# American Institute of Mathematical Sciences

July  2013, 33(7): 3085-3108. doi: 10.3934/dcds.2013.33.3085

## Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems

 1 School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007 2 Department of Mathematics, Shanghai Normal University, Shanghai 200234

Received  January 2012 Revised  November 2012 Published  January 2013

This paper is concerned with bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. By analyzing the multiplicities of the zeroes of the slow divergence integrals and their complete unfolding, the upper bounds of canard limit cycles bifurcating from the suitable limit periodic sets through respectively the generic Hopf breaking mechanism, the generic jump breaking mechanism and a succession of the Hopf and jump mechanisms in these polynomial Liénard systems are obtained.
Citation: Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
##### References:
 [1] P. De, Maesschalck and F. Dumortier, Classical Liénard equations of degess $n\geq6$ can have $[\frac{n-1}{2}]+2$ limit cycles,, J. Differential Equations, 250 (2011), 2162. doi: 10.1016/j.jde.2010.12.003. Google Scholar [2] P. De, Maesschalck and F. Dumortier, Bifurcations of multiple relaxation oscillations in polynomial Liénard equations,, Proc. Amer. Math. Soc., 139 (2011), 2073. doi: 10.1090/S0002-9939-2010-10610-X. Google Scholar [3] F. Dumortier, Slow divergence integral and balanced canard solutions,, Qual. Theory Dyn. Syst., 10 (2011), 65. doi: 10.1007/s12346-011-0038-9. Google Scholar [4] 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 [5] F. Dumortier and R. Roussarie, Multiple canard cycles in generalized Liénard equations,, J. Differential Equations, 174 (2001), 1. doi: 10.1006/jdeq.2000.3947. Google Scholar [6] F. Dumortier and R. Roussarie, Bifurcation of relaxation oscillations in dimension two,, Discrete Contin. Dyn. Syst., 19 (2007), 631. doi: 10.3934/dcds.2007.19.631. Google Scholar [7] F. Dumortier and R. Roussarie, Canard cycles with two breaking parameters,, Discrete Contin. Dyn. Syst., 17 (2007), 787. doi: 10.3934/dcds.2007.17.787. Google Scholar [8] F. Dumortier and R. Roussarie, Multi-layer canard cycles and translated power functions,, J. Differential Equations, 244 (2008), 1329. doi: 10.1016/j.jde.2007.08.013. Google Scholar [9] M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities,", in: Graduate Texts in Math., (1973). Google Scholar [10] M. Han, P. Bi and D. Xiao, Bifurcation of limit cycles and separatrix loops in singular Liénard systems,, Chaos Solitons Fractals, 20 (2004), 529. doi: 10.1016/S0960-0779(03)00412-0. Google Scholar [11] C. Li and J. Llibre, Uniqueness of limit cycles for Liénard equations of degree four,, J. Differential Equations, 252 (2012), 3142. doi: 10.1016/j.jde.2011.11.002. Google Scholar [12] A. Lins, W. de Melo and C.C. Pugh, On Liénard's equations,, in, (1977), 335. Google Scholar [13] J. Llibre, A survey on the limit cycles of the generalized polynomial Liénard differential equations,, in, (2008), 224. Google Scholar [14] L. Mamouhdi and R. Roussarie, Canard cycles of finite codimension with two breaking parameters,, Qual. Theory Dyn. Syst., 11 (2012), 167. doi: 10.1007/s12346-011-0061-x. Google Scholar [15] R. Roussarie, Putting a boundary to the space of Liénard equations,, Discrete Contin. Dyn. Syst., 17 (2007), 441. doi: 10.3934/dcds.2007.17.441. Google Scholar [16] Y. Tian and M. Han, Hopf bifurcation for two types of Liénard systems,, J. Differential Equations, 251 (2011), 834. doi: 10.1016/j.jde.2011.05.029. Google Scholar [17] J. Wang and D. Xiao, On the number of the limit cycles in small perturbation of a class of hyper-ellipic Hamiltonian systems with one nilponent saddle,, J. Differential Equations, 250 (2011), 2227. doi: 10.1016/j.jde.2010.11.004. Google Scholar [18] Z. Zhang, T. Ding, W. Huang and Z. Dong, "Qualitative Theory of Differential Equations,", Science Publisher, (1985). Google Scholar

show all references

##### References:
 [1] P. De, Maesschalck and F. Dumortier, Classical Liénard equations of degess $n\geq6$ can have $[\frac{n-1}{2}]+2$ limit cycles,, J. Differential Equations, 250 (2011), 2162. doi: 10.1016/j.jde.2010.12.003. Google Scholar [2] P. De, Maesschalck and F. Dumortier, Bifurcations of multiple relaxation oscillations in polynomial Liénard equations,, Proc. Amer. Math. Soc., 139 (2011), 2073. doi: 10.1090/S0002-9939-2010-10610-X. Google Scholar [3] F. Dumortier, Slow divergence integral and balanced canard solutions,, Qual. Theory Dyn. Syst., 10 (2011), 65. doi: 10.1007/s12346-011-0038-9. Google Scholar [4] 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 [5] F. Dumortier and R. Roussarie, Multiple canard cycles in generalized Liénard equations,, J. Differential Equations, 174 (2001), 1. doi: 10.1006/jdeq.2000.3947. Google Scholar [6] F. Dumortier and R. Roussarie, Bifurcation of relaxation oscillations in dimension two,, Discrete Contin. Dyn. Syst., 19 (2007), 631. doi: 10.3934/dcds.2007.19.631. Google Scholar [7] F. Dumortier and R. Roussarie, Canard cycles with two breaking parameters,, Discrete Contin. Dyn. Syst., 17 (2007), 787. doi: 10.3934/dcds.2007.17.787. Google Scholar [8] F. Dumortier and R. Roussarie, Multi-layer canard cycles and translated power functions,, J. Differential Equations, 244 (2008), 1329. doi: 10.1016/j.jde.2007.08.013. Google Scholar [9] M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities,", in: Graduate Texts in Math., (1973). Google Scholar [10] M. Han, P. Bi and D. Xiao, Bifurcation of limit cycles and separatrix loops in singular Liénard systems,, Chaos Solitons Fractals, 20 (2004), 529. doi: 10.1016/S0960-0779(03)00412-0. Google Scholar [11] C. Li and J. Llibre, Uniqueness of limit cycles for Liénard equations of degree four,, J. Differential Equations, 252 (2012), 3142. doi: 10.1016/j.jde.2011.11.002. Google Scholar [12] A. Lins, W. de Melo and C.C. Pugh, On Liénard's equations,, in, (1977), 335. Google Scholar [13] J. Llibre, A survey on the limit cycles of the generalized polynomial Liénard differential equations,, in, (2008), 224. Google Scholar [14] L. Mamouhdi and R. Roussarie, Canard cycles of finite codimension with two breaking parameters,, Qual. Theory Dyn. Syst., 11 (2012), 167. doi: 10.1007/s12346-011-0061-x. Google Scholar [15] R. Roussarie, Putting a boundary to the space of Liénard equations,, Discrete Contin. Dyn. Syst., 17 (2007), 441. doi: 10.3934/dcds.2007.17.441. Google Scholar [16] Y. Tian and M. Han, Hopf bifurcation for two types of Liénard systems,, J. Differential Equations, 251 (2011), 834. doi: 10.1016/j.jde.2011.05.029. Google Scholar [17] J. Wang and D. Xiao, On the number of the limit cycles in small perturbation of a class of hyper-ellipic Hamiltonian systems with one nilponent saddle,, J. Differential Equations, 250 (2011), 2227. doi: 10.1016/j.jde.2010.11.004. Google Scholar [18] Z. Zhang, T. Ding, W. Huang and Z. Dong, "Qualitative Theory of Differential Equations,", Science Publisher, (1985). Google Scholar
 [1] Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070 [2] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [3] Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465 [4] Armengol Gasull, Hector Giacomini. Upper bounds for the number of limit cycles of some planar polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217 [5] Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047 [6] Salomón Rebollo-Perdomo, Claudio Vidal. Bifurcation of limit cycles for a family of perturbed Kukles differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4189-4202. doi: 10.3934/dcds.2018182 [7] Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043 [8] C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289 [9] Maoan Han, Tonghua Zhang. Some bifurcation methods of finding limit cycles. Mathematical Biosciences & Engineering, 2006, 3 (1) : 67-77. doi: 10.3934/mbe.2006.3.67 [10] Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393 [11] 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 [12] Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236 [13] Maoan Han. On some properties and limit cycles of Lienard systems. Conference Publications, 2001, 2001 (Special) : 426-434. doi: 10.3934/proc.2001.2001.426 [14] Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071 [15] Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447 [16] Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557 [17] Amelia Álvarez, José-Luis Bravo, Manuel Fernández. The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1493-1501. doi: 10.3934/cpaa.2009.8.1493 [18] Michael Y. Li, Xihui Lin, Hao Wang. Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 747-760. doi: 10.3934/dcdsb.2014.19.747 [19] Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703 [20] Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627

2018 Impact Factor: 1.143

## Metrics

• HTML views (0)
• Cited by (3)

• on AIMS