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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.56 no.1 Bogotá Jan./June 2022 Epub Jan 02, 2024
https://doi.org/10.15446/recolma.v56n1.105621
Original article
On the invariant rational curves of a certain family of polynomial differential equations
Sobre las curvas racionales invariantes de cierta familia de ecuaciones diferenciales
1 Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México
In this work, we present sufficient conditions to determine if the limit cycles of certain differential systems in the plane are algebraic or not. In particular, we obtain criteria such that the limit cycles of equations derived from predatory prey models with rational functional response are necessarily transcendental ovals.
Keywords: algebraic limit-cycles; Puiseux series; Newton polygon; predator-prey models; functional-response
En este trabajo presentamos condiciones necesarias y suficientes para determinar si los ciclos límite de ciertas ecuaciones diferenciales en el plano son algebraicos o no. Particularmente, obtenemos criterios para que ciclos límite de ciertas ecuaciones derivadas de modelos depredador - presa con ciertos funcionales racionales de respuesta sean necesariamente óvalos trascendentes.
Palabras clave: ciclos límte algebraicos; series de Puiseux; polígono de Newton; modelos depredador-presa; funcional de respuesta
References
1. J. Cano, An extension of the Newton-Puiseux polygon construction to give solutions of Pfaffian forms, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 1, 125-142. [ Links ]
2. M. V. Demina, Invariant algebraic curves for Liénard dynamical systems revisited, Applied Mathematics Letters 84 (2018), 42-48. [ Links ]
3. M. V. Demina, Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems, Phys. Lett. A 382 (2018), no. 20, 1353-1360. [ Links ]
4. H. Díaz-Marín and O. Osuna, Non-algebraic limit cycles in Holling type III zooplankton-phytoplankton models, CUBO, A Mathematical Journal 23 (2021), no. 03, 343-355. [ Links ]
5. A. Ferragut and A. Gasull, Non-algebraic oscillations for predator-prey models, New Trends in Dynamical Systems. Salou, 2012, vol. EXTRA Publ. Mat., Universitat Autònoma de Barcelona, Departament de Matemàtiques, 2014, pp. 195-207. [ Links ]
6. I. Garcia, Transcendental limit cycles via the structure of arbitrary degree invariant algebraic curves of polynomial planar vector felds, Rocky Mountain J. Math. 35 (2005), 501-515. [ Links ]
7. I. A. García and J. Giné, Non-algebraic invariant curves for polynomial planar vector fields, Discrete and Continuous Dynamical Systems 10 (2004), no. 3, 755-768. [ Links ]
8. A. Gasull, H. Giacomini, and J. Torregrosa, Explicit non-algebraic limit cycles for polynomial systems, J. Comput. Appl. Math 20 (2007), 448-457. [ Links ]
9. J. Giné and M. Grau, Coexistence of algebraic and non-algebraic limit cycles explicitly given using Riccati equations, Nonlinearity 19 (2006), no. 8, 1939-1950. [ Links ]
10. J. Giné and J. Llibre, Formal Weierstrass nonintegrability criterion for some classes of polynomial differential systems in C2, International Journal of Bifurcation and Chaos 30 (2020), no. 4, 2050064. [ Links ]
11. J. Giné and J. Llibre , Strongly formal weierstrass non-integrability for polynomial differential systems in c2, Electronic Journal of Qualitative Theory of Differential Equations 1 (2020), no. 1-16. [ Links ]
12. M. Hayashi, On polynomial Liénard systems which have invariant algebraic curves, Funkcial. Ekvac. 39 (1996), no. 3, 403-408. [ Links ]
13. E. Hille, Ordinary Differential Equations in the Complex Domain, Dover Publications, Inc., Mineola, NY, 1976. [ Links ]
14. E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. [ Links ]
15. K. Odani, The limit cycle of the van der Pol equation is not algebraic, Journal of Differential Equations 115 (1995), no. 1, 146-152. [ Links ]
16. O. Osuna, S. Rebollo-Perdomo, and G. Villasenor-Aguilar, On a class of invariant algebraic curves for Kukles systems, Electronic Journal of Qualitative Theory of Differential Equations 2016 (2016), no. 61, 1-12. [ Links ]
17. A. Rojas-Palma and E. González-Olivares, Gause type predator-prey models with a generalized rational non-monotonic functional response, Proceedings of the 14th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2014, vol. 1, 2014. [ Links ]
Received: June 05, 2021; Accepted: July 05, 2022
This is an open-access article distributed under the terms of the Creative Commons Attribution License
