By O. Axelsson, L.S. Frank and A. Van Der Sluis (Eds.)
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Additional info for Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Proceedings ofthe Conference on Analytical and Numerical Approachesto Asymptotic Problems
S . F r a n k , A . ) ON THE QUESTION OF THE EXISTENCE AND NATURE OF HOMOGENEOUS-CENTER TARGET PATTERNS IN THE BELOUSOV-ZHABOTINSKII REAGENT Paul C. Fife' Mathematics Department University of Arizona Tucson, Arizona USA This continues previous work by Tyson and the author on the application of multiple-scaling techniques to the modeling and analysis of the expanding concentric rings of chemical activity seen in the Belousov-Zhabotinskii reagent. The model is based on "Oregonator" kinetics. Previous work with these kinetics concentrated on heterogeneous-center patterns, induced by an external particle or other stimulus; this explores the possibility of homogeneous center structures, more controversial but reportedly observed.
K = x ... , , T~ in the 10,2nl. Then we immediately have the following statement: Y. a) In order the 211-periodic solutions yo,yl of equations ( 4 . +1 t i o n s ( 5 . 8 ) , k = 1. o e O L S u b s t i t u t i n g i n Q . [ y 1 and p , c y 1 t h e d e r i v a t i v e s S . and S , f o r U . ( E ) and V . ( E l , 3 0. 11, w e o b t a i n f o r E 0 , . . , E 2 e the required q u a s i - l i n e a r s y s t e m o f e q u a t i o n s . Note t h a t t h e i n t e g r a l s i n ( 6 . 6 ) can be w r i t t e n i n t h e form, which d o e s n o t r e q u i r e t h a t t h e p e r i o d s w i t h r e s p e c t t o t h e a r g u 2 ments ' T ~ , .
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.