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]

J. Differential Equations, 250 (2011), 2162-2176. doi: 10.1016/j.jde.2010.12.003.  Google Scholar

[2]

Proc. Amer. Math. Soc., 139 (2011), 2073-2085. doi: 10.1090/S0002-9939-2010-10610-X.  Google Scholar

[3]

Qual. Theory Dyn. Syst., 10 (2011), 65-85. doi: 10.1007/s12346-011-0038-9.  Google Scholar

[4]

Proc. Amer. Math. Soc., 135 (2007), 1895-1904. doi: 10.1090/S0002-9939-07-08688-1.  Google Scholar

[5]

J. Differential Equations, 174 (2001), 1-29. doi: 10.1006/jdeq.2000.3947.  Google Scholar

[6]

Discrete Contin. Dyn. Syst., 19 (2007), 631-674. doi: 10.3934/dcds.2007.19.631.  Google Scholar

[7]

Discrete Contin. Dyn. Syst., 17 (2007), 787-806. doi: 10.3934/dcds.2007.17.787.  Google Scholar

[8]

J. Differential Equations, 244 (2008), 1329-1358. doi: 10.1016/j.jde.2007.08.013.  Google Scholar

[9]

in: Graduate Texts in Math., vol.14, Springer-Verlag, New York, 1973.  Google Scholar

[10]

Chaos Solitons Fractals , 20 (2004), 529-546. doi: 10.1016/S0960-0779(03)00412-0.  Google Scholar

[11]

J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.  Google Scholar

[12]

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]

in "Proceedings of Mathematical models in Engineering, Biology and Medicine Conference on Boundary Value Problems, (2008), 224-233.  Google Scholar

[14]

Qual. Theory Dyn. Syst., 11 (2012), 167-198. doi: 10.1007/s12346-011-0061-x.  Google Scholar

[15]

Discrete Contin. Dyn. Syst., 17 (2007), 441-448. doi: 10.3934/dcds.2007.17.441.  Google Scholar

[16]

J. Differential Equations , 251 (2011), 834-859. doi: 10.1016/j.jde.2011.05.029.  Google Scholar

[17]

J. Differential Equations , 250 (2011), 2227-2243. doi: 10.1016/j.jde.2010.11.004.  Google Scholar

[18]

Science Publisher, 1985 (in chinese); Transl. Math. Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 1994.  Google Scholar

show all references

References:
[1]

J. Differential Equations, 250 (2011), 2162-2176. doi: 10.1016/j.jde.2010.12.003.  Google Scholar

[2]

Proc. Amer. Math. Soc., 139 (2011), 2073-2085. doi: 10.1090/S0002-9939-2010-10610-X.  Google Scholar

[3]

Qual. Theory Dyn. Syst., 10 (2011), 65-85. doi: 10.1007/s12346-011-0038-9.  Google Scholar

[4]

Proc. Amer. Math. Soc., 135 (2007), 1895-1904. doi: 10.1090/S0002-9939-07-08688-1.  Google Scholar

[5]

J. Differential Equations, 174 (2001), 1-29. doi: 10.1006/jdeq.2000.3947.  Google Scholar

[6]

Discrete Contin. Dyn. Syst., 19 (2007), 631-674. doi: 10.3934/dcds.2007.19.631.  Google Scholar

[7]

Discrete Contin. Dyn. Syst., 17 (2007), 787-806. doi: 10.3934/dcds.2007.17.787.  Google Scholar

[8]

J. Differential Equations, 244 (2008), 1329-1358. doi: 10.1016/j.jde.2007.08.013.  Google Scholar

[9]

in: Graduate Texts in Math., vol.14, Springer-Verlag, New York, 1973.  Google Scholar

[10]

Chaos Solitons Fractals , 20 (2004), 529-546. doi: 10.1016/S0960-0779(03)00412-0.  Google Scholar

[11]

J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.  Google Scholar

[12]

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]

in "Proceedings of Mathematical models in Engineering, Biology and Medicine Conference on Boundary Value Problems, (2008), 224-233.  Google Scholar

[14]

Qual. Theory Dyn. Syst., 11 (2012), 167-198. doi: 10.1007/s12346-011-0061-x.  Google Scholar

[15]

Discrete Contin. Dyn. Syst., 17 (2007), 441-448. doi: 10.3934/dcds.2007.17.441.  Google Scholar

[16]

J. Differential Equations , 251 (2011), 834-859. doi: 10.1016/j.jde.2011.05.029.  Google Scholar

[17]

J. Differential Equations , 250 (2011), 2227-2243. doi: 10.1016/j.jde.2010.11.004.  Google Scholar

[18]

Science Publisher, 1985 (in chinese); Transl. Math. Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 1994.  Google Scholar

[1]

Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1077-1089. doi: 10.3934/cpaa.2021007

[2]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208

[3]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[4]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[5]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[6]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365

[7]

Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294

[8]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021015

[9]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[10]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011

[11]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003

[12]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026

[13]

Raz Kupferman, Cy Maor. The emergence of torsion in the continuum limit of distributed edge-dislocations. Journal of Geometric Mechanics, 2015, 7 (3) : 361-387. doi: 10.3934/jgm.2015.7.361

[14]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021040

[15]

Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037

[16]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029

[17]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[18]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190

[19]

Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021048

[20]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]