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

Algebraic limit cycles for quadratic polynomial differential systems

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.

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. ChavarrigaH. 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. ChavarrigaH. 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. ChavarrigaJ. 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. ChristopherJ. 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. CollG. 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. DumortierJ. 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. ChavarrigaH. 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. ChavarrigaH. 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. ChavarrigaJ. 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. ChristopherJ. 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. CollG. 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. DumortierJ. 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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