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Journal of Systems Science and Complexity

, Volume 31, Issue 6, pp 1698–1708 | Cite as

Computation of Invariant Curves and Identifying the Type of Critical Point

  • Junhai Zhi
  • Yufu Chen
Article
  • 2 Downloads

Abstract

This paper considers two dimensional systems which have purely imaginary eigenvalues. In order to obtain more propositions of invariant curves, the authors transform the real systems into complex differential systems by using a suitable linear transformation. The authors also propose an algorithm to compute exponential factors. An improved method of constructing integrating factor by using all invariant curves is presented and can be used in determining the type of the equilibrium points.

Keywords

Complex differential system equilibrium point exponential factor invariant curve 

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References

  1. [1]
    Liu Y, Li J, and Huang W, Planar Dynamical Systems: Selected Classical Problems, Science Press, Beijing, 2014.CrossRefzbMATHGoogle Scholar
  2. [2]
    Wang D, Mechanical manipulation for a class of differential systems, Journal of Symbolic Computation, 1991, 12(2): 233–254.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Du N and Chen S, Formal integrating factors as a method of distinguishing between a center and a focus, Journal of Mathematics, 1997, 17(2): 231–239.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Zhi J and Chen Y, A method to distinguishing between the center and the focus, Journal of Systems Science and Mathematical Sciences, 2017, 37(3): 863–869.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Christopher C, Invariant algebraic curves and conditions for centre, Proceedings of the Royal Society of Edinburgh, 1994, 124(6): 1209–1229.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Christopher C, Llibre J, and Pereira J V, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific Journal of Mathematics, 2007, 229(229): 63–117.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Pereira J V, Vector fields, invariant varieties and linear systems, Annales de L’Institut Fourier, 2001, 51(5): 1385–1405.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Pearson J M, Lloyd N G, and Christopher C J, Algorithmic derivation of centre conditions, SIAM Review, 1996, 38(4): 619–636.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Darboux G. Mémoire sur leséquations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math., 1878, 2: 151–200.Google Scholar
  10. [10]
    Prelle M J and Singer M F, Elementary first integrals of differential equations, Transactions of the American Mathematical Society, 1983, 279: 613–636.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Llibre J and Zhang X, Darboux theory of integrability in Cn taking into account the multiplicity, J. Differ. Equ., 2009, 246(2): 541–551.CrossRefzbMATHGoogle Scholar
  12. [12]
    Singer M F, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 1992, 333: 673–688.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Chèze G. Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time, Journal of Complexity, 2010, 27(2): 246–262.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Antoni F and Armengol G, Seeking Darboux polynomials, Springer-Verlag, New York, 2015, 139(1): 167–186.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Lloyd N G and Pearson J M, Symmetry in planar dynamical systems, Journal of Symbolic Computation, 2002, 33(3): 357–366MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Zhang X, Integrability of Dynamical Systems: Algebra and Analysis, Springer, Singapore, 2017.CrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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