The Euler-Jacobi formula and the planar quadratic-quartic polynomial differential systems
Authors:
Jaume Llibre and Claudia Valls
Journal:
Proc. Amer. Math. Soc. 149 (2021), 135-141
MSC (2010):
Primary 34A05; Secondary 34C05, 37C10
DOI:
https://doi.org/10.1090/proc/15257
Published electronically:
October 16, 2020
MathSciNet review:
4172592
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: The Euler-Jacobi formula provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar quadratic-quartic polynomial differential systems when these systems have eight finite singular points.
- V. Arnold, A. Varchenkom, and S. Goussein-Zude, Singularités des applications différentialbes, Mir, Moscow, 1982.
- A. N. Berlinskiĭ, On the behavior of the integral curves of a differential equation, Izv. Vysš. Učebn. Zaved. Matematika 1960 (1960), no. 2 (15), 3–18 (Russian). MR 0132249
- Carmen Chicone and Jing Huang Tian, On general properties of quadratic systems, Amer. Math. Monthly 89 (1982), no. 3, 167–178. MR 645790, DOI https://doi.org/10.2307/2320199
- Anna Cima, Armengol Gasull, and Francesc Mañosas, Some applications of the Euler-Jacobi formula to differential equations, Proc. Amer. Math. Soc. 118 (1993), no. 1, 151–163. MR 1150647, DOI https://doi.org/10.1090/S0002-9939-1993-1150647-9
- Anna Cima and Jaume Llibre, Configurations of fans and nests of limit cycles for polynomial vector fields in the plane, J. Differential Equations 82 (1989), no. 1, 71–97. MR 1023302, DOI https://doi.org/10.1016/0022-0396%2889%2990168-X
- A. G. Hovanskiĭ, The index of a polynomial vector field, Funktsional. Anal. i Prilozhen. 13 (1979), no. 1, 49–58, 96 (Russian). MR 527521
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 34A05, 34C05, 37C10
Retrieve articles in all journals with MSC (2010): 34A05, 34C05, 37C10
Additional Information
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
MR Author ID:
115015
ORCID:
0000-0002-9511-5999
Email:
jllibre@mat.uab.cat
Claudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal
MR Author ID:
636500
Email:
cvalls@math.ist.utl.pt
Keywords:
Euler-Jacobi formula,
singular points,
topological index,
polynomial differential systems
Received by editor(s):
March 21, 2019
Received by editor(s) in revised form:
June 10, 2019
Published electronically:
October 16, 2020
Additional Notes:
The first author was supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911.
The second author was partially supported by FCT/Portugal through UID/MAT/04459/2013.
Communicated by:
Wenxian Shen
Article copyright:
© Copyright 2020
American Mathematical Society
