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March 25, 2015 By

Center Problem for Odes with Coefficients Generating the Group of Rectangular Paths

C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (2) 2009, pp. 33–44

June 30, 2009

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

Abstract/Résumé:

We solve an analog of the Poincaré Center-Focus problem for the class of ordinary differential equations \(\frac{dv}{dx}=\sum_{i=1}^{\infty}a_{i}(x)\,v^{i+1}\), such that the first integrals of vectors of their coefficients determine rectangular paths in finite-dimensional complex vector spaces. In particular, we prove that all centers of such equations are universal, i.e., are determined by means of certain composition conditions. Also, we solve the Bautin problem on the number of periodic solutions with sufficiently small initial values for finite-dimensional families of such equations.

Nous résolvons l’analogue du problème du centre-foyer de Poincaré pour les équations différentielles ordinaires \(\frac{dv}{dx}=\sum_{i=1}^{\infty}a_{i}(x)\,v^{i+1}\) telles que les premières intégrales des vecteurs de leurs coefficients déterminent des chemins rectangulaires dans les espaces vectoriels complexes de dimension finie. En particulier, nous démontrons que tous les centres de telles équations sont universelles, c’est-à-dire déterminées par certaines conditions composées. De plus, nous résolvons le problème de Bautin sur le nombre de solutions périodiques ayant des valeurs initiales suffisamment petites pour les familles de dimension finie de telles équations..

Keywords: center problem, iterated integrals, the group of rectangular paths
AMS Subject Classification: Normal forms; center manifold theory; bifurcation theory 37L10

[This journal is open access except for the current year and the preceding 5 years]

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