Algebraic limit cycles for quadratic polynomial differential systems

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August 2018, 23(6): 2475-2485. doi: 10.3934/dcdsb.2018070

Algebraic limit cycles for quadratic polynomial differential systems

Jaume Llibre ^1,, and Claudia Valls ^2,

1.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2.	Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

* Corresponding author: Jaume Llibre

Received July 2017 Published August 2018 Early access January 2018

We prove that for a quadratic polynomial differential system having three pairs of diametrally opposite equilibrium points at infinity that are positively rationally independent, has at most one algebraic limit cycle. Our result provides a partial positive answer to the following conjecture: Quadratic polynomial differential systems have at most one algebraic limit cycle.

Keywords: Algebraic limit cycle, quadratic polynomial differential system, quadratic polynomial vector field.

Mathematics Subject Classification: Primary: 37D99.

Citation: Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070

References:

[1]	J. Chavarriga, H. Giacomini and M. Grau, Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems, Bull. Sci. Math., 129 (2005), 99-126. doi: 10.1016/j.bulsci.2004.09.002.
[2]	J. Chavarriga, H. Giacomini and J. Llibre, Uniqueness of algebraic limit cycles for quadratic systems, J. Math. Anal. Appl., 261 (2001), 85-99. doi: 10.1006/jmaa.2001.7476.
[3]	J. Chavarriga and J. Llibre, Invariant algebraic curves and rational first integrals planar polynomial vector fields, J. Differential Equations, 169 (2001), 1-16. doi: 10.1006/jdeq.2000.3891.
[4]	J. Chavarriga, J. Llibre and J. Sorolla, Algebraic limit cycles of degree four for quadratic systems, J. Differential Equations, 200 (2004), 206-244. doi: 10.1016/j.jde.2004.01.003.
[5]	L. S. Chen, Uniqueness of the limit cycle of a quadratic system in the plane, Acta Math. Sinica, 20 (1977), 11-13.
[6]	C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy. Soc. Edinburhgh, 124A (1994), 1209-1229. doi: 10.1017/S0308210500030213.
[7]	C. Christopher, J. Llibre and G. Swirszcz, Invariant algebraic curves of large degree for quadratic systems, J. Math. Anal. Appl., 303 (2005), 206-244. doi: 10.1016/j.jmaa.2004.08.042.
[8]	B. Coll and J. Llibre, Limit cycles for a quadratic systems with an invariant straight line and some evolution of phase portraits, Colloquia Mathematica Societatis Janos Bolyai, 53 (1988), 111-123.
[9]	B. Coll, G. Gasull and J. Llibre, Quadratic systems with a unique finite rest point, Publicacions Matematiques, 32 (1988), 199-259. doi: 10.5565/PUBLMAT_32288_08.
[10]	W. A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics Reported, 2 (1989), 61-88.
[11]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, (2006).
[12]	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.
[13]	R. M. Evdokimenco, Behavior of integral curves of a dynamic system, Differential Equations, 9 (1974), 1095-1103.
[14]	R. M. Evdokimenco, Investigation in the large of a dynamic systems with a given integral curve, Differential Equations, 15 (1979), 215-221.
[15]	V. F. Filiptsov, Algebraic limit cycles, Differencial?nye Uravnenija, 9 (1973), 1281-1288.
[16]	D. Hilbert, Mathematische Probleme, in Lecture, Second Internat. Congr. Math., Paris, 1900, in Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl., 1900, pp. 253-297; English transl. in Bull. Amer. Math. Soc., 8 (1902), 437-479.
[17]	J. Llibre, Integrability of polynomial differential systems, in Handbook of Differential Equations, Ordinary Differential Equations, Eds. A. Cañada, P. Drabek and A. Fonda, Elsevier, 1 (2004), 437-479.
[18]	J. Llibre and D. Schlomiuk, On the limit cycles bifurcating from an ellipse of a quadratic center, Discrete Contin. Dyn. Syst. Series B, 35 (2015), 1091-1102.
[19]	J. Llibre and G. Swirszcz, Classification of quadratic systems admitting the existence of an algebraic limit cycle, Bull. Sci. Math., 131 (2007), 405-421. doi: 10.1016/j.bulsci.2006.03.014.
[20]	J. Llibre and C. Valls, Quadratic polynomial differential systems with one pair of singular points at infinity have at most one algebraic limit cycle, to appear in Proc. Edinburgh Math. Soc.
[21]	J. Llibre and C. Valls, Quadratic polynomial differential systems with two pairs of singular points at infinity have at most one algebraic limit cycle, to appear in Geometria Dedicata.
[22]	B. Shen, The problem of the existence of limit cycles and separatrix cycles of cubic curves in quadratic systems, Chinese Ann. Math. Ser. A, 12 (1991), 382-389.
[23]	A. I. Yablonskii, Limit cycles of a certain differential equations, DifferentialEquations, 2 (1966), 193-239.
[24]	Q. Yuan-Xun, On the algebraic limit cycles of second degree of the differential equation $dy/dx=\sum_{0 ≤ i+j ≤ 2} a_{ij} x^i y^j/\sum_{0 ≤ i+j ≤ 2} b_{ij} x^i y^j$, Acta Math. Sinica, 8 (1958), 23-35.
[25]	X. Zhang, Invariant algebraic curves and rational first integrals of holomorphic foliations in CP(2), Sci. China Ser. A, 46 (2003), 271-279. doi: 10.1360/03ys9029.

show all references

References:

[1]	J. Chavarriga, H. Giacomini and M. Grau, Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems, Bull. Sci. Math., 129 (2005), 99-126. doi: 10.1016/j.bulsci.2004.09.002.
[2]	J. Chavarriga, H. Giacomini and J. Llibre, Uniqueness of algebraic limit cycles for quadratic systems, J. Math. Anal. Appl., 261 (2001), 85-99. doi: 10.1006/jmaa.2001.7476.
[3]	J. Chavarriga and J. Llibre, Invariant algebraic curves and rational first integrals planar polynomial vector fields, J. Differential Equations, 169 (2001), 1-16. doi: 10.1006/jdeq.2000.3891.
[4]	J. Chavarriga, J. Llibre and J. Sorolla, Algebraic limit cycles of degree four for quadratic systems, J. Differential Equations, 200 (2004), 206-244. doi: 10.1016/j.jde.2004.01.003.
[5]	L. S. Chen, Uniqueness of the limit cycle of a quadratic system in the plane, Acta Math. Sinica, 20 (1977), 11-13.
[6]	C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy. Soc. Edinburhgh, 124A (1994), 1209-1229. doi: 10.1017/S0308210500030213.
[7]	C. Christopher, J. Llibre and G. Swirszcz, Invariant algebraic curves of large degree for quadratic systems, J. Math. Anal. Appl., 303 (2005), 206-244. doi: 10.1016/j.jmaa.2004.08.042.
[8]	B. Coll and J. Llibre, Limit cycles for a quadratic systems with an invariant straight line and some evolution of phase portraits, Colloquia Mathematica Societatis Janos Bolyai, 53 (1988), 111-123.
[9]	B. Coll, G. Gasull and J. Llibre, Quadratic systems with a unique finite rest point, Publicacions Matematiques, 32 (1988), 199-259. doi: 10.5565/PUBLMAT_32288_08.
[10]	W. A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics Reported, 2 (1989), 61-88.
[11]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, (2006).
[12]	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.
[13]	R. M. Evdokimenco, Behavior of integral curves of a dynamic system, Differential Equations, 9 (1974), 1095-1103.
[14]	R. M. Evdokimenco, Investigation in the large of a dynamic systems with a given integral curve, Differential Equations, 15 (1979), 215-221.
[15]	V. F. Filiptsov, Algebraic limit cycles, Differencial?nye Uravnenija, 9 (1973), 1281-1288.
[16]	D. Hilbert, Mathematische Probleme, in Lecture, Second Internat. Congr. Math., Paris, 1900, in Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl., 1900, pp. 253-297; English transl. in Bull. Amer. Math. Soc., 8 (1902), 437-479.
[17]	J. Llibre, Integrability of polynomial differential systems, in Handbook of Differential Equations, Ordinary Differential Equations, Eds. A. Cañada, P. Drabek and A. Fonda, Elsevier, 1 (2004), 437-479.
[18]	J. Llibre and D. Schlomiuk, On the limit cycles bifurcating from an ellipse of a quadratic center, Discrete Contin. Dyn. Syst. Series B, 35 (2015), 1091-1102.
[19]	J. Llibre and G. Swirszcz, Classification of quadratic systems admitting the existence of an algebraic limit cycle, Bull. Sci. Math., 131 (2007), 405-421. doi: 10.1016/j.bulsci.2006.03.014.
[20]	J. Llibre and C. Valls, Quadratic polynomial differential systems with one pair of singular points at infinity have at most one algebraic limit cycle, to appear in Proc. Edinburgh Math. Soc.
[21]	J. Llibre and C. Valls, Quadratic polynomial differential systems with two pairs of singular points at infinity have at most one algebraic limit cycle, to appear in Geometria Dedicata.
[22]	B. Shen, The problem of the existence of limit cycles and separatrix cycles of cubic curves in quadratic systems, Chinese Ann. Math. Ser. A, 12 (1991), 382-389.
[23]	A. I. Yablonskii, Limit cycles of a certain differential equations, DifferentialEquations, 2 (1966), 193-239.
[24]	Q. Yuan-Xun, On the algebraic limit cycles of second degree of the differential equation $dy/dx=\sum_{0 ≤ i+j ≤ 2} a_{ij} x^i y^j/\sum_{0 ≤ i+j ≤ 2} b_{ij} x^i y^j$, Acta Math. Sinica, 8 (1958), 23-35.
[25]	X. Zhang, Invariant algebraic curves and rational first integrals of holomorphic foliations in CP(2), Sci. China Ser. A, 46 (2003), 271-279. doi: 10.1360/03ys9029.

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