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Center conditions for a class of planar rigid polynomial differential systems
On the limit cycles bifurcating from an ellipse of a quadratic center
1. | Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia |
2. | Département de Mathématiques et Statistique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, H3C 3J7, Canada |
References:
[1] |
J. C. Artés, J. Llibre and D. Schlomiuk, The geometry of the quadratic differential systems with a weak focus of second order, International J. of Bifurcation and Chaos, 16 (2006), 3127-3194.
doi: 10.1142/S0218127406016720. |
[2] |
N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type, Math. USSR-Sb., 100 (1954), 397-413. |
[3] |
J. Chavarriga, H. Giacomini and J. Llibre, Uniqueness of algebraic limit cycles for quadratic systems, J. Math. Anal. and Appl., 261 (2001), 85-99.
doi: 10.1006/jmaa.2001.7476. |
[4] |
J. Chavarriga, J. Llibre and J. Moulin Ollagnier, On a result of Darboux, LMS J. of Computation and Mathematics, 4 (2001), 197-210.
doi: 10.1112/S1461157000000863. |
[5] |
J. Chavarriga, J. Llibre and J. Sorolla, Algebraic limit cycles of degree $4$ of quadratic systems, J. Differential Equations, 200 (2004), 206-244.
doi: 10.1016/j.jde.2004.01.003. |
[6] |
C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2007. |
[7] |
C. Christopher, J. Llibre and G. Świrszcz, Invariant algebraic curves of large degree for quadratic systems, J. Math. Anal. Appl., 303 (2005), 450-461.
doi: 10.1016/j.jmaa.2004.08.042. |
[8] |
C. Christopher, J. Llibre, C. Pantazi and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems, J. of Physics A: Math. Gen., 35 (2002), 2457-2476.
doi: 10.1088/0305-4470/35/10/310. |
[9] |
B. Coll, A. Ferragut and J. Llibre, Polynomial inverse integrating factors of quadratic differential systems, Nonlinear Analysis Series A: Theory, Methods & Applications, 73 (2010), 881-914.
doi: 10.1016/j.na.2010.04.004. |
[10] |
W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293-304.
doi: 10.1016/0022-0396(66)90070-2. |
[11] |
G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math. 2ème série, 2 (1878), 60-96; 123-144; 151-200. |
[12] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-Verlag, New York, 2006. |
[13] |
F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133.
doi: 10.1006/jdeq.1994.1061. |
[14] |
R. M. Evdokimenco, Construction of algebraic paths and the qualitative investigation in the large of the properties of integral curves of a system of differential equations, Differential Equations, 6 (1970), 1349-1358. |
[15] |
R. M. Evdokimenco, Behavior of integral curves of a dynamic system, Differential Equations, 9 (1974), 1095-1103. |
[16] |
R. M. Evdokimenco, Investigation in the large of a dynamic system, Differential Equations, 15 (1979), 215-221. |
[17] |
V. F. Filiptsov, Algebraic limit cycles, Differential Equations, 9 (1973), 983-986. |
[18] |
H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996), 501-516.
doi: 10.1088/0951-7715/9/2/013. |
[19] |
P. Hartmann, Ordinary Differential Equations, SIAM Edition, 2002.
doi: 10.1137/1.9780898719222. |
[20] |
D. Hilbert, Mathematische probleme, Bull. Amer. Math. Soc., 8 (1902), 437-479.
doi: 10.1007/978-3-662-25726-5_19. |
[21] |
W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, (1911), 1446-1457 (Dutch). |
[22] |
W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20 (1912), 1354-1365; 21, 27-33 (Dutch). |
[23] |
J. Llibre, Open problems on the algebraic limit cycles of planar polynomial vector fields, Bulletin of Academy of Sciences of Moldova (Matematica), 1 (2008), 19-26. |
[24] |
J. Llibre, R. Ramírez and N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations, 248 (2010), 1401-1409.
doi: 10.1016/j.jde.2009.11.023. |
[25] |
J. Llibre, R. Ramírez and N. Sadovskaia, On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves, J. Differential Equations, 250 (2011), 983-999.
doi: 10.1016/j.jde.2010.06.009. |
[26] |
J. Llibre and G. Rodríguez, Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations, 198 (2004), 374-380.
doi: 10.1016/j.jde.2003.10.008. |
[27] |
J. Llibre and D. Schlomiuk, The geometry of differential quadratic systems with a weak focus of third order, Canadian J. of Math., 56 (2004), 310-343.
doi: 10.4153/CJM-2004-015-2. |
[28] |
J. Llibre and G. Swirszcz, Relationships between limit cycles and algebraic invariant curves for quadratic systems, J. Differential Equations, 229 (2006), 529-537.
doi: 10.1016/j.jde.2006.03.013. |
[29] |
J. Llibre and G. Swirszcz, Classification of quadratic systems admitting the existence of an algebraic limit cycle, Bull. des Sciences Mathemàtiques, 131 (2007), 405-421.
doi: 10.1016/j.bulsci.2006.03.014. |
[30] |
M. Ndiaye, Le Problème du Centre Pour des Systèmes Dynamiques Polynomiaux à Deux Dimensions, Ph.D thesis, Université de Tours, 1996. |
[31] |
D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc., 338 (1993), 799-841.
doi: 10.1090/S0002-9947-1993-1106193-6. |
[32] |
D. Schlomiuk, Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields, in Bifurcations and Periodic Orbits of Vector Fields, NATO ASI Series, Series C-Vol, 408 (1993), 429-467. |
[33] |
D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Expositiones Mathematicae, 8 (1990), 3-25. |
[34] |
A. I. Yablonskii, Limit cycles of a certain differential equations, Differential Equations, 2 (1966), 335-344 (In Russian). |
[35] |
Ye Yanqian et al., Theory of Limit Cycles, Translations of Math. Monographs, Vol. 66, Amer. Math. Soc, Providence, 1986. |
[36] |
Y.-S. Ch'in, On the algebraic limit cycles of second degree of the differential equation $dy/dx= \sum_{0\le i+j \le 2} a_{ij} x^i y^j/ \sum_{0\le i+j \le 2} b_{ij} x^i y^j$, Acta Math. Sinica, 7 (1958), 934-935. |
[37] |
X. Zhang, The 16th Hilbert problem on algebraic limit cycles, J. Differential Equations, 251 (2011), 1778-1789.
doi: 10.1016/j.jde.2011.06.008. |
show all references
References:
[1] |
J. C. Artés, J. Llibre and D. Schlomiuk, The geometry of the quadratic differential systems with a weak focus of second order, International J. of Bifurcation and Chaos, 16 (2006), 3127-3194.
doi: 10.1142/S0218127406016720. |
[2] |
N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type, Math. USSR-Sb., 100 (1954), 397-413. |
[3] |
J. Chavarriga, H. Giacomini and J. Llibre, Uniqueness of algebraic limit cycles for quadratic systems, J. Math. Anal. and Appl., 261 (2001), 85-99.
doi: 10.1006/jmaa.2001.7476. |
[4] |
J. Chavarriga, J. Llibre and J. Moulin Ollagnier, On a result of Darboux, LMS J. of Computation and Mathematics, 4 (2001), 197-210.
doi: 10.1112/S1461157000000863. |
[5] |
J. Chavarriga, J. Llibre and J. Sorolla, Algebraic limit cycles of degree $4$ of quadratic systems, J. Differential Equations, 200 (2004), 206-244.
doi: 10.1016/j.jde.2004.01.003. |
[6] |
C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2007. |
[7] |
C. Christopher, J. Llibre and G. Świrszcz, Invariant algebraic curves of large degree for quadratic systems, J. Math. Anal. Appl., 303 (2005), 450-461.
doi: 10.1016/j.jmaa.2004.08.042. |
[8] |
C. Christopher, J. Llibre, C. Pantazi and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems, J. of Physics A: Math. Gen., 35 (2002), 2457-2476.
doi: 10.1088/0305-4470/35/10/310. |
[9] |
B. Coll, A. Ferragut and J. Llibre, Polynomial inverse integrating factors of quadratic differential systems, Nonlinear Analysis Series A: Theory, Methods & Applications, 73 (2010), 881-914.
doi: 10.1016/j.na.2010.04.004. |
[10] |
W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293-304.
doi: 10.1016/0022-0396(66)90070-2. |
[11] |
G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math. 2ème série, 2 (1878), 60-96; 123-144; 151-200. |
[12] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-Verlag, New York, 2006. |
[13] |
F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133.
doi: 10.1006/jdeq.1994.1061. |
[14] |
R. M. Evdokimenco, Construction of algebraic paths and the qualitative investigation in the large of the properties of integral curves of a system of differential equations, Differential Equations, 6 (1970), 1349-1358. |
[15] |
R. M. Evdokimenco, Behavior of integral curves of a dynamic system, Differential Equations, 9 (1974), 1095-1103. |
[16] |
R. M. Evdokimenco, Investigation in the large of a dynamic system, Differential Equations, 15 (1979), 215-221. |
[17] |
V. F. Filiptsov, Algebraic limit cycles, Differential Equations, 9 (1973), 983-986. |
[18] |
H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996), 501-516.
doi: 10.1088/0951-7715/9/2/013. |
[19] |
P. Hartmann, Ordinary Differential Equations, SIAM Edition, 2002.
doi: 10.1137/1.9780898719222. |
[20] |
D. Hilbert, Mathematische probleme, Bull. Amer. Math. Soc., 8 (1902), 437-479.
doi: 10.1007/978-3-662-25726-5_19. |
[21] |
W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, (1911), 1446-1457 (Dutch). |
[22] |
W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20 (1912), 1354-1365; 21, 27-33 (Dutch). |
[23] |
J. Llibre, Open problems on the algebraic limit cycles of planar polynomial vector fields, Bulletin of Academy of Sciences of Moldova (Matematica), 1 (2008), 19-26. |
[24] |
J. Llibre, R. Ramírez and N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations, 248 (2010), 1401-1409.
doi: 10.1016/j.jde.2009.11.023. |
[25] |
J. Llibre, R. Ramírez and N. Sadovskaia, On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves, J. Differential Equations, 250 (2011), 983-999.
doi: 10.1016/j.jde.2010.06.009. |
[26] |
J. Llibre and G. Rodríguez, Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations, 198 (2004), 374-380.
doi: 10.1016/j.jde.2003.10.008. |
[27] |
J. Llibre and D. Schlomiuk, The geometry of differential quadratic systems with a weak focus of third order, Canadian J. of Math., 56 (2004), 310-343.
doi: 10.4153/CJM-2004-015-2. |
[28] |
J. Llibre and G. Swirszcz, Relationships between limit cycles and algebraic invariant curves for quadratic systems, J. Differential Equations, 229 (2006), 529-537.
doi: 10.1016/j.jde.2006.03.013. |
[29] |
J. Llibre and G. Swirszcz, Classification of quadratic systems admitting the existence of an algebraic limit cycle, Bull. des Sciences Mathemàtiques, 131 (2007), 405-421.
doi: 10.1016/j.bulsci.2006.03.014. |
[30] |
M. Ndiaye, Le Problème du Centre Pour des Systèmes Dynamiques Polynomiaux à Deux Dimensions, Ph.D thesis, Université de Tours, 1996. |
[31] |
D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc., 338 (1993), 799-841.
doi: 10.1090/S0002-9947-1993-1106193-6. |
[32] |
D. Schlomiuk, Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields, in Bifurcations and Periodic Orbits of Vector Fields, NATO ASI Series, Series C-Vol, 408 (1993), 429-467. |
[33] |
D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Expositiones Mathematicae, 8 (1990), 3-25. |
[34] |
A. I. Yablonskii, Limit cycles of a certain differential equations, Differential Equations, 2 (1966), 335-344 (In Russian). |
[35] |
Ye Yanqian et al., Theory of Limit Cycles, Translations of Math. Monographs, Vol. 66, Amer. Math. Soc, Providence, 1986. |
[36] |
Y.-S. Ch'in, On the algebraic limit cycles of second degree of the differential equation $dy/dx= \sum_{0\le i+j \le 2} a_{ij} x^i y^j/ \sum_{0\le i+j \le 2} b_{ij} x^i y^j$, Acta Math. Sinica, 7 (1958), 934-935. |
[37] |
X. Zhang, The 16th Hilbert problem on algebraic limit cycles, J. Differential Equations, 251 (2011), 1778-1789.
doi: 10.1016/j.jde.2011.06.008. |
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