Advanced Search
Article Contents
Article Contents

Evaluating cyclicity of cubic systems with algorithms of computational algebra

Abstract / Introduction Related Papers Cited by
  • We describe an algorithmic approach to studying limit cycle bifurcations in a neighborhood of an elementary center or focus of a polynomial system. Using it we obtain an upper bound for cyclicity of a family of cubic systems. Then using a theorem by Christopher [3] we study bifurcation of limit cycles from each component of the center variety. We obtain also the sharp bound for the cyclicity of a generic time-reversible cubic system.
    Mathematics Subject Classification: Primary: 34C07; Secondary: 34C23.


    \begin{equation} \\ \end{equation}
  • [1]

    N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik N. S., 30 (1952), 181-196; Translations Amer. Math. Soc., 100 (1954), 181-196.


    B. Buchberger, An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symbolic Comput., 41 (2006), 475-511.doi: 10.1016/j.jsc.2005.09.007.


    C. Christopher, Estimating limit cycles bifurcations, in "Trends in Mathematics, Differential Equations with Symbolic Computations" (Eds. D. Wang and Z. Zheng), Birkhäuser-Verlag, (2005), 23-36.doi: 10.1007/3-7643-7429-2_2.


    C. J. Christopher and C. Rousseau, Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in $\mathbbC^2$, Publ. Mat., 45 (2001), 95-123.doi: 10.5565/PUBLMAT_45101_04.


    D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms," Springer-Verlag, New York, 1992.doi: 10.1216/rmjm/1181071923.


    A. Cima, A. Gasull, V. Mañosa and F. Mañosas, Algebraic properties of the Liapunov and periodic constants, Rocky Mountain J. Math., 27 (1997), 471-501.doi: 10.1216/rmjm/1181071923.


    W. Decker, S. Laplagne, G. Pfister and H. A. Schönemann, Singular 3-1 library for computing the primary decomposition and radical of ideals, primdec.lib, 2010.


    J.-P. Françoise and Y. Yomdin, Bernstein inequalities and applications to analytic geometry and differential equations, J. Functional Analysis, 146 (1997), 185-205.doi: 10.1006/jfan.1996.3029.


    P. Gianni, B. Trager and G. Zacharias, Gröbner bases and primary decomposition of polynomials, J. Symbolic Comput., 6 (1988), 146-167.doi: 10.1016/S0747-7171(88)80040-3.


    J. Giné, On some open problems in planar differential systems and Hilbert's 16th problem, Chaos Solitons Fractals, 31 (2007), 1118-1134.doi: 10.1016/j.chaos.2005.10.057.


    G.-M. Greuel and G. Pfister, "A Singular Introduction to Commutative Algebra," Springer-Verlag, New York, 2002.


    W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3-1-2. A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern (2010). Available from: http://www.singular.uni-kl.de.


    M. Han, H. Zang and T. Zhang, A new proof to Bautin's theorem, Chaos Solitons Fractals, 31 (2007), 218-223.doi: 10.1016/j.chaos.2005.09.051.


    Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations," Graduate Studies in Mathematics, 86, (American Mathematical Society, Providence), 2008.


    A. Jarrah, R. Laubenbacher and V. G. Romanovski, The Sibirsky component of the center variety of polynomial differential systems, J. Symb. Comput., 35 (2003), 577-589.doi: 10.1016/S0747-7171(03)00016-6.


    V. Levandovskyy, A. Logar and V. G. Romanovski, The cyclicity of a cubic system, Open Syst. Inf. Dyn., 16 (2009), 429-439.doi: 10.1142/S1230161209000323.


    V. Levandovskyy, V. G. Romanovski and D. S. Shafer, The cyclicity of a cubic system with nonradical Bautin ideal, J. Differential Equations, 246 (2009), 1274-1287.doi: 10.1016/j.jde.2008.07.026.


    Y.-R. Liu and J.-B. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1989), 10-23.


    J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.doi: 10.1142/S0218127403006352.


    N. G. Lloyd, J. M. Pearson and V. G. Romanovsky, Computing integrability conditions for a cubic differential system, Comput. Math. Appl., 32 (1996), 99-107.doi: 10.1016/S0898-1221(96)00188-5.


    V. G. Romanovski, Time-reversibility in 2-Dim systems, Open Syst. Inf. Dyn., 15 (2008), 359-370.doi: 10.1142/S1230161208000249.


    V. G. Romanovski and D. S. Shafer, Time-reversibility in two-dimensional polynomial systems, in "Trends in Mathematics, Differential Equations with Symbolic Computations" (Eds. D. Wang and Z. Zheng), Birkhäuser-Verlag, (2005), 67-83.doi: 10.1007/3-7643-7429-2_5.


    V. G. Romanovski and D. S. Shafer, "The Center and Cyclicity Problems: A Computational Algebra Approach," Birkhäuser Boston, Inc., Boston, MA, 2009.


    R. Roussarie, "Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem," Progress in Mathematics, 164, Birkhäuser, Basel, 1998.


    K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differ. Uravn. (Russian), 1 (1965), 53-66; Differ. Equ. (English translation), 1 (1965), 36-47.


    S. Yakovenko, A geometric proof of the Bautin theorem, Concerning the Hilbert Sixteenth Problem. Advances in Mathematical Sciences, Vol. 23; Amer. Math. Soc. Transl., 165 (1995), 203-219.


    H. Żołądek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273.doi: 10.1006/jdeq.1994.1049.


    H. Żołądek, On a certain generalization of Bautin's theorem, Nonlinearity, 7 (1994), 273-279.doi: 10.1088/0951-7715/7/1/013.

  • 加载中

Article Metrics

HTML views() PDF downloads(124) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint