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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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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).

[12]

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

[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. 

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[24]

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

[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.

[26]

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

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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).

[12]

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

[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. 

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[24]

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

[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.

[26]

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

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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