Abstract
A phase portrait of a vector field on a plane is called completely symmetric if it is invariant with respect to the group consisting of four involutionsi 1,i 2,i 1 i 2,id. The simplest example is a local center defined by the germ of an analytic vector field with a non-degenerate linear approximation. By the Poincare-Lyapunov theorem such a center is diffeomorphic to the center defined by the vector field\(\dot x_1 = x_2 ,\dot x_2 = - x_1 \) and consequently it is is completely symmetric. The paper is devoted to the classification of completely symmetric centers defined by germs of vector fields with a nilpotent linear approximation and by germs of vector fields with zero 2-jet and generic 3-jet.
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Dedicated to Jorge Sotomayor on his 60th birthday
This research was supported by the Fund for the Promotion of Research at the Technion. I am thankful to A. Davydov, M.A. Teixeira and S. Yakovenko for discussing local classification problems involving involutions. This paper would never have been written if J. Llibre did not explain me the world of centers and various approaches for their analysis.
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Zhitomirskii, M. Completely symmetric centers. Qual. Th. Dyn. Syst 4, 455–470 (2004). https://doi.org/10.1007/BF02970870
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DOI: https://doi.org/10.1007/BF02970870