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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 and 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.

[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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