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On the Integrability of Quasihomogeneous Systems and Quasidegenerate Infinity Systems

Abstract

The integrability of quasihomogeneous systems is considered, and the properties of the first integrals and the inverse integrating factors of such systems are shown. By solving the systems of ordinary differential equations which are established by using the vector fields of the quasihomogeneous systems, one can obtain an inverse integrating factor of the systems. Moreover, the integrability of a class of systems (quasidegenerate infinity systems) which generalize the so-called degenerate infinity vector fields is considered, and a method how to obtain an inverse integrating factor of the systems from the first integrals of the corresponding quasihomogeneous systems is shown.

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References

  1. 1.

    Chavarriga J, García IA, Giné J: On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity. International Journal of Bifurcation and Chaos 2001,11(3):711–722. 10.1142/S0218127401002390

    MATH  MathSciNet  Article  Google Scholar 

  2. 2.

    Goriely A: Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations. Journal of Mathematical Physics 1996,37(4):1871–1893. 10.1063/1.531484

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    García IA: On the integrability of quasihomogeneous and related planar vector fields. International Journal of Bifurcation and Chaos 2003,13(4):995–1002. 10.1142/S021812740300700X

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Llibre J, Zhang X: Polynomial first integrals for quasi-homogeneous polynomial differential systems. Nonlinearity 2002,15(4):1269–1280. 10.1088/0951-7715/15/4/313

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Hu Y, Guan K: Techniques for searching first integrals by Lie group and application to gyroscope system. Science in China. Series A 2005,48(8):1135–1143. 10.1360/04ys0141

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    Hu Y, Yang X: A method for obtaining first integrals and integrating factors of autonomous systems and application to Euler-Poisson equations. Reports on Mathematical Physics 2006,58(1):41–50. 10.1016/S0034-4877(06)80039-X

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Guan K, Liu S, Lei J: The Lie algebra admitted by an ordinary differential equation system. Annals of Differential Equations 1998,14(2):131–142.

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Olver PJ: Applications of Lie Groups to Differential Equations. 2nd edition. Springer, New York, NY, USA; 1989.

    MATH  Google Scholar 

  9. 9.

    Chavarriga J, Giacomini H, Giné J, Llibre J: Darboux integrability and the inverse integrating factor. Journal of Differential Equations 2003,194(1):116–139. 10.1016/S0022-0396(03)00190-6

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Chavarriga J, García IA: Integrability and explicit solutions in some Bianchi cosmological dynamical systems. Journal of Nonlinear Mathematical Physics 2001,8(1):96–105. 10.2991/jnmp.2001.8.1.9

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

    Gasull A, Prohens R: Quadratic and cubic systems with degenerate infinity. Journal of Mathematical Analysis and Applications 1996,198(1):25–34. 10.1006/jmaa.1996.0065

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Pearson JM, Lloyd NG, Christopher CJ: Algorithmic derivation of centre conditions. SIAM Review 1996,38(4):619–636. 10.1137/S0036144595283575

    MATH  MathSciNet  Article  Google Scholar 

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Correspondence to Yanxia Hu.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Hu, Y. On the Integrability of Quasihomogeneous Systems and Quasidegenerate Infinity Systems. Adv Differ Equ 2007, 098427 (2007). https://doi.org/10.1155/2007/98427

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Vector Field