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January  2019, 39(1): 483-502. doi: 10.3934/dcds.2019020

Uniqueness of limit cycles for quadratic vector fields

José Luis Bravo , , Manuel Fernández , Ignacio Ojeda and  Fernando Sánchez

Departamento de Matemáticas, Universidad de Extremadura, Badajoz 06006, Spain

* Corresponding author: J. L. Bravo

Received  April 2018 Published  October 2018

Fund Project: 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).

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.

Keywords: Abel equation, closed solution, periodic solution, limit cycle, algebraic variety.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34A34, 37C27, 37G15.
Citation: José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020
References:
[1]

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.  Google Scholar

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O. Bachmann, G.-M. Greuel, C. Lossen, G. Pfister and H. Schönemann, A Singular Introduction to Commutative Algebra, Springer, Berlin, 2007. Google Scholar

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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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

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L. A. Cherkas, Number of limit cycles of an autonomous second-order system, Diff. Eq., 5 (1976), 666-668.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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). Google Scholar

[12]

G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953), 15-31.  doi: 10.2307/1969724.  Google Scholar

[13]

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.   Google Scholar

[14]

A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745.  doi: 10.1142/S0218127406017130.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

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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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

D. Mumford, Algebraic Geometry I: Complex Projective Varieties, Reprint of the 1976 Edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[22]

A. A. Panov, The number of periodic solutions of polynomial differential equations, Math. Notes, 64 (1998), 622-628.  doi: 10.1007/BF02316287.  Google Scholar

[23]

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.  Google Scholar

[24]

V. A. Pliss, Non-Local Problems of the Theory of Oscillations, Academic Press, New York, 1966.  Google Scholar

[25]

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.  Google Scholar

[26]

J. Sotomayor, Curvas Definidas Por Equaçöes Diferenciais no Plano, IMPA, Rio de Janeiro, 1981.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

O. Bachmann, G.-M. Greuel, C. Lossen, G. Pfister and H. Schönemann, A Singular Introduction to Commutative Algebra, Springer, Berlin, 2007. Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

L. A. Cherkas, Number of limit cycles of an autonomous second-order system, Diff. Eq., 5 (1976), 666-668.   Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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). Google Scholar

[12]

G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953), 15-31.  doi: 10.2307/1969724.  Google Scholar

[13]

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.   Google Scholar

[14]

A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Int. J. Bif. Chaos, 16 (2006), 3737-3745.  doi: 10.1142/S0218127406017130.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

D. Mumford, Algebraic Geometry I: Complex Projective Varieties, Reprint of the 1976 Edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[22]

A. A. Panov, The number of periodic solutions of polynomial differential equations, Math. Notes, 64 (1998), 622-628.  doi: 10.1007/BF02316287.  Google Scholar

[23]

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.  Google Scholar

[24]

V. A. Pliss, Non-Local Problems of the Theory of Oscillations, Academic Press, New York, 1966.  Google Scholar

[25]

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.  Google Scholar

[26]

J. Sotomayor, Curvas Definidas Por Equaçöes Diferenciais no Plano, IMPA, Rio de Janeiro, 1981.  Google Scholar

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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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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