# 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, 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), 2162-2176. 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), 2073-2085. 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), 65-85. 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), 1895-1904. 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), 1-29. 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), 631-674. 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), 787-806. 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), 1329-1358. 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., vol.14, Springer-Verlag, New York, 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), 529-546. 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), 3142-3162. 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 "Geometry and Topology, Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976, Lecture Notes in Math., vol. 597", Springer, Berlin, (1977), 335-357.  Google Scholar [13] J. Llibre, A survey on the limit cycles of the generalized polynomial Liénard differential equations, in "Proceedings of Mathematical models in Engineering, Biology and Medicine Conference on Boundary Value Problems, (2008), 224-233.  Google Scholar [14] L. Mamouhdi and R. Roussarie, Canard cycles of finite codimension with two breaking parameters, Qual. Theory Dyn. Syst., 11 (2012), 167-198. 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), 441-448. 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), 834-859. 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), 2227-2243. 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 (in chinese); Transl. Math. Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 1994.  Google Scholar

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##### 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), 2162-2176. 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), 2073-2085. 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), 65-85. 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), 1895-1904. 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), 1-29. 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), 631-674. 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), 787-806. 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), 1329-1358. 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., vol.14, Springer-Verlag, New York, 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), 529-546. 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), 3142-3162. 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 "Geometry and Topology, Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976, Lecture Notes in Math., vol. 597", Springer, Berlin, (1977), 335-357.  Google Scholar [13] J. Llibre, A survey on the limit cycles of the generalized polynomial Liénard differential equations, in "Proceedings of Mathematical models in Engineering, Biology and Medicine Conference on Boundary Value Problems, (2008), 224-233.  Google Scholar [14] L. Mamouhdi and R. Roussarie, Canard cycles of finite codimension with two breaking parameters, Qual. Theory Dyn. Syst., 11 (2012), 167-198. 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), 441-448. 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), 834-859. 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), 2227-2243. 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 (in chinese); Transl. Math. Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 1994.  Google Scholar
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