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February  2015, 35(2): 669-701. doi: 10.3934/dcds.2015.35.669

Bifurcation values for a family of planar vector fields of degree five

Johanna D. García-Saldaña 1, , Armengol Gasull 1, and  Hector Giacomini 2,

1. 

Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain
2. 

Laboratoire de Mathématique et Physique Théorique, C.N.R.S. UMR 7350, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont,37200 Tours

Received  February 2013 Revised  February 2014 Published  September 2014

We study the number of limit cycles and the bifurcation diagram in the Poincaré sphere of a one-parameter family of planar differential equations of degree five $\dot {\bf x}=X_b({\bf x})$ which has been already considered in previous papers. We prove that there is a value $b^*>0$ such that the limit cycle exists only when $b\in(0,b^*)$ and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length $27/1000$ where $b^*$ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic curves without contact for the flow of the differential equation. These curves are obtained using analytic information about the separatrices of the infinite critical points of the vector field. To prove that the Bendixson--Dulac Theorem works we develop a method for studying whether one-parameter families of polynomials in two variables do not vanish based on the computation of the so called double discriminant.
Keywords: Polynomial planar system, double discriminant., phase portrait on the Poincaré sphere, Dulac function, bifurcation, uniqueness of limit cycles.
Mathematics Subject Classification: Primary: 34C07; Secondary: 34C23, 34C25, 37C27, 13P1.
Citation: Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669
References:
[1]

M. J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3103-3118. doi: 10.1142/S0218127411030416.  Google Scholar

[2]

J. G. Alcazar, J. Schicho and J. R. Sendra, A delineability-based method for computing critical sets of algebraic surfaces, J. Symbolic Comput., 42 (2007), 678-691. doi: 10.1016/j.jsc.2007.02.001.  Google Scholar

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G. A. Baker and P. Graves-Morris, Padé Approximants, Second edition, Encyclopedia of Mathematics and its Applications, 59, Cambridge University Press, Cambridge, 1996.  Google Scholar

[5]

L. A. Cherkas, The Dulac function for polynomial autonomous systems on a plane, (Russian) Differ. Uravn., 33 (1997), 689-699, 719; translation in Differential Equations, 33 (1997), 692-701.  Google Scholar

[6]

L. A. Cherkas, A. A. Grin and K. R. Schneider, Dulac-Cherkas functions for generalized Liénard systems, Electron. J. Qual. Theory Differ. Equ., 35 (2011), 23 pp.  Google Scholar

[7]

C. Chicone, Ordinary Differential Equations with Applications, Second edition, Texts in Applied Mathematics, 34, Springer, New York, 2006.  Google Scholar

[8]

D. Cox, J. Little and D. O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics, 185, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4757-6911-1.  Google Scholar

[9]

G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953), 15-31. doi: 10.2307/1969724.  Google Scholar

[10]

F. Dumortier, Singularities of vector fields on the plane, J. Differential Equations, 23 (1977), 53-106. doi: 10.1016/0022-0396(77)90136-X.  Google Scholar

[11]

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

[12]

A. Gasull and H. Giacomini, A new criterion for controlling the number of limit cycles of some generalized Liénard equations, J. Differential Equations, 185 (2002), 54-73. doi: 10.1006/jdeq.2002.4172.  Google Scholar

[13]

A. Gasull and H. Giacomini, Upper bounds for the number of limit cycles through linear differential equations, Pacific J. Math., 226 (2006), 277-296. doi: 10.2140/pjm.2006.226.277.  Google Scholar

[14]

A. Gasull and H. Giacomini, Upper bounds for the number of limit cycles of some planar polynomial differential systems, Discrete Contin. Dyn. Syst., 27 (2010), 217-229. doi: 10.3934/dcds.2010.27.217.  Google Scholar

[15]

A. Gasull, H. Giacomini and J. Torregrosa, Some results on homoclinic and heteroclinic connections in planar systems, Nonlinearity, 23 (2010), 2977-3001. doi: 10.1088/0951-7715/23/12/001.  Google Scholar

[16]

A. Gasull, H. Giacomini and J. Torregrosa, Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations, Discrete Contin. Dyn. Syst., 33 (2013), 3567-3582. doi: 10.3934/dcds.2013.33.3567.  Google Scholar

[17]

M. Han and T. Qian, Uniqueness of periodic solutions for certain second-order equations, Acta Math. Sin. (Engl. Ser.), 20 (2004), 247-254. doi: 10.1007/s10114-003-0300-4.  Google Scholar

[18]

W. Krandick and K. Mehlhorn, New bounds for the Descartes method, J. Symbolic Comput., 41 (2006), 49-66. doi: 10.1016/j.jsc.2005.02.004.  Google Scholar

[19]

D. Lazard and S. McCallum, Iterated discriminants, J. Symbolic Comput., 44 (2009), 1176-1193. doi: 10.1016/j.jsc.2008.05.006.  Google Scholar

[20]

N. G. Lloyd, A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc. (2), 20 (1979), 277-286. doi: 10.1112/jlms/s2-20.2.277.  Google Scholar

[21]

A. M. Lyapunov, Stability of Motion, Mathematics in Science and Engineering, 30, Academic Press, New York, London, 1966.  Google Scholar

[22]

L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math. Soc., 76 (1954), 127-148. doi: 10.1090/S0002-9947-1954-0060657-0.  Google Scholar

[23]

D. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81. doi: 10.1090/S0002-9939-1975-0356138-6.  Google Scholar

[24]

L. M. Perko, Rotated vector fields and the global behavior of limit cycles for a class of quadratic systems in the plane, J. Differential Equations, 18 (1975), 63-86. doi: 10.1016/0022-0396(75)90081-9.  Google Scholar

[25]

L. M. Perko, Global families of limit cycles of planar analytic systems, Trans. Amer. Math. Soc., 322 (1990), 627-656. doi: 10.1090/S0002-9947-1990-0998357-4.  Google Scholar

[26]

L. M. Perko, Bifurcation of limit cycles, in Bifurcations of Planar Vector Fields (Luminy, 1989), Lecture Notes in Math., 1455, Springer, Berlin, 1990, 315-333. doi: 10.1007/BFb0085398.  Google Scholar

[27]

L. M. Perko, Differential Equations and Dynamical Systems, Second edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-0249-0.  Google Scholar

[28]

J. Pettigrew and J. A. G. Roberts, Characterizing singular curves in parametrized families of biquadratics, J. Phys. A, 41 (2008), 115203, 28 pp. doi: 10.1088/1751-8113/41/11/115203.  Google Scholar

[29]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Translated from the German by R. Bartels, W. Gautschi and C. Witzgall, Springer-Verlag, New York-Heidelberg, 1980.  Google Scholar

[30]

X. Wang, J. Jiang and P. Yan, Analysis of global bifurcation for a class of systems of degree five, J. Math. Anal. Appl., 222 (1998), 305-318. doi: 10.1006/jmaa.1997.5546.  Google Scholar

[31]

K. Yamato, An effective method of counting the number of limit cycles, Nagoya Math. J., 76 (1979), 35-114.  Google Scholar

show all references

References:
[1]

M. J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3103-3118. doi: 10.1142/S0218127411030416.  Google Scholar

[2]

J. G. Alcazar, J. Schicho and J. R. Sendra, A delineability-based method for computing critical sets of algebraic surfaces, J. Symbolic Comput., 42 (2007), 678-691. doi: 10.1016/j.jsc.2007.02.001.  Google Scholar

[3]

A. A. Andronov, E. A. Leontovich, I. I Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, John Wiley & Sons, New York, 1973.  Google Scholar

[4]

G. A. Baker and P. Graves-Morris, Padé Approximants, Second edition, Encyclopedia of Mathematics and its Applications, 59, Cambridge University Press, Cambridge, 1996.  Google Scholar

[5]

L. A. Cherkas, The Dulac function for polynomial autonomous systems on a plane, (Russian) Differ. Uravn., 33 (1997), 689-699, 719; translation in Differential Equations, 33 (1997), 692-701.  Google Scholar

[6]

L. A. Cherkas, A. A. Grin and K. R. Schneider, Dulac-Cherkas functions for generalized Liénard systems, Electron. J. Qual. Theory Differ. Equ., 35 (2011), 23 pp.  Google Scholar

[7]

C. Chicone, Ordinary Differential Equations with Applications, Second edition, Texts in Applied Mathematics, 34, Springer, New York, 2006.  Google Scholar

[8]

D. Cox, J. Little and D. O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics, 185, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4757-6911-1.  Google Scholar

[9]

G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953), 15-31. doi: 10.2307/1969724.  Google Scholar

[10]

F. Dumortier, Singularities of vector fields on the plane, J. Differential Equations, 23 (1977), 53-106. doi: 10.1016/0022-0396(77)90136-X.  Google Scholar

[11]

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

[12]

A. Gasull and H. Giacomini, A new criterion for controlling the number of limit cycles of some generalized Liénard equations, J. Differential Equations, 185 (2002), 54-73. doi: 10.1006/jdeq.2002.4172.  Google Scholar

[13]

A. Gasull and H. Giacomini, Upper bounds for the number of limit cycles through linear differential equations, Pacific J. Math., 226 (2006), 277-296. doi: 10.2140/pjm.2006.226.277.  Google Scholar

[14]

A. Gasull and H. Giacomini, Upper bounds for the number of limit cycles of some planar polynomial differential systems, Discrete Contin. Dyn. Syst., 27 (2010), 217-229. doi: 10.3934/dcds.2010.27.217.  Google Scholar

[15]

A. Gasull, H. Giacomini and J. Torregrosa, Some results on homoclinic and heteroclinic connections in planar systems, Nonlinearity, 23 (2010), 2977-3001. doi: 10.1088/0951-7715/23/12/001.  Google Scholar

[16]

A. Gasull, H. Giacomini and J. Torregrosa, Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations, Discrete Contin. Dyn. Syst., 33 (2013), 3567-3582. doi: 10.3934/dcds.2013.33.3567.  Google Scholar

[17]

M. Han and T. Qian, Uniqueness of periodic solutions for certain second-order equations, Acta Math. Sin. (Engl. Ser.), 20 (2004), 247-254. doi: 10.1007/s10114-003-0300-4.  Google Scholar

[18]

W. Krandick and K. Mehlhorn, New bounds for the Descartes method, J. Symbolic Comput., 41 (2006), 49-66. doi: 10.1016/j.jsc.2005.02.004.  Google Scholar

[19]

D. Lazard and S. McCallum, Iterated discriminants, J. Symbolic Comput., 44 (2009), 1176-1193. doi: 10.1016/j.jsc.2008.05.006.  Google Scholar

[20]

N. G. Lloyd, A note on the number of limit cycles in certain two-dimensional systems, J. London Math. Soc. (2), 20 (1979), 277-286. doi: 10.1112/jlms/s2-20.2.277.  Google Scholar

[21]

A. M. Lyapunov, Stability of Motion, Mathematics in Science and Engineering, 30, Academic Press, New York, London, 1966.  Google Scholar

[22]

L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math. Soc., 76 (1954), 127-148. doi: 10.1090/S0002-9947-1954-0060657-0.  Google Scholar

[23]

D. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81. doi: 10.1090/S0002-9939-1975-0356138-6.  Google Scholar

[24]

L. M. Perko, Rotated vector fields and the global behavior of limit cycles for a class of quadratic systems in the plane, J. Differential Equations, 18 (1975), 63-86. doi: 10.1016/0022-0396(75)90081-9.  Google Scholar

[25]

L. M. Perko, Global families of limit cycles of planar analytic systems, Trans. Amer. Math. Soc., 322 (1990), 627-656. doi: 10.1090/S0002-9947-1990-0998357-4.  Google Scholar

[26]

L. M. Perko, Bifurcation of limit cycles, in Bifurcations of Planar Vector Fields (Luminy, 1989), Lecture Notes in Math., 1455, Springer, Berlin, 1990, 315-333. doi: 10.1007/BFb0085398.  Google Scholar

[27]

L. M. Perko, Differential Equations and Dynamical Systems, Second edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-0249-0.  Google Scholar

[28]

J. Pettigrew and J. A. G. Roberts, Characterizing singular curves in parametrized families of biquadratics, J. Phys. A, 41 (2008), 115203, 28 pp. doi: 10.1088/1751-8113/41/11/115203.  Google Scholar

[29]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Translated from the German by R. Bartels, W. Gautschi and C. Witzgall, Springer-Verlag, New York-Heidelberg, 1980.  Google Scholar

[30]

X. Wang, J. Jiang and P. Yan, Analysis of global bifurcation for a class of systems of degree five, J. Math. Anal. Appl., 222 (1998), 305-318. doi: 10.1006/jmaa.1997.5546.  Google Scholar

[31]

K. Yamato, An effective method of counting the number of limit cycles, Nagoya Math. J., 76 (1979), 35-114.  Google Scholar

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