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Bifurcation values for a family of planar vector fields of degree five
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 |
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. |
[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. |
[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. |
[4] |
G. A. Baker and P. Graves-Morris, Padé Approximants, Second edition, Encyclopedia of Mathematics and its Applications, 59, Cambridge University Press, Cambridge, 1996. |
[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. |
[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. |
[7] |
C. Chicone, Ordinary Differential Equations with Applications, Second edition, Texts in Applied Mathematics, 34, Springer, New York, 2006. |
[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. |
[9] |
G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953), 15-31.
doi: 10.2307/1969724. |
[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. |
[11] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Universitext, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
D. Lazard and S. McCallum, Iterated discriminants, J. Symbolic Comput., 44 (2009), 1176-1193.
doi: 10.1016/j.jsc.2008.05.006. |
[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. |
[21] |
A. M. Lyapunov, Stability of Motion, Mathematics in Science and Engineering, 30, Academic Press, New York, London, 1966. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
K. Yamato, An effective method of counting the number of limit cycles, Nagoya Math. J., 76 (1979), 35-114. |
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. |
[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. |
[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. |
[4] |
G. A. Baker and P. Graves-Morris, Padé Approximants, Second edition, Encyclopedia of Mathematics and its Applications, 59, Cambridge University Press, Cambridge, 1996. |
[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. |
[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. |
[7] |
C. Chicone, Ordinary Differential Equations with Applications, Second edition, Texts in Applied Mathematics, 34, Springer, New York, 2006. |
[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. |
[9] |
G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math., 57 (1953), 15-31.
doi: 10.2307/1969724. |
[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. |
[11] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Universitext, 2006. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
D. Lazard and S. McCallum, Iterated discriminants, J. Symbolic Comput., 44 (2009), 1176-1193.
doi: 10.1016/j.jsc.2008.05.006. |
[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. |
[21] |
A. M. Lyapunov, Stability of Motion, Mathematics in Science and Engineering, 30, Academic Press, New York, London, 1966. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
K. Yamato, An effective method of counting the number of limit cycles, Nagoya Math. J., 76 (1979), 35-114. |
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