi- librium, J. Diff. Eq. 14 (1973), 431-441. nSral de la stabiliti? Studies 11,Princeton University, 1947. C. van der, On the computation of 4th order coefficients of a normalized Hamiltonian at 1:l resonance, Preprint, Utrecht 1980. [ 71 Moser, J . , Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Corn. Pure Appl. Math. 2 (1976), 727-747. [ 81 Sanders, J . A . , On the theory of nonlinear resonance, Thesis, Utrecht 1978.

T i n g s i m p l e , b u t t e d i o u s c a l c u l a t i o n s , b a s e d on t h e f o r m u l a s ( 5 . 6 ) and on t h e definition of two-gap c a s e : P . and 1 Q. I' we g i v e t h e a v e r a g e d q u a n t i t i e s < P . > f o r t h e 3 3 . DOBROKHOTOV 22 and V . P S YU . MASLOV Due t o t h e s e formulas and t a k i n g i n t o a c c o u n t ( 4 . 9 ) i t i s easy t o r e p r e s e n t t h e e q u a t i o n s ( 6 . 10) - where R . a r e a n a l y t i c f u n c t i o n s i n t h e arguments C , G , U .

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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.)


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