Skip to main content
Log in

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

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Algaba, A., Fuentes, N., García, C.: Centers of quasi-homogeneous polynomial planar systems. Nonlinear Anal. Real World Appl. 13, 419–431 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  3. Algaba, A., García, C., Teixeira, M.A.: Reversibility and quasi-homogeneous normal forms of vector fields. Nonlinear Anal. 73, 510–525 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Argemí, J.: Sur les points singuliers multiples de systémes dynamiques dans \(\mathbb{R}^{2}\). Annali di Matematica Pura ed Applicata Serie IV 79, 35–70 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aziz, W., Llibre, J., Pantazi, C.: Centers of quasi-homogeneous polynomial differential equations of degree three. Adv. Math. 254, 233–250 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Collins, C.B.: Algebraic classification of homogeneous polynomial vector fields in the plane. Jpn. J. Indus. Appl. Math. 13, 63–91 (1996)

    Article  MATH  Google Scholar 

  10. Cuesta, N.: Aritmética de las sucesiones \(6n-1,\,6n+1\) y de los primos gemelos. Collect. Math. 37, 211–227 (1986)

    MathSciNet  Google Scholar 

  11. Date, T.: Classification and analysis of two-dimensional homogeneous quadratic differential equations systems. J. Differ. Equ. 32, 311–334 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. García, I.: On the integrability of quasihomogeneous and related planar vector fields. Int. J. Bifurc. Chaos 13, 995–1002 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gavrilov, L., Giné, J., Grau, M.: On the cyclicity of weight-homogeneous centers. J. Differ. Equ. 246, 3126–3135 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol. 19. World Scientific Publishing Co. Inc, River Edge (2001)

    MATH  Google Scholar 

  17. Guy, R.K.: Unsolved Problems in Number Theory, 3rd edn. Springer, New York (2004)

    Book  MATH  Google Scholar 

  18. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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 quasi-homogeneous centers. J. Dyn. Diff. Equ. 21, 133–152 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liang, H., Huang, J., Zhao, Y.: Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems. Nonlinear Dyn. 78, 1659–1681 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Llibre, J., Zhang, X.: Polynomial first integrals for quasi-homogeneous polynomial differential systems. Nonlinearity 15, 1269–1280 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lyagina, L.S.: The integral curves of the equation \(y^{\prime }=(ax^{2}+bxy+cy^{2})/(dx^{2}+Exy+fy^{2})\) (Russian). Usp. Mat. Nauk 6–2(42), 171–183 (1951)

    MathSciNet  MATH  Google Scholar 

  28. Markus, L.: Quadratic differential equations and non-associative algebras. Ann. Math. Stud. 45, 185–213 (1960)

    MathSciNet  MATH  Google Scholar 

  29. Newton, T.A.: Two dimensional homogeneous quadratic differential systems. SIAM Rev. 20, 120–138 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Oliveira, R., Zhao, Y.: Structural stability of planar quasihomogeneous vector fields. Qual. Theory Dyn. Syst. 13, 39–72 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. OliveiraeSilva, T., Herzog, S., Pardi, S.: Empirical verification of the even par Golbach conjecture and computation of prime gaps up to \(4\cdot 10^{8}\). Math. Comput. 83, 2033–2060 (2014)

    Article  MATH  Google Scholar 

  32. Sibirskii, K.S., Vulpe, N.I.: Geometric classification of quadratic differential systems. Differ. Equ. 13, 548–556 (1977)

    MathSciNet  MATH  Google Scholar 

  33. Tang, Y., Wang, L., Zhang, X.: Center of planar quintic quasi-homogeneous polynomial differential systems. Discrete Contin. Dyn. Syst. 35, 2177–2191 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Vdovina, E.V.: Classification of singular points of the equation \(y^{\prime }=(a_{0}x^{2}+a_{1}xy+a_{2}y^{2})/(b_{0}x^{2}+b_{1}xy+b_{2}y^{2})\) by Forster’s method (Russian). Differ. Uravn. 20, 1809–1813 (1984)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are partially supported by a Grant Number MTM2014-56953-P of the Spanish Government.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jesús S. Pérez del Río.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

García, B., Lombardero, A. & Pérez del Río, J.S. Classification and Counting of Planar Quasi-Homogeneous Differential Systems Through Their Weight Vectors. Qual. Theory Dyn. Syst. 17, 541–561 (2018). https://doi.org/10.1007/s12346-017-0253-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-017-0253-0

Keywords

Navigation