Cyclicity of some Liénard Systems

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

Cyclicity of some Liénard Systems

Na Li ^1, , Maoan Han ^2, and Valery G. Romanovski ^1,

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

Abstract
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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.

Keywords: Limit cycle, Liénard system., cyclicity.

Mathematics Subject Classification: Primary: 34C07; Secondary: 37G1.

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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