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November  2015, 14(6): 2127-2150. doi: 10.3934/cpaa.2015.14.2127

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

Received  March 2014 Revised  June 2015 Published  September 2015

The Liénard system and its generalizations are important models of nonlinear oscillators. We study small-amplitude limit cycles of two families of Liénard systems and find exact number of such limit cycles bifurcating from a center or focus at the origin for these families, thus obtaining the precise bound for cyclicity of the families.
Citation: Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
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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