Classification and Counting of Planar Quasi-Homogeneous Differential Systems Through Their Weight Vectors

  1. García, Belén 1
  2. Antón Lombardero 1
  3. Jesús S. Pérez del Río 1
  1. 1 Universidad de Oviedo
    info

    Universidad de Oviedo

    Oviedo, España

    ROR https://ror.org/006gksa02

Revue:
Qualitative theory of dynamical systems

ISSN: 1575-5460

Année de publication: 2018

Volumen: 17

Número: 3

Pages: 541-561

Type: Article

DOI: 10.1007/S12346-017-0253-0 DIALNET GOOGLE SCHOLAR lock_openAccès ouvert editor

D'autres publications dans: Qualitative theory of dynamical systems

Résumé

The quasi-homogeneous systems have important properties and they have been studied from various points of view. In this work, we provide the classification of quasi-homogeneous systems on the basis of the weight vector concept, especially in terms of the minimum weight vector, which is proved to be unique for any quasi-homogeneous system. Later we obtain the exact number of different forms of non-homogeneous quasi-homogeneous systems of arbitrary degree, proving a nice relation between this number and Euler’s totient function. Finally, we provide software implementations for some of the above results, and also for the algorithm, recently published by García et al., that generates all the quasi-homogeneous systems.

€ Voir financement

Information sur le financement

Financeurs

  • Spanish Government
    • MTM2014-56953-P

Références bibliographiques

  • 1. Algaba, A., Fuentes, N., García, C.: Centers of quasi-homogeneous polynomial planar systems. Nonlinear Anal. Real World Appl. 13, 419–431 (2012)
  • 2. Algaba, A., García, C., Reyes, M.: Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations. Appl. Math. Comput. 215, 314–323 (2009)
  • 3. Algaba, A., García, C., Teixeira, M.A.: Reversibility and quasi-homogeneous normal forms of vector fields. Nonlinear Anal. 73, 510–525 (2010)
  • 4. Argemí, J.: Sur les points singuliers multiples de systémes dynamiques dans R2. Annali di Matematica Pura ed Applicata Serie IV 79, 35–70 (1968)
  • 5. Aziz, W., Llibre, J., Pantazi, C.: Centers of quasi-homogeneous polynomial differential equations of degree three. Adv. Math. 254, 233–250 (2014)
  • 6. Cairó, L., Llibre, J.: Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3. J. Math. Anal. Appl. 331, 1284–1298 (2007)
  • 7. Cima, A., Llibre, J.: Algebraic and topological classification of the homogeneous cubic systems in the plane. J. Math. Anal. Appl. 147, 420–448 (1990)
  • 8. Coll, B., Gasull, A., Prohens, R.: Differential equations defined by the sum of two quasi-homogeneous vector fields. Can. J. Math. 49, 212–231 (1997)
  • 9. Collins, C.B.: Algebraic classification of homogeneous polynomial vector fields in the plane. Jpn. J. Indus. Appl. Math. 13, 63–91 (1996)
  • 10. Cuesta, N.: Aritmética de las sucesiones 6n − 1, 6n + 1 y de los primos gemelos. Collect. Math. 37, 211–227 (1986)
  • 11. Date, T.: Classification and analysis of two-dimensional homogeneous quadratic differential equations systems. J. Differ. Equ. 32, 311–334 (1979)
  • 12. García, B., Llibre, J., Pérez del Río, J.S.: Planar quasi-homogeneous polynomial differential systems and their integrability. J. Differ. Equ. 255, 3185–3204 (2013)
  • 13. García, I.: On the integrability of quasihomogeneous and related planar vector fields. Int. J. Bifurc. Chaos 13, 995–1002 (2003)
  • 14. Gavrilov, L., Giné, J., Grau, M.: On the cyclicity of weight-homogeneous centers. J. Differ. Equ. 246, 3126–3135 (2009)
  • 15. Giné, J., Grau, M., Llibre, J.: Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems. Discrete Contin. Dyn. Syst. 33, 4531–4547 (2013)
  • 16. Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol. 19. World Scientific Publishing Co. Inc, River Edge (2001)
  • 17. Guy, R.K.: Unsolved Problems in Number Theory, 3rd edn. Springer, New York (2004)
  • 18. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)
  • 19. Hilton, P., Pedersen, J., Walden, B.: Paper-Folding, Polygons, Complete Symbols, and the Euler Totient Function:An Ongoing Saga Connecting Geometry, Algebra, and Numer Theory, Advances in Algebra and Combinatorics, pp. 157–178. World Sci Publications, Hacckensaack (2008)
  • 20. Hu, Y.: On the integrability of quasihomogeneous systems and quasidegenerate infinity systems. Adv. Differ. Equ. (2007) (Art ID. 98427)
  • 21. Li, W., Llibre, J., Yang, J., Zhang, Z.: Limit cycles bifurcating from the period annulus of quasihomogeneous centers. J. Dyn. Diff. Equ. 21, 133–152 (2009)
  • 22. Liang, H., Huang, J., Zhao, Y.: Classification of global phase portraits of planar quartic quasihomogeneous polynomial differential systems. Nonlinear Dyn. 78, 1659–1681 (2014)
  • 23. Llibre, J., Lopes, B.D., de Moraes, J.R.: Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers of weight-degree 2. Appl. Math. Comput. 274, 47–54 (2016)
  • 24. Llibre, J., Pérez del Río, J.S., Rodríguez, J.A.: Structural stability of planar homogeneous polynomial vector fields. Applications to critical points and to infinity. J. Differ. Equ. 125, 490–520 (1996)
  • 25. Llibre, J., Pessoa, C.: On the centers of the weight-homogeneous polynomial vector fields on the plane. J. Math. Anal. Appl. 359, 722–730 (2009)
  • 26. Llibre, J., Zhang, X.: Polynomial first integrals for quasi-homogeneous polynomial differential systems. Nonlinearity 15, 1269–1280 (2002)
  • 27. Lyagina, L.S.: The integral curves of the equation y = (ax2 + bxy + cy2)/(dx2 + Exy + f y2) (Russian). Usp. Mat. Nauk 6–2(42), 171–183 (1951)
  • 28. Markus, L.: Quadratic differential equations and non-associative algebras. Ann. Math. Stud. 45, 185– 213 (1960)
  • 29. Newton, T.A.: Two dimensional homogeneous quadratic differential systems. SIAM Rev. 20, 120–138 (1978)
  • 30. Oliveira, R., Zhao, Y.: Structural stability of planar quasihomogeneous vector fields. Qual. Theory Dyn. Syst. 13, 39–72 (2014)
  • 31. OliveiraeSilva, T., Herzog, S., Pardi, S.: Empirical verification of the even par Golbach conjecture and computation of prime gaps up to 4 · 108. Math. Comput. 83, 2033–2060 (2014)
  • 32. Sibirskii, K.S., Vulpe, N.I.: Geometric classification of quadratic differential systems. Differ. Equ. 13, 548–556 (1977)
  • 33. Tang, Y., Wang, L., Zhang, X.: Center of planar quintic quasi-homogeneous polynomial differential systems. Discrete Contin. Dyn. Syst. 35, 2177–2191 (2015)
  • 34. Tsygvintsev, A.: On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations. J. Phys. A Math. Gen. 34, 2185–2193 (2001)
  • 35. Vdovina, E.V.: Classification of singular points of the equation y = (a0x2 + a1xy + a2 y2)/(b0x2 + b1xy + b2 y2) by Forster’s method (Russian). Differ. Uravn. 20, 1809–1813 (1984)
  • 36. Yoshida, H.: A criterion for the non-existence of an additional analytic integral in Hamiltonian systems with n degrees of freedom. Phys. Lett. A 141, 108–112 (1989)
La source de données: Dialnet