Reversibility and quasi-homogeneous normal forms of vector fields

• Published in: Nonlinear analysis Vol. 73; no. 2; pp. 510 - 525
• Main Authors: Algaba, A., García, C., Teixeira, M.A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 15-07-2010; Elsevier
• Subjects: Algorithms; Center; Exact sciences and technology; Mathematical analysis; Mathematics; Nonlinearity; Normal form; Quasi-homogeneous system; Reversibility; Sciences and techniques of general use; Singularity; Triangles; Vectors (mathematics); Singularity; Reversibility; Center; Quasi-homogeneous system; Normal form; Nonlinear analysis; Mathematical model; Vector field
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2010.03.046

This paper uses tools in quasi-homogeneous normal form theory to discuss certain aspects of reversible vector fields around an equilibrium point. Our main result provides an algorithm, via Lie Triangle, that detects the non-reversibility of vector fields. That is, it is possible to decide whether a planar center is not reversible. Some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields. We find several families of nilpotent centers which are non-reversible.