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Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane
Cyclicity of some Liénard Systems
1. | Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China, China |
2. | Department of Mathematics, Shanghai Normal University, Shanghai 200234 |
References:
[1] |
A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Mayer, Theory of bifurcations of dynamic systems on a plane, New York: Wiley, 1973. |
[2] |
N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik N. S., 30 (1952), 181-196; Translations Amer. Math. Soc., 100 (1954), 181-196. |
[3] |
T. R. Blows and N. G. Lloyd, The number of small-amplitude limit cycles of Liénard equations, Math. Proc. Cambridge Philos. Soc., 95 (1984), 359-366.
doi: 10.1017/S0305004100061636. |
[4] |
A. Buică and J. Llibre, Limit cycles of a perturbed cubic polynomial differential center, Chaos Solitons & Fractals, 32 (2007), 1059-1069.
doi: 10.1016/j.chaos.2005.11.060. |
[5] |
L. A. Cherkas, Conditions for a Liénard equation to have a center, Differ. Uravn., 12 (1976), 292-298; Differ. Equ., 12 (1976), 201-206. |
[6] |
C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Transactions Amer. Math. Soc., 312 (1989), 319-329.
doi: 10.2307/2000999. |
[7] |
C. Christopher, Estimating limit cycles bifurcations, in Trends in Mathematics, Differential Equations with Symbolic Computations (Eds. D. Wang and Z. Zheng), Birkhäuser-Verlag, (2005), 23-36.
doi: 10.1007/3-7643-7429-2_2. |
[8] |
F. Dumortier, D. Panazzolo and R. Roussarie, More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc., 135 (2007), 1895-1904.
doi: 10.1090/S0002-9939-07-08688-1. |
[9] |
B. Ferčec and A. Mahdi, Center conditions and cyclicity for a family of cubic systems: Computer algebra approach, Mathematics and Computers in Simulation, 87 (2013), 55-67.
doi: 10.1016/j.matcom.2013.02.003. |
[10] |
A. Gasull, J. T. Lázaro and J. Torregrosa, Upper bounds for the number of zeroes for some Abelian integrals, Nonlinear Anal., 75 (2012), 5169-5179.
doi: 10.1016/j.na.2012.04.033. |
[11] |
A. Gasull, C. Li and J. Torregrosa, Limit cycles appearing from the perturbation of a system with a multiple line of critical points, Nonlinear Anal., 75 (2012), 278-285.
doi: 10.1016/j.na.2011.08.032. |
[12] |
A. Gasull, R. Prohens and J. Torregrosa, Bifurcation of limit cycles from a polynomial non-global center, J. Dynam. Differential Equations, 20 (2008), 945-960.
doi: 10.1007/s10884-008-9112-7. |
[13] |
A. Gasull and J. Torregrosa, Small-amplitude limit cycles in Liénard systems via multiplicity, J. Differential Equations, 159 (1999), 186-211.
doi: 10.1006/jdeq.1999.3649. |
[14] |
B. Coll, F. Dumortier and R. Prohens, Configurations of limit cycles in Liénard equations, J. Differential Equations, 255 (2013), 4169-4184.
doi: 10.1016/j.jde.2013.08.004. |
[15] |
M. A. Golberg, The derivative of a determinant, American Mathematical Monthly, (1972), 1124-1126. |
[16] |
R. C. Gunning and H. Rossi, Analvvytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. |
[17] |
M. Han, Liapunov constants and Hopf cyclicity of Liénard systems, Ann. Differential Equations, 15 (1999), 113-126. |
[18] |
M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Interna. J. Bifur. Chaos, 22 (2012), 1250296.
doi: 10.1142/S0218127412502963. |
[19] |
M. Han, Bifurcation Theory of Limit Cycles, Science Press, Beijing, 2013. |
[20] |
J. Jiang and M. Han, Limit cycles in two types of symmetric Liénard systems, Interna. J. Bifur. Chaos, 17 (2007), 2169-2174.
doi: 10.1142/S0218127407018300. |
[21] |
J. Jiang and M. Han, Small-amplitude limit cycles of some Liénard systems, Nonlinear Anal. TMA, 71 (2009), 6373-6377.
doi: 10.1016/j.na.2009.09.011. |
[22] |
V. Levandovskyy, G. Pfister and V. G. Romanovski, Evaluating cyclicity of cubic systems with algorithms of computational algebra, Communications in Pure and Applied Analysis, 11 (2012), 2023-2035.
doi: 10.3934/cpaa.2012.11.2023. |
[23] |
A. M. Liapunov, Stability of motion, with a contribution by V. A. Pliss and an introduction by V. P. Basov., Mathematics in Science and Engineering, 30 (1966). |
[24] |
A. Liénard, Etude des oscillations entretenues, Rev. gen. electr., 23 (1928), 901-912. |
[25] |
A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, in Geometry and Topology, Springer Berlin Heidelberg, (1977), 335-357. |
[26] |
J. Llibre, A. C. Mereu and M. A. Teixeira, Limit cycles of the generalized polynomial Liénard differential equations, Math. Proc. Cambridge Philos. Soc., 148 (2010), 363-383.
doi: 10.1017/S0305004109990193. |
[27] |
J. Llibre, J. S. Pérez del Río and J. A. Rodríguez, Averaging analysis of a perturbated quadratic center, Nonlinear Anal., 46 (2001), 45-51.
doi: 10.1016/S0362-546X(99)00444-7. |
[28] |
N. Lloyd and S. Lynch, Small-amplitude limit cycles of certain Liénard systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 418 (1988), 199-208. |
[29] |
A. L. Neto, On the number of solutions of the equation $\frac{dx}{dt} = \sum_{j = 0}^n a_j (t) x^j ,0 \leq t \leq 1$, for which $x(0)=x(1)$, Invent. Math., 59 (1980), no. 1, 67-76.
doi: 10.1007/BF01390315. |
[30] |
V. G. Romanovski, On the cyclicity of the equilibrium position of the center or focus type of a certain system, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 19, no. 4, 82-87 (in Russian); (1986): Vestnik Leningrad Univ. Math. 19, no. 4, 51-56 (English translation). |
[31] |
V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009.
doi: 10.1007/978-0-8176-4727-8. |
[32] |
R. Roussarie, A Note On Finite Cyclicity Property and Hilbert's 16th Problem, Lecture Notes in Mathematics, Vol. 1331, New York: Springer-Verlag, 1988.
doi: 10.1007/BFb0083072. |
[33] |
R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem, Progress in Mathematics, 164, Birkhäuser, Basel, 1998.
doi: 10.1007/978-3-0348-8798-4. |
[34] |
K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differ. Uravn. (Russian), 1 (1965), 53-66; Differ. Equ. (English translation), 1 (1965), 36-47. |
[35] |
Y. Tian and M. Han, Hopf bifurcations for two types of Liénard systems, J. Differential Equations, 251 (2011), 834-859.
doi: 10.1016/j.jde.2011.05.029. |
[36] |
G. Xiang and M. Han, Global bifurcation of limit cycles in a family of polynomial systems, J. Math. Anal. Appl., 295 (2004), 633-644.
doi: 10.1016/j.jmaa.2004.03.047. |
[37] |
G. Xiang and M. Han, Global bifurcation of limit cycles in a family of multiparameter system, Interna. J. Bifur. Chaos, 14 (2004), 3325-3335.
doi: 10.1142/S0218127404011144. |
[38] |
D. Yan and Y. Tian, Hopf cyclicity for a Liénard system, J. Zhejiang Univ.(Science edition), 38 (2011), 10-18. |
[39] |
H. Żołądek, On a certain generalization of Bautin's theorem, Nonlinearity, 7 (1994), 273-279. |
show all references
References:
[1] |
A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Mayer, Theory of bifurcations of dynamic systems on a plane, New York: Wiley, 1973. |
[2] |
N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik N. S., 30 (1952), 181-196; Translations Amer. Math. Soc., 100 (1954), 181-196. |
[3] |
T. R. Blows and N. G. Lloyd, The number of small-amplitude limit cycles of Liénard equations, Math. Proc. Cambridge Philos. Soc., 95 (1984), 359-366.
doi: 10.1017/S0305004100061636. |
[4] |
A. Buică and J. Llibre, Limit cycles of a perturbed cubic polynomial differential center, Chaos Solitons & Fractals, 32 (2007), 1059-1069.
doi: 10.1016/j.chaos.2005.11.060. |
[5] |
L. A. Cherkas, Conditions for a Liénard equation to have a center, Differ. Uravn., 12 (1976), 292-298; Differ. Equ., 12 (1976), 201-206. |
[6] |
C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Transactions Amer. Math. Soc., 312 (1989), 319-329.
doi: 10.2307/2000999. |
[7] |
C. Christopher, Estimating limit cycles bifurcations, in Trends in Mathematics, Differential Equations with Symbolic Computations (Eds. D. Wang and Z. Zheng), Birkhäuser-Verlag, (2005), 23-36.
doi: 10.1007/3-7643-7429-2_2. |
[8] |
F. Dumortier, D. Panazzolo and R. Roussarie, More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc., 135 (2007), 1895-1904.
doi: 10.1090/S0002-9939-07-08688-1. |
[9] |
B. Ferčec and A. Mahdi, Center conditions and cyclicity for a family of cubic systems: Computer algebra approach, Mathematics and Computers in Simulation, 87 (2013), 55-67.
doi: 10.1016/j.matcom.2013.02.003. |
[10] |
A. Gasull, J. T. Lázaro and J. Torregrosa, Upper bounds for the number of zeroes for some Abelian integrals, Nonlinear Anal., 75 (2012), 5169-5179.
doi: 10.1016/j.na.2012.04.033. |
[11] |
A. Gasull, C. Li and J. Torregrosa, Limit cycles appearing from the perturbation of a system with a multiple line of critical points, Nonlinear Anal., 75 (2012), 278-285.
doi: 10.1016/j.na.2011.08.032. |
[12] |
A. Gasull, R. Prohens and J. Torregrosa, Bifurcation of limit cycles from a polynomial non-global center, J. Dynam. Differential Equations, 20 (2008), 945-960.
doi: 10.1007/s10884-008-9112-7. |
[13] |
A. Gasull and J. Torregrosa, Small-amplitude limit cycles in Liénard systems via multiplicity, J. Differential Equations, 159 (1999), 186-211.
doi: 10.1006/jdeq.1999.3649. |
[14] |
B. Coll, F. Dumortier and R. Prohens, Configurations of limit cycles in Liénard equations, J. Differential Equations, 255 (2013), 4169-4184.
doi: 10.1016/j.jde.2013.08.004. |
[15] |
M. A. Golberg, The derivative of a determinant, American Mathematical Monthly, (1972), 1124-1126. |
[16] |
R. C. Gunning and H. Rossi, Analvvytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. |
[17] |
M. Han, Liapunov constants and Hopf cyclicity of Liénard systems, Ann. Differential Equations, 15 (1999), 113-126. |
[18] |
M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Interna. J. Bifur. Chaos, 22 (2012), 1250296.
doi: 10.1142/S0218127412502963. |
[19] |
M. Han, Bifurcation Theory of Limit Cycles, Science Press, Beijing, 2013. |
[20] |
J. Jiang and M. Han, Limit cycles in two types of symmetric Liénard systems, Interna. J. Bifur. Chaos, 17 (2007), 2169-2174.
doi: 10.1142/S0218127407018300. |
[21] |
J. Jiang and M. Han, Small-amplitude limit cycles of some Liénard systems, Nonlinear Anal. TMA, 71 (2009), 6373-6377.
doi: 10.1016/j.na.2009.09.011. |
[22] |
V. Levandovskyy, G. Pfister and V. G. Romanovski, Evaluating cyclicity of cubic systems with algorithms of computational algebra, Communications in Pure and Applied Analysis, 11 (2012), 2023-2035.
doi: 10.3934/cpaa.2012.11.2023. |
[23] |
A. M. Liapunov, Stability of motion, with a contribution by V. A. Pliss and an introduction by V. P. Basov., Mathematics in Science and Engineering, 30 (1966). |
[24] |
A. Liénard, Etude des oscillations entretenues, Rev. gen. electr., 23 (1928), 901-912. |
[25] |
A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, in Geometry and Topology, Springer Berlin Heidelberg, (1977), 335-357. |
[26] |
J. Llibre, A. C. Mereu and M. A. Teixeira, Limit cycles of the generalized polynomial Liénard differential equations, Math. Proc. Cambridge Philos. Soc., 148 (2010), 363-383.
doi: 10.1017/S0305004109990193. |
[27] |
J. Llibre, J. S. Pérez del Río and J. A. Rodríguez, Averaging analysis of a perturbated quadratic center, Nonlinear Anal., 46 (2001), 45-51.
doi: 10.1016/S0362-546X(99)00444-7. |
[28] |
N. Lloyd and S. Lynch, Small-amplitude limit cycles of certain Liénard systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 418 (1988), 199-208. |
[29] |
A. L. Neto, On the number of solutions of the equation $\frac{dx}{dt} = \sum_{j = 0}^n a_j (t) x^j ,0 \leq t \leq 1$, for which $x(0)=x(1)$, Invent. Math., 59 (1980), no. 1, 67-76.
doi: 10.1007/BF01390315. |
[30] |
V. G. Romanovski, On the cyclicity of the equilibrium position of the center or focus type of a certain system, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 19, no. 4, 82-87 (in Russian); (1986): Vestnik Leningrad Univ. Math. 19, no. 4, 51-56 (English translation). |
[31] |
V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009.
doi: 10.1007/978-0-8176-4727-8. |
[32] |
R. Roussarie, A Note On Finite Cyclicity Property and Hilbert's 16th Problem, Lecture Notes in Mathematics, Vol. 1331, New York: Springer-Verlag, 1988.
doi: 10.1007/BFb0083072. |
[33] |
R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem, Progress in Mathematics, 164, Birkhäuser, Basel, 1998.
doi: 10.1007/978-3-0348-8798-4. |
[34] |
K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differ. Uravn. (Russian), 1 (1965), 53-66; Differ. Equ. (English translation), 1 (1965), 36-47. |
[35] |
Y. Tian and M. Han, Hopf bifurcations for two types of Liénard systems, J. Differential Equations, 251 (2011), 834-859.
doi: 10.1016/j.jde.2011.05.029. |
[36] |
G. Xiang and M. Han, Global bifurcation of limit cycles in a family of polynomial systems, J. Math. Anal. Appl., 295 (2004), 633-644.
doi: 10.1016/j.jmaa.2004.03.047. |
[37] |
G. Xiang and M. Han, Global bifurcation of limit cycles in a family of multiparameter system, Interna. J. Bifur. Chaos, 14 (2004), 3325-3335.
doi: 10.1142/S0218127404011144. |
[38] |
D. Yan and Y. Tian, Hopf cyclicity for a Liénard system, J. Zhejiang Univ.(Science edition), 38 (2011), 10-18. |
[39] |
H. Żołądek, On a certain generalization of Bautin's theorem, Nonlinearity, 7 (1994), 273-279. |
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