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

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

2001-01-23

arxiv.org/abs/math-ph/0101022v3

Physics

Mathematical Physics

Shortened, streamlined version (with revised title) posted on 15 july 2002; now 35 pages

Scientific paper

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. Poincar\'e and Lie renormalized forms [{\it Lett. Math. Phys.} {\bf 42} (1997), 103-114; {\it Ann. Inst. H. Poincar\'e (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 Poincar\'e 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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