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March 25, 2015 By

Free Subgroups of the Group of Formal Power Series and the Center Problem for ODEs

C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (4) 2009, pp. 97–106

December 30, 2009

Alexander Brudnyi, Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4; email: albru@math.ucalgary.ca

Abstract/Résumé:

The paper belongs to the area related to the famous Poincaré center-focus problem and contains a new necessary and sufficient condition for existence of a center for ordinary differential equations with coefficients derived algebraically from a certain “basic” class. This class consists of families of equations \(\frac{dv}{dx} = \sum_{j=1}^{\infty} a_j (x) \,v^{j+1}\) whose first return maps generate free subgroups of the group of formal power series. It is shown that such families form a sufficiently “massive” subset in the set of all possible equations as above. The paper contains various characterizations of this “basic” class. It follows the lines of the author’s approach to the center-focus problem (involving modern algebraic techniques) that already deepened the understanding of the problem.

Cet article porte sur le fameux problème du centre-foyer de Poincaré et contient une nouvelle condition nécessaire et suffisante pour l’existence d’un centre pour les équations diffèrentielles ordinaires avec des coéfficients derivés algébriquement d’une certaine classe de “base”. Cette classe consiste en des familles d’équations de la forme \(\frac{dv}{dx} = \sum_{j=1}^{\infty} a_j (x) \,v^{j+1}\) dont les premières fonctions de retour engendrent des sous groupes libres d’un groupe de séries entières formelles. On démontre que de telles familles forment un sous ensemble suffisamment “massif” dans l’ensemble de toutes les équations possible ci-dessus. L’article contient des diverses caractérisations de cette classe de “base”. Il poursuit les directions de l’auteur sur le problème du centre-foyer (selon les techniques algébraiques modernes) qui ont déjà approfondies les connaissances du problème.

Keywords: center problem, free group, the group of formal power series
AMS Subject Classification: Normal forms; center manifold theory; bifurcation theory 37L10

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform classification Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup elliptic curves fixed point Fourier transform function fields. functoriality general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping numerical range orthogonal polynomials Predual space prime number property SP Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces uniqueness Weak Markov set Whitney problems

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