Nondegenerate and Nilpotent Centers for a Cubic System of Differential Equations

  1. Algaba, Antonio 1
  2. García, Cristóbal 1
  3. Giné, Jaume 2
  1. 1 Universidad de Huelva
    info

    Universidad de Huelva

    Huelva, España

    ROR https://ror.org/03a1kt624

  2. 2 Universitat de Lleida
    info

    Universitat de Lleida

    Lleida, España

    ROR https://ror.org/050c3cw24

Journal:
Qualitative theory of dynamical systems

ISSN: 1575-5460

Year of publication: 2019

Volume: 18

Issue: 1

Pages: 333-345

Type: Article

DOI: 10.1007/S12346-018-0301-4 DIALNET GOOGLE SCHOLAR lock_openOpen access editor

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Abstract

We consider the autonomous system of differential equations of the form x˙=P1(x,y)+P2(x,y),y˙=Q1(x,y)+Q3(x,y),where Pi and Qi are homogeneous polynomials of degree i. For such systems we provide the necessary and sufficient conditions to have a center at the origin. In fact this family only has nondegenerate and nilpotent centers.

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Bibliographic References

  • 1. Álvarez, M.J., Gasull, A.: Monodromy and stability for nilpotent critical points. Int. J. Bifurcat Chaos 15(4), 1253–1265 (2005)
  • 2. Álvarez, M.J., Gasull, A.: Generating limit cycles from a nilpotent critical point via normal forms. J. Math. Anal. Appl. 318(1), 271–287 (2006)
  • 3. Algaba, A., Checa, I., García, C., Gamero, E.: On orbital-reversibility for a class of planar dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 20(1), 229–239 (2015)
  • 4. Algaba, A., García, C., Giné, J.: Nilpotent centres via inverse integrating factors. Eur. J. Appl. Math. 27(5), 781–795 (2016)
  • 5. Algaba, A., García, C., Reyes, M.: Characterization of a monodromic singular point of a planar vector field. Nonlinear Anal. 74(16), 5402–5414 (2011)
  • 6. Algaba, A., García, C., Reyes, M.: A new algorithm for determining the monodromy of a planar differential system. Appl. Math. Comput. 237, 419–429 (2014)
  • 7. Andreev, A.: Investigations of the behavior of the integral curves of a system of two differential equations in the neighborhood of a singular point. Transl. Am. Math. Soc. 8, 187–207 (1958)
  • 8. Berthier, M., Moussu, R.: Réversibilité et classification des centres nilpotents. Ann. Inst. Fourier (Grenoble) 44, 465–494 (1994)
  • 9. Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: Local analytic integrability for nilpotent centers. Ergod. Theory Dyn. Syst. 23(2), 417–428 (2003)
  • 10. Decker, W., Laplagne, S., Pfister, G., Schonemann, H.A.: SINGULAR 3-1 library for computing the prime decomposition and radical of ideals. primdec.lib (2010)
  • 11. Dumortier, F.: Singularities of vector fields on the plane. J. Differ. Equ. 23(1), 53–106 (1977)
  • 12. Farr, W.W., Li, C., Labouriau, I.S., Langford, W.F.: Degenerate Hopf-bifurcation formulas and Hilbert’s 16th problem. SIAM J. Math. Anal. 20, 13–29 (1989)
  • 13. Gasull, A., Llibre, J., Mañosa, V., Mañosas, F.: The focus-centre problem for a type of degenerate system. Nonlinearity 13, 699–729 (2000)
  • 14. García, I., Giné, J., Grau, M.: A necessary condition in the monodromy problem for analytic differential equations on the plane. J. Symb. Comput. 41(9), 943–958 (2006)
  • 15. Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decompositions of polynomials. J. Symb. Comput. 6, 146–167 (1988)
  • 16. Giné, J.: The nondegenerate center problem and the inverse integrating factor. Bull. Sci. Math. 130(2), 152–161 (2006)
  • 17. Giné, J.: The center problem for a linear center perturbed by homogeneous polynomials. Acta Math. Sin. (Engl. Ser.) 22(6), 1613–1620 (2006)
  • 18. Giné, J.: Reduction of integrable planar polynomial differential systems. Appl. Math. Lett. 25(11), 1862–1865 (2012)
  • 19. Giné, J.: Center conditions and limit cycles for BiLiénard systems. Electron. J. Differ. Equ. 2017(86), 1–5 (2017)
  • 20. Giné, J., De Prada, P.: The nondegenerate center problem in certain families of planar differential systems. Int. J. Bifurcat. Chaos 19(1), 435–443 (2009)
  • 21. Giné, J., Llibre, J.: On the mechanisms for producing linear type centers in polynomial differential systems. Mosc. Math. J. 18(3), 1–12 (2018)
  • 22. Giné, J., Peralta-Salas, D.: Existence of inverse integrating factors and Lie symmetries for degenerate planar centers. J. Differ. Equ. 252(1), 344–357 (2012)
  • 23. Giné, J., Santallusia, X.: On the Poincaré–Liapunov constants and the Poincaré series. Appl. Math. (Warsaw) 28(1), 17–30 (2001)
  • 24. Greuel, G.M., Pfister, G., Schönemann, H.A.: SINGULAR 3.0 A computer algebra system for polynomial computations. In: Centre for Computer Algebra, University of Kaiserlautern (2005). http:// www.singular.uni-kl.de
  • 25. Mañosa, V.: On the center problem for degenerate singular points of planar vector fields. Int. J. Bifurcat. Chaos 12(4), 687–707 (2002)
  • 26. Mazzi, L., Sabatini, M.: A characterization of centres via first integrals. J. Differ. Equ. 76, 222–237 (1988)
  • 27. Moussu, R.: Symétrie et forme normale des centres et foyers dégénérés. Ergod. Theory Dyn. Syst. 2, 241–251 (1982)
  • 28. Reeb, G.: Sur certaines propriétés topologiques des variétés feuilletées (French) Publ. Inst. Math. Univ. Strasbourg 11, pp. 5–89, 155–156. Actualités Sci. Ind., no. 1183 Hermann & Cie., Paris (1952)
  • 29. Romanovski, V.G., Prešern,M.: An approach to solving systems of polynomials via modular arithmetics with applications. J. Comput. Appl. Math. 236(2), 196–208 (2011)
  • 30. Singer, M.F.: Liouvillian first integrals of differential equations. Trans. Am. Math. Soc. 333, 673–688 (1992)
  • 31. Wang, P.S., Guy, M.J.T., Davenport, J.H.: P-adic reconstruction of rational numbers. ACM SIGSAM Bull. 16(2), 2–3 (1982)
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