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Poincare' and Lie renormalized forms for regular singular points of vector fields in the plane

Giuseppe Gaeta
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Abstract

We discuss the local behaviour of vector fields in the plane $\R^2$ around a regular singular point, using recently introduced reduced normal forms, i.e. Poincaré and Lie renormalized forms [{\it Lett. Math. Phys.} {\bf 42} (1997), 103-114; {\it Ann. Inst. H. Poincaré (Phys. Theo.)} {\bf 70} (1999), 461-514; {\it Lett. Math. Phys.} {\bf 57} (2001), 41-60]. We give a complete classification, and provide explicit formulas, using Poincaré renormalized forms for non-degenerate cases, and Lie ones for certain degenerate cases. Both schemes are completely algorithmic, prove to be easy to implement, and only require to solve linear equations.

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Giuseppe Gaeta. 2002-07-15. Poincare' and Lie renormalized forms for regular singular points of vector fields in the plane. https://arxiv.org/abs/math-ph/0101022

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