ANALYTICAL INTEGRABILITY OF PERTURBATIONS OF DEGENERATE QUADRATIC SYSTEMS

Antonio Algaba; Cristóbal García; Manuel Reyes; Jaume Giné

doi:10.11948/20230188

2024 Volume 14 Issue 2

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Antonio Algaba, Cristóbal García, Manuel Reyes, Jaume Giné. ANALYTICAL INTEGRABILITY OF PERTURBATIONS OF DEGENERATE QUADRATIC SYSTEMS[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 864-885. doi: 10.11948/20230188

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Antonio Algaba, Cristóbal García, Manuel Reyes, Jaume Giné. ANALYTICAL INTEGRABILITY OF PERTURBATIONS OF DEGENERATE QUADRATIC SYSTEMS[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 864-885. doi: 10.11948/20230188

ANALYTICAL INTEGRABILITY OF PERTURBATIONS OF DEGENERATE QUADRATIC SYSTEMS

1.
Departamento de Ciencias Integradas, Center of Advanced Studies in Physics, Mathematics and Computation, Universidad de Huelva, 21071, Spain
2.
Departament de Matemàtica, Universitat de Lleida, Av. Jaume II, 69, 25001 Lleida, Catalonia, Spain

Author Bio: Email: algaba@uhu.es(A. Algaba); Email: cristoba@uhu.es(C. García); Email: colume@uhu.es(M. Reyes)
Corresponding author: Email: jaume.gine@udl.cat(J. Giné)
Fund Project: This work has been partially supported by Ministerio de Ciencia, Innovación y Universidades, Spain (project PGC2018-096265-B-I00) and by Consejería de Economía, Innovación, Ciencia y Empleo de la Junta de Andalucía, Spain (projects FQM-276, UHU-1260150 and P12-FQM-1658). The fourth author is partially supported by the Agencia Estatal de Investigación grant PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya) grant number 2021SGR 01678

Abstract

Abstract

We consider analytic perturbations of quadratic homogeneous differential systems having an isolated singularity at the origin. Here we characterize the analytically integrable perturbations of quadratic homogeneous systems of the form $ (\dot{x}, \dot{y})^T=f_1(P_1, Q_1)^T $ with $ f_1(x, y) $ a non-zero linear homogeneous polynomial and $ P_1(x, y), Q_1(x, y) $ non-zero linear homogeneous polynomials without common factors. We prove that all systems are orbitally equivalent to their quasi-homogeneous leading terms with respect to a certain type but not necessarily to the homogeneous leading terms. This result completes the previous results for the analytic perturbations of irreducible quadratic systems with analytic first integral which are orbitally equivalent to the homogeneous leading term, i.e. all are homogenizable.
- Analytic integrability /
- quadratic differential systems /
- degenerate singular points /
- orbitally equivalence
MSC: 34C05, 34A05, 34C20, 34C14

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References

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