Abstract
In this paper, we consider the family of generalized Abel equations of the form
where A, B are trigonometric polynomials and \(m,n\in \mathbb {N}\). We characterize the existence of non-trivial limit cycles in this family, in terms of the trigonometric monomials.
Similar content being viewed by others
References
Álvarez, A., Bravo, J.L., Fernández, M.: The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign. Commun. Pure Appl. Anal. 8–5, 1493–1501 (2009)
Álvarez, A., Bravo, J.L., Fernández, M.: Limit cycles of Abel equations of the first kind. J. Math. Anal. Appl. 423(1), 734–745 (2015)
Álvarez, M.J., Bravo, J.L., Fernández, M.: Uniqueness of limit cycles for polynomial first-order differential equations. J. Math. Anal. Appl. 360(1), 168–189 (2009)
Álvarez, M.J., Bravo, J.L., Fernández, M.: Abel-like differential equations with a unique limit cycle. Nonlinear Anal. 74, 3694–3702 (2011)
Álvarez, M.J., Bravo, J.L., Fernández, M.: Existence of non-trivial limit cycles in Abel equations with symmetries. Nonlinear Anal. 84, 18–28 (2013)
Álvarez, M.J., Bravo, J.L., Fernández, M., Prohens, R.: Centers and limit cycles for a family of Abel equations. J. Math. Anal. Appl. 453, 485–501 (2017)
Álvarez, M.J., Bravo, J.L., Fernández, M., Prohens, R.: Alien limit cycles in Abel equations. J. Math. Anal. Appl. 482(1), 123525 (2020)
Álvarez, M.J., Gasull, A., Giacomini, H.: A new uniqueness criterion for the number of periodic orbits of Abel equations. J. Differ. Equ. 234, 161–176 (2007)
Alwash, M.A.M.: Periodic solutions of Abel differential equations. J. Math. Anal. Appl. 329, 1161–1169 (2007)
Alwash, M.A.M.: Periodic solutions of polynomial non-autonomous differential equations. Electron. J. Differ. Equ. 2005(84), 1–8 (2005)
Benardete, D.M., Noonburg, V.W., Pollina, B.: Qualitative tools for studying periodic solutions and bifurcations as applied to the periodically harvested logistic equation. Am. Math. Mon. 115(3), 202–219 (2008)
Bravo, J.L., Fernández, M.: Limit cycles of non-autonomous scalar ODEs with two summands. Commun. Pure Appl. Anal. 12–2, 1091–1102 (2013)
Bravo, J.L., Fernández, M., Gasull, A.: Limit cycles for some Abel equations with coefficients without fixed signs. Int. J. Bifurc. Chaos 19–11, 389–3876 (2009)
Bravo, J.L., Torregrosa, J.: Abel-like equations with no periodic solutions. J. Math. Anal. Appl. 342, 931–942 (2008)
Briskin, M., Françoise, J.P., Yomdin, Y.: Center conditions II: parametric and model center problems. Isr. J. Math. 118, 61–82 (2000)
Cherkas, L.A.: Number of limit cycles of an autonomous second-order system. Differ. Uravn. 12, 944–946 (1975)
Gasull, A., Guillamon, A.: Limit cycles for generalized Abel equations. Int. J. Bifurc. Chaos 16, 3737–3745 (2006)
Gasull, A., Llibre, J.: Limit cycles for a class of Abel equations. SIAM J. Math. Anal. 21–5, 1235–1244 (1990)
Gasull, A., Prohens, R., Torregrosa, J.: Limit cycles for rigid cubic systems. J. Math. Anal. Appl. 303, 391–404 (2005)
Gasull, A., Torregrosa, J.: Some results on rigid systems. In: International Conference on Differential Equations (Equadiff-2003), pp. 340–345. World Scientific Publishing, Hackensack, NJ (2005)
Huang, J., Liang, H.: A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities. J. Math. Anal. Appl. 457(1), 498–521 (2018)
Huang, J., Liang, H.: A geometric criterion for equation \(\dot{x} = \sum _{i=0}^m a_i(t)x^i\) having at most m isolated periodic solutions. Preprint, arXiv:1606.04776
Huang, J., Liang, H.: Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves. Nonlinear Differ. Equ. Appl. 24, 47 (2017). https://doi.org/10.1007/s00030-017-0469-3
Huang, J., Liang, H., Llibre, J.: Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities. J. Differ. Equ. 265(9), 3888–3913 (2018)
Huang, J., Zhao, Y.: Periodic solutions for equation \(x^{\prime }=A(t)x^m+B(t)x^n+C(t)x^l\) with \(A(t)\) and \(B(t)\) changing signs. J. Differ. Equ. 253, 73–99 (2012)
Lloyd, N.G.: A note on the number of limit cycles in certain two-dimensional systems. J. Lond. Math. Soc. 20, 277–286 (1979)
Lins-Neto, A.: On the number of solutions of the equation \(\frac{d x}{dt}=\sum _{j=0}^n a_j(t)x^j\), \(0\le t\le 1\), for which \(x(0)=x(1)\). Inv. Math. 59, 67–76 (1980)
Panov, A.A.: The number of periodic solutions of polynomial differential equations. Math. Notes 64–5, 622–628 (1998)
Pliss, V.A.: Non Local Problems of the Theory of Oscillations. Academic Press, New York (1966)
Smale, S.: Mathematical problems for the next century. Math. Intell. 20, 7–15 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The four authors are partially supported by MINECO/FEDER Grant No. MTM2017-83568-P. J.L.B. and M.F. are also partially supported by the Junta de Extremadura/FEDER Grant Nos. GR18023 and IB18023.
Rights and permissions
About this article
Cite this article
Álvarez, M.J., Bravo, J.L., Fernández, M. et al. Characterization of the Existence of Non-trivial Limit Cycles for Generalized Abel Equations. Qual. Theory Dyn. Syst. 20, 15 (2021). https://doi.org/10.1007/s12346-021-00450-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-021-00450-4