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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. S. Chen, Uniqueness of the limit cycle of a quadratic system in the plane, Acta Math. Sinica, 20 (1977), 11-13.   Google Scholar

[6]

C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy. Soc. Edinburhgh, 124A (1994), 1209-1229.  doi: 10.1017/S0308210500030213.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

W. A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics Reported, 2 (1989), 61-88.   Google Scholar

[11]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, (2006).   Google Scholar

[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.   Google Scholar

[13]

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

[14]

R. M. Evdokimenco, Investigation in the large of a dynamic systems with a given integral curve, Differential Equations, 15 (1979), 215-221.   Google Scholar

[15]

V. F. Filiptsov, Algebraic limit cycles, Differencial?nye Uravnenija, 9 (1973), 1281-1288.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.   Google Scholar

[23]

A. I. Yablonskii, Limit cycles of a certain differential equations, DifferentialEquations, 2 (1966), 193-239.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. S. Chen, Uniqueness of the limit cycle of a quadratic system in the plane, Acta Math. Sinica, 20 (1977), 11-13.   Google Scholar

[6]

C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy. Soc. Edinburhgh, 124A (1994), 1209-1229.  doi: 10.1017/S0308210500030213.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

W. A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics Reported, 2 (1989), 61-88.   Google Scholar

[11]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, (2006).   Google Scholar

[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.   Google Scholar

[13]

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

[14]

R. M. Evdokimenco, Investigation in the large of a dynamic systems with a given integral curve, Differential Equations, 15 (1979), 215-221.   Google Scholar

[15]

V. F. Filiptsov, Algebraic limit cycles, Differencial?nye Uravnenija, 9 (1973), 1281-1288.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.   Google Scholar

[23]

A. I. Yablonskii, Limit cycles of a certain differential equations, DifferentialEquations, 2 (1966), 193-239.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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