Reversibility and quasi-homogeneous normal forms of vector fields
The authors use tools of quasi-homogeneous normal form theory to study the reversibility of a vector field. The property of reversibility (time-reversal symmetry) of a planar vector field is a sufficient condition for a monodromic singular point to be a center. This provides a strong motivation to study the reversibility of a vector field since the center problem is one of the main open problem in the qualitative theory of planar dynamical systems. In fact, for any planar system having a non-degenerate (respectively, nilpotent) center, there exists a analytic transformation such that the system becomes reversible (respectively, orbitally reversible). Hence, the reversibility problem can provide new information in the center problem. The main results of the paper are the following: There are given necessary conditions for the reversibility of a vector field using reversible generators, see Theorem 2.1. It provides an algorithm, via Lie triangle, that detects the non-reversibility of vector fields, see Theorem 3.19. These results are applied to several examples of nilpotent and degenerate centers. The paper provides families of nilpotent centers which are non-reversible.
scientific article; zbMATH DE number 5720180