CENTERS FOR THE KUKLES HOMOGENEOUS SYSTEMS WITH EVEN DEGREE

Jaume Giné; Jaume Llibre; Claudia Valls

doi:10.11948/2017093

2017 Volume 7 Issue 4

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Jaume Giné, Jaume Llibre, Claudia Valls. CENTERS FOR THE KUKLES HOMOGENEOUS SYSTEMS WITH EVEN DEGREE[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1534-1548. doi: 10.11948/2017093

Citation:

Jaume Giné, Jaume Llibre, Claudia Valls. CENTERS FOR THE KUKLES HOMOGENEOUS SYSTEMS WITH EVEN DEGREE[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1534-1548. doi: 10.11948/2017093

CENTERS FOR THE KUKLES HOMOGENEOUS SYSTEMS WITH EVEN DEGREE

1 Departament de Matemàtica, Universitat de Lleida, Avda. Jaume Ⅱ, 69;25001 Lleida, Catalonia, Spain;
2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain;
3 Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal

Abstract

Abstract

For the polynomial differential system Ẋ=-y, ẏ=x + Q_n(x,y), where Q_n(x,y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n ≥ 2 has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all n even. More precisely, we prove both conjectures in the case n=4 and for n ≥ 6 even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of n odd was studied in[8].
- Center-focus problem /
- isochronous center /
- Poincaré-Liapunov constants /
- Gröbner basis of polynomial systems
MSC: 34C05;37C10

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References

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