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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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Extra info for Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems

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17) we have for small H . , = exp(i B . +4U~HPP COS(Tk+CoP) + O(H 2 +... + 4U1Hll 2 11 + H P2e ) and hence, as it was in one-dimensional case, the functions H . ,P Ill have the sense of "nonlinear amplitudes". Nevertheless in this case there exists such a set of values iB . H . = e 1k/2, P 1 k > j 2 1, that define the "interaction' 0 s the components L Ik with different phases of the solution Y ~ ( T ~ ~ . . , T ~For ) . 12) by solving which with respect to V following relations: G = 9 (UIH,C) , H .

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. Van der Meer [ 6 1 has computed the relevant part of its Taylor expansion and checked that it is nondegenerate. Remark 5. If one prescribes the period w of the sought periodic solution then it is easy to show the existence of w-dependent diffeomorphisms which bring the periodic solutions into normal form.

And H = e has n o t 1 kj s u c h a s i m p l e form, as ( 6 . 8 ) , it i s a l s o c o n v e n i e n t f o r some c a l c u l a t i o n s . , 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 .

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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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