In this paper we consider real differential systems which at singular points have purely imaginary eigenvalues. We show that the real invariant algebraic curves of the system are not identically zero in some deleted neighborhood U°(O). A method for seeking local first integrals or integrating factors is proposed, which can be used in determining the types of the equilibrium points.
ZHI Junhai
,
CHEN Yufu
. Constructing integrating factor to distinguish between the center and the focus[J]. Journal of University of Chinese Academy of Sciences, 2018
, 35(5)
: 577
-581
.
DOI: 10.7523/j.issn.2095-6134.2018.05.001
[1] Wang D. Mechanical manipulation for a class of differential systems[J]. Journal of Symbolic Computation, 1991, 12(2):233-254.
[2] Zhi J, Chen Y. A Method to distinguishing between the center and the focus[J]. Journal of Systems Science and Mathematical Sciences, 2017, 37(3):863-869.
[3] Darboux. Memoire sur les equations differentielles algebriques du premier ordre et du premier degre (Melanges)[J]. Bull Sci Math, 1878, 2(2):151-200.
[4] Prelle M J, Singer M F. Elementary first integrals of differential equations[J]. Transactions of the American Mathematical Society, 1983, 279(1):215-229.
[5] Singer M F. Liouvillian first integrals of differential equations[J]. Trans Amer Math Soc, 1992, 333:673-688.
[6] Christopher C. Liouvillian first integrals of second order polynomials differential equations[J]. Electronic Journal of Differential Equations, 1999, 1999(19):1-7.
[7] Christopher C. Invariant algebraic curves and conditions for centre[J]. Proceedings of the Royal Society of Edinburgh, 1994, 124(6):1209-1229.
[8] Pearson J M, Lloyd N G, Christopher C J. Algorithmic derivation of centre conditions[J]. Siam Review, 1996, 38(4):619-636.
[9] Pereira J V. Vector fields, invariant varieties and linear systems[J]. Annales de LInstitut Fourier, 2001, 51(5):1385-1405.
[10] Christopher C, Llibre J, Pereira J V. Multiplicity of invariant algebraic curves in polynomial vector fields[J]. Pacific Journal of Mathematics, 2007, 229(229):63-117.
[11] Garcia I A, Grau M. A Survey on the inverse integrating factor[J]. Qualitative Theory of Dynamical Systems, 2010, 9(1):115-166.
[12] Gine J. Dulac functions of planar vector fields[J]. Qualitative Theory of Dynamical Systems, 2014, 13(1):121-128.
