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Uniqueness of limit cycles for quadratic vector fields

  • * Corresponding author: J. L. Bravo

    * Corresponding author: J. L. Bravo 

The first two authors were partially supported by AEI/FEDER UE grant number MTM 2011-22751 and Junta de Extremadura grant GR15055 (Junta de Extremadura/FEDER funds). The third author was partially supported by the research group FQM-024 (Junta de Extremadura/FEDER funds) and by the project MTM2015-65764-C3-1-P (MINECO/FEDER, UE). The fourth author was partially supported by Junta de Extremadura grant GR15055 (Junta de Extremadura/FEDER funds).

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  • This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as $x' = a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$, $y' = x+a_1 y + a_2x^2+(2 a_3+a_4)xy -a_2y^2$. In particular, we study the semi-varieties defined in terms of the parameters $a_1, a_2, ..., a_6$ where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.

    Mathematics Subject Classification: Primary: 34C25; Secondary: 34A34, 37C27, 37G15.


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  • Table 1.  Codimensions of the semi-varieties.

    Case Point $c_p$ $c_I$
    1a) $a_1=1$, $a_2=0$, $a_3=1$, $a_4=-3$, $a_5=0$, $a_6=0$. 4 4
    1b) $a_1=1$, $a_2=1$, $a_3=-1$, $a_4=2$, $a_5=-4$, $a_6=1$. 4 4
    2) $a_1=-1$, $a_2=\sqrt{14}$, $a_3=-2$,
    $a_4=-1$, $a_5=-3 \sqrt{14}$, $a_6=0$.
    3 3
    3a) $a_1=-1$, $a_2=(201 + 2 \sqrt{1509})/58$,
    $a_3=(-33 + 4 \sqrt{1509})/58$, $a_4=-1$,
    $a_5=-16$, $a_6=(-201 - 2 \sqrt{1509})/58)$
    3 3
    3b) $a_1=0$, $a_2=0$, $a_3=0$, $a_4=1$, $a_5=-2$, $a_6=-1$. 5 5
    4) $a_1=1$, $a_2=0$, $a_3=1/3$, $a_4=-1$, $a_5=-1$, $a_6=0$. 3 3
    5a) $a_1=0$, $a_2=1$, $a_3=-15/16$,
    $a_4=-53/16$, $a_5=(-941 - 31 \sqrt{7913})/512$, $a_6=1$.
    1 1
    5b) $a_1=0$, $a_2=(4096 - 7 \sqrt{1726})/16384$, $a_3=0$,
    $a_4=-(58339673 + 28672 \sqrt{1726})/94666752$,
    $a_5=-1$, $a_6=-2889/16384$.
    2 2
    5c) $a_1=1$, $a_2=4$, $a_3=-12$, $a_4=30$, $a_5=-15$, $a_6=1/2$ * 2
    5d) $a_1=0$, $a_2=\sqrt{185}/32$, $a_3=0$,
    $a_4=-1$, $a_5=-3 \sqrt{185}/32$, $a_6=-5/32$
    2 2
    5e) $a_1=0$, $a_2=2 \sqrt{2}$, $a_3=-1$, $a_4=0$, $a_5=-9 \sqrt{2}$, $a_6=8$ * 2
    5f) $a_1=0$, $a_2=2/3$, $a_3=0$, $a_4=-1$, $a_5=-2$, $a_6=-1/3$ 3 2
    5g) $a_1=0$, $a_2=1$, $a_3=-9/2$, $a_4=15/2$, $a_5=-15$, $a_6=8$ * 2
    5h) $a_1=0$, $a_2=1$, $a_3=-8$, $a_4=35/2$, $a_5=-15$, $a_6=9/2$ * 2
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  •   M. J. Álvarez , A. Gasull  and  H. Giacomini , A new uniqueness criterion for the number of periodic orbits of Abel equations, J. Differential Equations, 234 (2007) , 161-176.  doi: 10.1016/j.jde.2006.11.004.
      M. A. M. Alwash  and  N. G. Lloyd , Nonautonomous equations related to polynomial two dimensional systems, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987) , 129-152.  doi: 10.1017/S0308210500021971.
      A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press (A division of John Wiley & Sons), Israel Program for Scientific Translations Jerusalem-London, 1973.
      O. Bachmann, G.-M. Greuel, C. Lossen, G. Pfister and H. Schönemann, A Singular Introduction to Commutative Algebra, Springer, Berlin, 2007.
      N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or centre type, American Math. Soc. Translation, 1954 (1954), 19pp.
      J. L. Bravo , M. Fernández  and  A. Gasull , Limit cycles for some Abel equations having coefficients without fixed signs, Int. J. Bif. Chaos, 19 (2009) , 3869-3876.  doi: 10.1142/S0218127409025195.
      J. L. Bravo  and  J. Torregrosa , Abel-like equations with no periodic solutions, J. Math. Anal. Appl., 342 (2008) , 931-942.  doi: 10.1016/j.jmaa.2007.12.060.
      L. A. Cherkas , Number of limit cycles of an autonomous second-order system, Diff. Eq., 5 (1976) , 666-668. 
      B. Coll , A. Gasull  and  J. Llibre , Some theorems on the existence, uniqueness and non existence of limit cycles for quadratic systems, J. Differential Equations, 67 (1987) , 372-399.  doi: 10.1016/0022-0396(87)90133-1.
      D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Second Edition, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1997. doi: 10.1007/978-3-319-16721-3.
      W. Decker, G. M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-0 — A Computer Algebra System for Polynomial Computations, http://www.singular.uni-kl.de (2016).
      G. F. D. Duff , Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953) , 15-31.  doi: 10.2307/1969724.
      H. Dulac , Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singulier un centre, Bull. Soc. Math. France, 32 (1908) , 230-252. 
      A. Gasull  and  A. Guillamon , Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006) , 3737-3745.  doi: 10.1142/S0218127406017130.
      A. Gasull  and  J. Llibre , Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990) , 1235-1244.  doi: 10.1137/0521068.
      P. Gianni , B. Trager  and  G. Zacharias , Gröbner bases and primary decomposition of polynomial ideals, Computational Aspects of Commutative Algebra, J. Symbolic Comput., 6 (1988) , 149-167.  doi: 10.1016/S0747-7171(88)80040-3.
      J. Huang  and  Y. Zhao , Periodic solutions for equation $x' = A(t)x^m + B(t)x^n + C(t)x^l$ with $A(t)$ and $B(t)$ changing signs, J. Differential Equations, 253 (2012) , 73-99.  doi: 10.1016/j.jde.2012.03.021.
      A. Lins Neto , On the number of solutions of the equation $\frac{d x}{dt} = \sum_{j = 0}^n a_j(t)x^j$, $0≤ t≤ 1$, for which $x(0) = x(1)$, Inv. Math., 59 (1980) , 67-76.  doi: 10.1007/BF01390315.
      J. Llibre  and  Xiang Zhang , The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, (2017) , 1-14.  doi: 10.1017/S0308210517000221.
      N. G. Lloyd , A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc., 20 (1979) , 277-286.  doi: 10.1112/jlms/s2-20.2.277.
      D. Mumford, Algebraic Geometry I: Complex Projective Varieties, Reprint of the 1976 Edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
      A. A. Panov , The number of periodic solutions of polynomial differential equations, Math. Notes, 64 (1998) , 622-628.  doi: 10.1007/BF02316287.
      L. M. Perko, Differential Equations and Dynamical Systems, Third edition, Texts in Applied Mathematics 7, Springer–Verlag, New York [etc.], 2001. doi: 10.1007/978-1-4613-0003-8.
      V. A. Pliss, Non-Local Problems of the Theory of Oscillations, Academic Press, New York, 1966.
      V. G. Romanovski and D. S. Shafer, The Centre and Cyclicity Problems. A Computational Algebra Approach, Birkhäuser, 2009. doi: 10.1007/978-0-8176-4727-8.
      J. Sotomayor, Curvas Definidas Por Equaçöes Diferenciais no Plano, IMPA, Rio de Janeiro, 1981.
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