# American Institute of Mathematical Sciences

March  2021, 20(3): 1077-1089. doi: 10.3934/cpaa.2021007

## Rational limit cycles of Abel equations

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain 2 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal

* Corresponding author

Received  June 2020 Revised  November 2020 Published  March 2021 Early access  January 2021

Fund Project: The first author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants 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 is partially supported by FCT/Portugal through UID/MAT/04459/2019

We deal with Abel equations $dy/dx = A(x) y^2 + B(x) y^3$, where $A(x)$ and $B(x)$ are real polynomials. We prove that these Abel equations can have at most two rational limit cycles and we characterize when this happens. Moreover we provide examples of these Abel equations with two nontrivial rational limit cycles.

Citation: Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1077-1089. doi: 10.3934/cpaa.2021007
##### References:
 [1] A. Álvarez, J. L. Bravo and M. Fernández, The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign, Commun. Pure Appl. Anal., 8 (2009), 1493-1501. doi: 10.3934/cpaa.2009.8.1493. [2] M. J. Álvarez, J. L. Bravo and M. Fernández, Existence of non-trivial limit cycles in Abel equations with symmetries, Nonlinear Anal., 84 (2013), 18-28. doi: 10.1016/j.na.2013.02.001. [3] A. Álvarez, J. L. Bravo and M. Fernández, Limit cycles of Abel equations of the first kind, J. Math. Anal. Appl., 423 (2015), 734-745. doi: 10.1016/j.jmaa.2014.10.019. [4] M. J. Álvarez, J. L. Bravo, M. Fernández and R. Prohens, Centers and limit cycles for a family of Abel equations, J. Math. Anal. Appl., 453 (2017), 485-501. doi: 10.1016/j.jmaa.2017.04.017. [5] M. J. Álvarez, J. L. Bravo, M. Fernández and R.Prohens, Alien limit cycles in Abel equations, J. Math. Anal. Appl., 482 (2020), 123525, 20 pp. doi: 10.1016/j.jmaa.2019.123525. [6] M. J. Álvarez, A. Gasull and J. Yu, Lower bounds for the number of limit cycles of trigonometric Abel equations, J. Math. Anal. Appl., 342 (2008), 682-693. doi: 10.1016/j.jmaa.2007.12.016. [7] M. A. M Alwash and N. G. Lloyd, Non-autonomous equations related to polylnomial two-dimensional systems, P. Roy. Soc. Edinb. A, 105 (1987), 129-152. doi: 10.1017/S0308210500021971. [8] M. Blinov, M. Briskin and Y. Yomdin, Center conditions: parametric and model center problems, Israel J. Math., 118 (2000), 61-108. doi: 10.1007/BF02803517. [9] J. L. Bravo, M. Fernández and A. Gasull, Limit cycles for some Abel equations having coefficients without fixed signs, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 3869-3876. doi: 10.1142/S0218127409025195. [10] J. L. Bravo, M. Fernández and A. Gasull, Stability of singular limit cycles for Abel equations, Discret. Contin. Dyn. S., 35 (2015), 1873-1890. doi: 10.3934/dcds.2015.35.1873. [11] E. Fossas, J. M. Olm and H. Sira-Ramírez, Iterative approximation of limit cycles for a class of Abel equations, Phys. D, 237 (2008), 3159-3164. doi: 10.1016/j.physd.2008.05.011. [12] J. P. Françoise, Local bifurcations of limit cycles, Abel equations and Liénard systems, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Kluwer Acad. Publ., Dordrecht, 2004. doi: 10.1007/978-94-007-1025-2_4. [13] J. P. Françoise, Integrability and limit cycles for Abel equations, Banach Center Publ., Warsaw, 2011. doi: 10.4064/bc94-0-11. [14] A. Gasull and J. Llibre, Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990), 1235-1244. doi: 10.1137/0521068. [15] A. Gasull, From Abel's differential equations to Hilbert's sixteenth problem, (Catalan), Butl. Soc. Catalana Mat., 28 (2013), 123-146. [16] J. Giné, M. Grau and J. Llibre, On the polynomial limit cycles of polynomial differential equations, Israel J. Math., 181 (2011), 461-475. doi: 10.1007/s11856-011-0019-3. [17] J. Huang and H. Liang, Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves, Nonlinear Differ. Equ. Appl., 24 (2017), 31 pp. doi: 10.1007/s00030-017-0469-3. [18] Y. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity, 13 (2000), 1337-1342. doi: 10.1088/0951-7715/13/4/319. [19] C. Liu, C. Li, X. Wang and J. Wu, On the rational limit cycles of Abel equations, Chaos, Solitons and Fractals, 110 (2018), 28-32. doi: 10.1016/j.chaos.2018.03.004. [20] N. G. Lloyd, A note on the number of limit cycles in certain two-dimensional systems, J London Math Soc., 20 (1979), 277-286. doi: 10.1112/jlms/s2-20.2.277. [21] A. L. Neto, On the number of solutions of the equation $\frac{dx}{dt} = \sum_{j = 0}^n a_j (t) x^j$, $0 \le t \le 1$, for which $x(0) = x(1)$, Invent. Math., 59 (1980), 67-76. doi: 10.1007/BF01390315. [22] P. Torres, Existence of closed solutions for a polynomial first order differential equation, J. Math. Anal. Appl., 328 (2007), 1108-1116. doi: 10.1016/j.jmaa.2006.05.078. [23] G. D. Wang and W. C. Chen, The number of closed solutions to the Abel equation and its application, (Chinese), J. Systems Sci. Math. Sci., 25 (2005), 693-699. [24] X. D. Xie and S. L. Cai, The number of limit cycles for the Abel equation and its application(Chinese), Gaoxiao Yingyong Shuxue Xuebao Ser. A, 9 (1994), 266-274. [25] J. F. Zhang, Limit cycles for a class of Abel equations with coefficients that change sign(Chinese), Chinese Ann. Math. Ser. A, 18 (1997), 271-278. [26] J. F. Zhang and X. X. Chen, Some criteria for limit cycles of a class of Abel equations(Chinese), J. Fuzhou Univ. Nat. Sci. Ed., 27 (1999), 9-11.

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##### References:
 [1] A. Álvarez, J. L. Bravo and M. Fernández, The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign, Commun. Pure Appl. Anal., 8 (2009), 1493-1501. doi: 10.3934/cpaa.2009.8.1493. [2] M. J. Álvarez, J. L. Bravo and M. Fernández, Existence of non-trivial limit cycles in Abel equations with symmetries, Nonlinear Anal., 84 (2013), 18-28. doi: 10.1016/j.na.2013.02.001. [3] A. Álvarez, J. L. Bravo and M. Fernández, Limit cycles of Abel equations of the first kind, J. Math. Anal. Appl., 423 (2015), 734-745. doi: 10.1016/j.jmaa.2014.10.019. [4] M. J. Álvarez, J. L. Bravo, M. Fernández and R. Prohens, Centers and limit cycles for a family of Abel equations, J. Math. Anal. Appl., 453 (2017), 485-501. doi: 10.1016/j.jmaa.2017.04.017. [5] M. J. Álvarez, J. L. Bravo, M. Fernández and R.Prohens, Alien limit cycles in Abel equations, J. Math. Anal. Appl., 482 (2020), 123525, 20 pp. doi: 10.1016/j.jmaa.2019.123525. [6] M. J. Álvarez, A. Gasull and J. Yu, Lower bounds for the number of limit cycles of trigonometric Abel equations, J. Math. Anal. Appl., 342 (2008), 682-693. doi: 10.1016/j.jmaa.2007.12.016. [7] M. A. M Alwash and N. G. Lloyd, Non-autonomous equations related to polylnomial two-dimensional systems, P. Roy. Soc. Edinb. A, 105 (1987), 129-152. doi: 10.1017/S0308210500021971. [8] M. Blinov, M. Briskin and Y. Yomdin, Center conditions: parametric and model center problems, Israel J. Math., 118 (2000), 61-108. doi: 10.1007/BF02803517. [9] J. L. Bravo, M. Fernández and A. Gasull, Limit cycles for some Abel equations having coefficients without fixed signs, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 3869-3876. doi: 10.1142/S0218127409025195. [10] J. L. Bravo, M. Fernández and A. Gasull, Stability of singular limit cycles for Abel equations, Discret. Contin. Dyn. S., 35 (2015), 1873-1890. doi: 10.3934/dcds.2015.35.1873. [11] E. Fossas, J. M. Olm and H. Sira-Ramírez, Iterative approximation of limit cycles for a class of Abel equations, Phys. D, 237 (2008), 3159-3164. doi: 10.1016/j.physd.2008.05.011. [12] J. P. Françoise, Local bifurcations of limit cycles, Abel equations and Liénard systems, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Kluwer Acad. Publ., Dordrecht, 2004. doi: 10.1007/978-94-007-1025-2_4. [13] J. P. Françoise, Integrability and limit cycles for Abel equations, Banach Center Publ., Warsaw, 2011. doi: 10.4064/bc94-0-11. [14] A. Gasull and J. Llibre, Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990), 1235-1244. doi: 10.1137/0521068. [15] A. Gasull, From Abel's differential equations to Hilbert's sixteenth problem, (Catalan), Butl. Soc. Catalana Mat., 28 (2013), 123-146. [16] J. Giné, M. Grau and J. Llibre, On the polynomial limit cycles of polynomial differential equations, Israel J. Math., 181 (2011), 461-475. doi: 10.1007/s11856-011-0019-3. [17] J. Huang and H. Liang, Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves, Nonlinear Differ. Equ. Appl., 24 (2017), 31 pp. doi: 10.1007/s00030-017-0469-3. [18] Y. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity, 13 (2000), 1337-1342. doi: 10.1088/0951-7715/13/4/319. [19] C. Liu, C. Li, X. Wang and J. Wu, On the rational limit cycles of Abel equations, Chaos, Solitons and Fractals, 110 (2018), 28-32. doi: 10.1016/j.chaos.2018.03.004. [20] N. G. Lloyd, A note on the number of limit cycles in certain two-dimensional systems, J London Math Soc., 20 (1979), 277-286. doi: 10.1112/jlms/s2-20.2.277. [21] A. L. Neto, On the number of solutions of the equation $\frac{dx}{dt} = \sum_{j = 0}^n a_j (t) x^j$, $0 \le t \le 1$, for which $x(0) = x(1)$, Invent. Math., 59 (1980), 67-76. doi: 10.1007/BF01390315. [22] P. Torres, Existence of closed solutions for a polynomial first order differential equation, J. Math. Anal. Appl., 328 (2007), 1108-1116. doi: 10.1016/j.jmaa.2006.05.078. [23] G. D. Wang and W. C. Chen, The number of closed solutions to the Abel equation and its application, (Chinese), J. Systems Sci. Math. Sci., 25 (2005), 693-699. [24] X. D. Xie and S. L. Cai, The number of limit cycles for the Abel equation and its application(Chinese), Gaoxiao Yingyong Shuxue Xuebao Ser. A, 9 (1994), 266-274. [25] J. F. Zhang, Limit cycles for a class of Abel equations with coefficients that change sign(Chinese), Chinese Ann. Math. Ser. A, 18 (1997), 271-278. [26] J. F. Zhang and X. X. Chen, Some criteria for limit cycles of a class of Abel equations(Chinese), J. Fuzhou Univ. Nat. Sci. Ed., 27 (1999), 9-11.
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