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March  2015, 35(3): 1091-1102. doi: 10.3934/dcds.2015.35.1091

On the limit cycles bifurcating from an ellipse of a quadratic center

Jaume Llibre 1, and  Dana Schlomiuk 2,

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

Received  June 2012 Revised  July 2014 Published  October 2014

It is well known that invariant algebraic curves of polynomial differential systems play an important role in questions regarding integrability of these systems. But do they also have a role in relation to limit cycles? In this article we show that not only they do have a role in the production of limit cycles in polynomial perturbations of such systems but that algebraic invariant curves can even generate algebraic limit cycles in such perturbations. We prove that when we perturb any quadratic system with an invariant ellipse surrounding a center (quadratic systems with center always have invariant algebraic curves and some of them have invariant ellipses) within the class of quadratic differential systems, there is at least one 1-parameter family of such systems having a limit cycle bifurcating from the ellipse. Therefore the cyclicity of the period annulus of such systems is at least one.
Keywords: quadratic vector fields, quadratic center, periodic orbit, Quadratic systems, bifurcation from center, inverse integrating factor., limit cycle, cyclicity of the period annulus.
Mathematics Subject Classification: Primary: 34C0.
Citation: Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091
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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[10]

W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293-304. doi: 10.1016/0022-0396(66)90070-2.  Google Scholar

[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. Google Scholar

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F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-Verlag, New York, 2006.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

[15]

R. M. Evdokimenco, Behavior of integral curves of a dynamic system, Differential Equations, 9 (1974), 1095-1103. Google Scholar

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R. M. Evdokimenco, Investigation in the large of a dynamic system, Differential Equations, 15 (1979), 215-221.  Google Scholar

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V. F. Filiptsov, Algebraic limit cycles, Differential Equations, 9 (1973), 983-986. Google Scholar

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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.  Google Scholar

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P. Hartmann, Ordinary Differential Equations, SIAM Edition, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

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D. Hilbert, Mathematische probleme, Bull. Amer. Math. Soc., 8 (1902), 437-479. doi: 10.1007/978-3-662-25726-5_19.  Google Scholar

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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). Google Scholar

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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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Ndiaye, Le Problème du Centre Pour des Systèmes Dynamiques Polynomiaux à Deux Dimensions, Ph.D thesis, Université de Tours, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Expositiones Mathematicae, 8 (1990), 3-25.  Google Scholar

[34]

A. I. Yablonskii, Limit cycles of a certain differential equations, Differential Equations, 2 (1966), 335-344 (In Russian).  Google Scholar

[35]

Ye Yanqian et al., Theory of Limit Cycles, Translations of Math. Monographs, Vol. 66, Amer. Math. Soc, Providence, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293-304. doi: 10.1016/0022-0396(66)90070-2.  Google Scholar

[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. Google Scholar

[12]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-Verlag, New York, 2006.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

R. M. Evdokimenco, Behavior of integral curves of a dynamic system, Differential Equations, 9 (1974), 1095-1103. Google Scholar

[16]

R. M. Evdokimenco, Investigation in the large of a dynamic system, Differential Equations, 15 (1979), 215-221.  Google Scholar

[17]

V. F. Filiptsov, Algebraic limit cycles, Differential Equations, 9 (1973), 983-986. Google Scholar

[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.  Google Scholar

[19]

P. Hartmann, Ordinary Differential Equations, SIAM Edition, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[20]

D. Hilbert, Mathematische probleme, Bull. Amer. Math. Soc., 8 (1902), 437-479. doi: 10.1007/978-3-662-25726-5_19.  Google Scholar

[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). Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Ndiaye, Le Problème du Centre Pour des Systèmes Dynamiques Polynomiaux à Deux Dimensions, Ph.D thesis, Université de Tours, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

D. Schlomiuk, J. Guckenheimer and R. Rand, Integrability of plane quadratic vector fields, Expositiones Mathematicae, 8 (1990), 3-25.  Google Scholar

[34]

A. I. Yablonskii, Limit cycles of a certain differential equations, Differential Equations, 2 (1966), 335-344 (In Russian).  Google Scholar

[35]

Ye Yanqian et al., Theory of Limit Cycles, Translations of Math. Monographs, Vol. 66, Amer. Math. Soc, Providence, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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