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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 & 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). Google Scholar [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. Google Scholar [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. doi: 10.1017/S0305004100061636. Google Scholar [4] A. Buică and J. Llibre, Limit cycles of a perturbed cubic polynomial differential center,, Chaos Solitons & Fractals, 32 (2007), 1059. doi: 10.1016/j.chaos.2005.11.060. Google Scholar [5] L. A. Cherkas, Conditions for a Liénard equation to have a center,, Differ. Uravn., 12 (1976), 292. Google Scholar [6] C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields,, Transactions Amer. Math. Soc., 312 (1989), 319. doi: 10.2307/2000999. Google Scholar [7] C. Christopher, Estimating limit cycles bifurcations,, in Trends in Mathematics, (2005), 23. doi: 10.1007/3-7643-7429-2_2. Google Scholar [8] F. Dumortier, D. Panazzolo and R. Roussarie, More limit cycles than expected in Liénard equations,, Proc. Amer. Math. Soc., 135 (2007), 1895. doi: 10.1090/S0002-9939-07-08688-1. Google Scholar [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. doi: 10.1016/j.matcom.2013.02.003. Google Scholar [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. doi: 10.1016/j.na.2012.04.033. Google Scholar [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. doi: 10.1016/j.na.2011.08.032. Google Scholar [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. doi: 10.1007/s10884-008-9112-7. Google Scholar [13] A. Gasull and J. Torregrosa, Small-amplitude limit cycles in Liénard systems via multiplicity,, J. Differential Equations, 159 (1999), 186. doi: 10.1006/jdeq.1999.3649. Google Scholar [14] B. Coll, F. Dumortier and R. Prohens, Configurations of limit cycles in Liénard equations,, J. Differential Equations, 255 (2013), 4169. doi: 10.1016/j.jde.2013.08.004. Google Scholar [15] M. A. Golberg, The derivative of a determinant,, American Mathematical Monthly, (1972), 1124. Google Scholar [16] R. C. Gunning and H. Rossi, Analvvytic Functions of Several Complex Variables,, Prentice-Hall, (1965). Google Scholar [17] M. Han, Liapunov constants and Hopf cyclicity of Liénard systems,, Ann. Differential Equations, 15 (1999), 113. Google Scholar [18] M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations,, Interna. J. Bifur. Chaos, 22 (2012). doi: 10.1142/S0218127412502963. Google Scholar [19] M. Han, Bifurcation Theory of Limit Cycles,, Science Press, (2013). Google Scholar [20] J. Jiang and M. Han, Limit cycles in two types of symmetric Liénard systems,, Interna. J. Bifur. Chaos, 17 (2007), 2169. doi: 10.1142/S0218127407018300. Google Scholar [21] J. Jiang and M. Han, Small-amplitude limit cycles of some Liénard systems,, Nonlinear Anal. TMA, 71 (2009), 6373. doi: 10.1016/j.na.2009.09.011. Google Scholar [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. doi: 10.3934/cpaa.2012.11.2023. Google Scholar [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). Google Scholar [24] A. Liénard, Etude des oscillations entretenues,, Rev. gen. electr., 23 (1928), 901. Google Scholar [25] A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, in, Geometry and Topology, (1977), 335. Google Scholar [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. doi: 10.1017/S0305004109990193. Google Scholar [27] J. Llibre, J. S. Pérez del Río and J. A. Rodríguez, Averaging analysis of a perturbated quadratic center,, Nonlinear Anal., 46 (2001), 45. doi: 10.1016/S0362-546X(99)00444-7. Google Scholar [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. Google Scholar [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), 67. doi: 10.1007/BF01390315. Google Scholar [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, 19 (1986), 82. Google Scholar [31] V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach,, Birkhäuser Boston, (2009). doi: 10.1007/978-0-8176-4727-8. Google Scholar [32] R. Roussarie, A Note On Finite Cyclicity Property and Hilbert's 16th Problem,, Lecture Notes in Mathematics, (1331). doi: 10.1007/BFb0083072. Google Scholar [33] R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem,, Progress in Mathematics, 164 (1998). doi: 10.1007/978-3-0348-8798-4. Google Scholar [34] K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point,, Differ. Uravn. (Russian), 1 (1965), 53. Google Scholar [35] Y. Tian and M. Han, Hopf bifurcations for two types of Liénard systems,, J. Differential Equations, 251 (2011), 834. doi: 10.1016/j.jde.2011.05.029. Google Scholar [36] G. Xiang and M. Han, Global bifurcation of limit cycles in a family of polynomial systems,, J. Math. Anal. Appl., 295 (2004), 633. doi: 10.1016/j.jmaa.2004.03.047. Google Scholar [37] G. Xiang and M. Han, Global bifurcation of limit cycles in a family of multiparameter system,, Interna. J. Bifur. Chaos, 14 (2004), 3325. doi: 10.1142/S0218127404011144. Google Scholar [38] D. Yan and Y. Tian, Hopf cyclicity for a Liénard system,, J. Zhejiang Univ.(Science edition), 38 (2011), 10. Google Scholar [39] H. Żołądek, On a certain generalization of Bautin's theorem,, Nonlinearity, 7 (1994), 273. Google Scholar

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). Google Scholar [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. Google Scholar [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. doi: 10.1017/S0305004100061636. Google Scholar [4] A. Buică and J. Llibre, Limit cycles of a perturbed cubic polynomial differential center,, Chaos Solitons & Fractals, 32 (2007), 1059. doi: 10.1016/j.chaos.2005.11.060. Google Scholar [5] L. A. Cherkas, Conditions for a Liénard equation to have a center,, Differ. Uravn., 12 (1976), 292. Google Scholar [6] C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields,, Transactions Amer. Math. Soc., 312 (1989), 319. doi: 10.2307/2000999. Google Scholar [7] C. Christopher, Estimating limit cycles bifurcations,, in Trends in Mathematics, (2005), 23. doi: 10.1007/3-7643-7429-2_2. Google Scholar [8] F. Dumortier, D. Panazzolo and R. Roussarie, More limit cycles than expected in Liénard equations,, Proc. Amer. Math. Soc., 135 (2007), 1895. doi: 10.1090/S0002-9939-07-08688-1. Google Scholar [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. doi: 10.1016/j.matcom.2013.02.003. Google Scholar [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. doi: 10.1016/j.na.2012.04.033. Google Scholar [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. doi: 10.1016/j.na.2011.08.032. Google Scholar [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. doi: 10.1007/s10884-008-9112-7. Google Scholar [13] A. Gasull and J. Torregrosa, Small-amplitude limit cycles in Liénard systems via multiplicity,, J. Differential Equations, 159 (1999), 186. doi: 10.1006/jdeq.1999.3649. Google Scholar [14] B. Coll, F. Dumortier and R. Prohens, Configurations of limit cycles in Liénard equations,, J. Differential Equations, 255 (2013), 4169. doi: 10.1016/j.jde.2013.08.004. Google Scholar [15] M. A. Golberg, The derivative of a determinant,, American Mathematical Monthly, (1972), 1124. Google Scholar [16] R. C. Gunning and H. Rossi, Analvvytic Functions of Several Complex Variables,, Prentice-Hall, (1965). Google Scholar [17] M. Han, Liapunov constants and Hopf cyclicity of Liénard systems,, Ann. Differential Equations, 15 (1999), 113. Google Scholar [18] M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations,, Interna. J. Bifur. Chaos, 22 (2012). doi: 10.1142/S0218127412502963. Google Scholar [19] M. Han, Bifurcation Theory of Limit Cycles,, Science Press, (2013). Google Scholar [20] J. Jiang and M. Han, Limit cycles in two types of symmetric Liénard systems,, Interna. J. Bifur. Chaos, 17 (2007), 2169. doi: 10.1142/S0218127407018300. Google Scholar [21] J. Jiang and M. Han, Small-amplitude limit cycles of some Liénard systems,, Nonlinear Anal. TMA, 71 (2009), 6373. doi: 10.1016/j.na.2009.09.011. Google Scholar [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. doi: 10.3934/cpaa.2012.11.2023. Google Scholar [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). Google Scholar [24] A. Liénard, Etude des oscillations entretenues,, Rev. gen. electr., 23 (1928), 901. Google Scholar [25] A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, in, Geometry and Topology, (1977), 335. Google Scholar [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. doi: 10.1017/S0305004109990193. Google Scholar [27] J. Llibre, J. S. Pérez del Río and J. A. Rodríguez, Averaging analysis of a perturbated quadratic center,, Nonlinear Anal., 46 (2001), 45. doi: 10.1016/S0362-546X(99)00444-7. Google Scholar [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. Google Scholar [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), 67. doi: 10.1007/BF01390315. Google Scholar [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, 19 (1986), 82. Google Scholar [31] V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach,, Birkhäuser Boston, (2009). doi: 10.1007/978-0-8176-4727-8. Google Scholar [32] R. Roussarie, A Note On Finite Cyclicity Property and Hilbert's 16th Problem,, Lecture Notes in Mathematics, (1331). doi: 10.1007/BFb0083072. Google Scholar [33] R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem,, Progress in Mathematics, 164 (1998). doi: 10.1007/978-3-0348-8798-4. Google Scholar [34] K. S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point,, Differ. Uravn. (Russian), 1 (1965), 53. Google Scholar [35] Y. Tian and M. Han, Hopf bifurcations for two types of Liénard systems,, J. Differential Equations, 251 (2011), 834. doi: 10.1016/j.jde.2011.05.029. Google Scholar [36] G. Xiang and M. Han, Global bifurcation of limit cycles in a family of polynomial systems,, J. Math. Anal. Appl., 295 (2004), 633. doi: 10.1016/j.jmaa.2004.03.047. Google Scholar [37] G. Xiang and M. Han, Global bifurcation of limit cycles in a family of multiparameter system,, Interna. J. Bifur. Chaos, 14 (2004), 3325. doi: 10.1142/S0218127404011144. Google Scholar [38] D. Yan and Y. Tian, Hopf cyclicity for a Liénard system,, J. Zhejiang Univ.(Science edition), 38 (2011), 10. Google Scholar [39] H. Żołądek, On a certain generalization of Bautin's theorem,, Nonlinearity, 7 (1994), 273. Google Scholar
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