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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities," in: Graduate Texts in Math., vol.14, Springer-Verlag, New York, 1973.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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), 2162-2176. doi: 10.1016/j.jde.2010.12.003.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

M. Golubitsky and V. Guillemin, "Stable Mappings and Their Singularities," in: Graduate Texts in Math., vol.14, Springer-Verlag, New York, 1973.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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