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A partial positive answer to a Lijun–Yun conjecture

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Abstract

In this paper, we prove a result about the vanishing of the moments for a class of Abel equation and we give a partial positive answer to a conjecture proposed by Lijun and Yun (J Math Anal Appl 261:100–112, 2001).

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References

  1. Algaba, A., García, C., Giné, J.: Nondegenerate and nilpotent centers for a cubic system of differential equations. Qual. Theory Dyn. Syst. 18(1), 333–345 (2019)

    Article  MathSciNet  Google Scholar 

  2. Alvarez, 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)

    Article  MathSciNet  Google Scholar 

  3. Alwash, M.A.M.: On the composition conjectures. Electron. J. Differ. Equ. 2003(69), 1–4 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Alwash, M.A.M., Lloyd, N.G.: Nonautonomous equations related to polynomial two-dimensional systems. Proc. R. Soc. Edinburgh 105A, 129–152 (1987)

    Article  MathSciNet  Google Scholar 

  5. Alwash, M.A.M.: The composition conjecture for Abel equation. Expo. Math. 27(3), 241–250 (2009)

    Article  MathSciNet  Google Scholar 

  6. Blinov, M., Briskin, M., Yomdin, Y.: Local Center Conditions for the Abel Equation and Cyclicity of Its Zero Solution, in Complex Analysis and Dynamical Systems II, Contemporary Mathematics 382, pp. 65–82. American Mathematical Society, Providence (2005). (MR 2007b:34079 Zbl 1095.34018)

    MATH  Google Scholar 

  7. Blinov, M.: Some computations around the center problem, related to the composition algebra of univariate polynomials. M.Sc. Thesis, Weizmann Institute of Science (1997)

  8. Blinov, M.: Center and Composition conditions for Abel Equation. Ph.D. thesis, Weizmann Institute of Science (2002)

  9. Briskin, M., Francoise, J.P., Yomdin, Y.: Center conditions, compositions of polynomials and moments on algebraic curve. Ergodic Theory Dyn. Syst. 19(5), 1201–1220 (1999)

    Article  MathSciNet  Google Scholar 

  10. Briskin, M., Francoise, J.P., Yomdin, Y.: Center condition II: parametric and model center problems. Isr. J. Math. 118, 61–82 (2000)

    Article  MathSciNet  Google Scholar 

  11. Briskin, M., Francoise, J.P., Yomdin, Y.: Center condition III: parametric and model center problems. Isr. J. Math. 118, 83–108 (2000)

    Article  MathSciNet  Google Scholar 

  12. Briskin, M., Francoise, J.P., Yomdin, Y.: Poincaré centre-focus problem. C. R. Acad. Sci. Paris Sér. I(326), 1295–1298 (1998)

    Article  Google Scholar 

  13. Briskin, M., Francoise, J.P., Yomdin, Y.: Generalized moments, center-focus conditions, and compositions of polynomials. Operator Theory, System Theory and Related Topics (Beer-Sheva/Rehovot, 1997), pp. 161–185. Basel, Birkhäuser (2001). (Oper. Theory Adv. Appl. 123 MR 2002e:34050 Zbl 1075.34509)

    Chapter  Google Scholar 

  14. Briskin, M., Roytwarf, N., Yomdin, Y.: Center conditions at infinity for Abel differential equations. Ann. Math. 172, 437–83 (2010)

    Article  MathSciNet  Google Scholar 

  15. Brudnyi, A.: An algebraic model for the center problem. Bull. Sci. Math. 128, 839–857 (2004)

    Article  MathSciNet  Google Scholar 

  16. Brudnyi, A.: On center sets of ODEs determined by moments of their coefficients. Bull. Sci. Math. 130, 33–48 (2006)

    Article  MathSciNet  Google Scholar 

  17. Carbonell, M., Llibre, J.: Hopf Bifurcation, Averaging Methods and Liapunov Quantities for Polynomial Systems with Homogeneous Nonlinearities, in Proceedings of European Conference on Iteration Theory, ECIT87, pp. 145–160. World Scientific, Singapore (1989)

    Google Scholar 

  18. Cherkas, I.A.: Number of limit cycles of an autonomous second-order system. Differ. Equ. 5, 666–668 (1976)

    MATH  Google Scholar 

  19. Cima, A., Gasull, A., Manosas, F.: Centers for trigonometric Abel equations. Qual. Theory Dyn. Syst. 11(1), 19–37 (2012)

    Article  MathSciNet  Google Scholar 

  20. Christopher, C.: Abel equations: composition conjectures and the model problem. Bull. Lond. Math. Soc. 32, 332–338 (2000)

    Article  MathSciNet  Google Scholar 

  21. Dulac, H.: Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre. Bull. Sci. Math. Sér. (2) 32, 230–252 (1908)

    MATH  Google Scholar 

  22. Giné, J., Grau, M., Llibre, J.: Universal centres and composition conditions. Proc. Lond. Math. Soc. (3) 106(3), 481–507 (2013)

    Article  MathSciNet  Google Scholar 

  23. Giné, J., Grau, M., Santallusia, X.: A counterexample to the composition condition conjecture for polynomial abel differential equations. Ergod. Theor. Dyn. Syst. (2018). https://doi.org/10.1017/etds.2018.16

    Article  MATH  Google Scholar 

  24. Giné, J., Grau, M., Santallusia, X.: The center problem and composition condition for Abel differential equations. Expo. Math. 34, 210–222 (2016)

    Article  MathSciNet  Google Scholar 

  25. Giné, J., Grau, M., Santallusia, X.: Universal centers in the cubic trigonometric Abel equation. Electron. J. Qual. Theory Differ. Equ. 1, 1–7 (2014)

    Article  MathSciNet  Google Scholar 

  26. Kaplansky, I.: Commutative Rings. Allyn and Bacon, Newton (1970)

    MATH  Google Scholar 

  27. Li, C., Li, W., Llibre, J., Zhang, Z.: On the limit cycles of polynomial differential systems with homogeneous nonlinearities. Proc. Edinburgh Math. Soc. 43, 529–543 (2000)

    Article  MathSciNet  Google Scholar 

  28. Li, C., Li, W., Llibre, J., Zhang, Z.: New families of centers and limit cycles for polynomial differential systems with homogeneous nonlinearities. Ann. Differ. Equ. 193, 302–317 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Liapunoff, A.: Problème général dela stabilité du mouvement. Annal. Fac. Sci. Touluose Sér 2(9), 204–477 (1907). (Reproduction in Annals of Mathematics Studies 17, Princeton: Princeton University Press, 1947, reprinted 1965, Kraus Reprint Corporation, New York.)

    Google Scholar 

  30. Lijun, Y., Yun, T.: Some new results on abel equations. J. Math. Anal. Appl. 261, 100–112 (2001)

    Article  MathSciNet  Google Scholar 

  31. Pakovich, F.: Solution of the parametric center problem for the Abel differential equation. J. Eur. Math. Soc. 19(8), 2343–2369 (2017)

    Article  MathSciNet  Google Scholar 

  32. Pakovich, F.: A counterexample to the composition conjecture. Proc. Am. Math. Soc 130, 3747–3749 (2002)

    Article  MathSciNet  Google Scholar 

  33. Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. J. Math. Pures Appl. (Sér. 3) 7, 375–422 (1881)

    MATH  Google Scholar 

  34. Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. J. Math. Pures Appl. (Sér. 3) 8, 251–296 (1882)

    MATH  Google Scholar 

  35. Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. J. Math. Pures Appl. (Sér. 4) 1, 167–244 (1885)

    MATH  Google Scholar 

  36. Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. J. Math. Pures Appl. (Sér. 4) 2, 151–217 (1886)

    MATH  Google Scholar 

  37. Reyn, J.W.: A bibliography of the qualitative theory of quadratic systems of differential equations in the plane. In: Report of the Faculty of Technical Mathematics and Information, Delft, 3rd edn, pp. 94–102 (1994)

  38. Yomdin, Y.: The center problem for the Abel equation, compositions of functions, and moment conditions. Moscow Math. J. 3, 1167–1195 (2003). (With an addendum by F. Pakovich)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the referees for several suggestions which helped us to improve the presentation of the paper. The authors were partially supported by FAPEMIG FORTIS-10254/2014.

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Correspondence to Alexandre M. Alves.

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Alves, A.M., Lemos, A. & de Araujo, A.L.A. A partial positive answer to a Lijun–Yun conjecture. Boll Unione Mat Ital 13, 49–59 (2020). https://doi.org/10.1007/s40574-019-00203-x

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