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The conditional variational principle for maps with the pseudo-orbit tracing property
Uniqueness of limit cycles for quadratic vector fields
Departamento de Matemáticas, Universidad de Extremadura, Badajoz 06006, Spain |
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.
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. |
Case | Point | |
|
1a) | |
4 | 4 |
1b) | |
4 | 4 |
2) | 3 | 3 | |
3a) | 3 | 3 | |
3b) | |
5 | 5 |
4) | |
3 | 3 |
5a) | 1 | 1 | |
5b) | 2 | 2 | |
5c) | |
* | 2 |
5d) | 2 | 2 | |
5e) | |
* | 2 |
5f) | |
3 | 2 |
5g) | |
* | 2 |
5h) | |
* | 2 |
Case | Point | |
|
1a) | |
4 | 4 |
1b) | |
4 | 4 |
2) | 3 | 3 | |
3a) | 3 | 3 | |
3b) | |
5 | 5 |
4) | |
3 | 3 |
5a) | 1 | 1 | |
5b) | 2 | 2 | |
5c) | |
* | 2 |
5d) | 2 | 2 | |
5e) | |
* | 2 |
5f) | |
3 | 2 |
5g) | |
* | 2 |
5h) | |
* | 2 |
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