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

Rational limit cycles of Abel equations

Jaume Llibre 1,, and  Claudia Valls 2,

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  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.

Keywords: Algebraic limit cycles, rational limit cycles, Abel equations.
Mathematics Subject Classification: Primary: 34C05, 34C07, 34C08.
Citation: Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. P. Françoise, Integrability and limit cycles for Abel equations, Banach Center Publ., Warsaw, 2011. doi: 10.4064/bc94-0-11.  Google Scholar

[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.  Google Scholar

[15]

A. Gasull, From Abel's differential equations to Hilbert's sixteenth problem, (Catalan), Butl. Soc. Catalana Mat., 28 (2013), 123-146.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. P. Françoise, Integrability and limit cycles for Abel equations, Banach Center Publ., Warsaw, 2011. doi: 10.4064/bc94-0-11.  Google Scholar

[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.  Google Scholar

[15]

A. Gasull, From Abel's differential equations to Hilbert's sixteenth problem, (Catalan), Butl. Soc. Catalana Mat., 28 (2013), 123-146.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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