American Institute of Mathematical Sciences

Advanced
May  2010, 9(3): 583-610. doi: 10.3934/cpaa.2010.9.583

Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes

Iliya D. Iliev 1, , Chengzhi Li 2, and  Jiang Yu 3,

1. 

Institute of Mathematics, Bulgarian Academy of Sciences, Bl. 8, 1113 Sofia, Bulgaria
2. 

LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
3. 

Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, China

Received  May 2009 Revised  October 2009 Published  January 2010

We study the bifurcations of limit cycles in a class of planar reversible quadratic systems whose critical points are a center, a saddle and two nodes, under small quadratic perturbations. By using the properties of related complete elliptic integrals and the geometry of some planar curves defined by them, we prove that at most two limit cycles bifurcate from the period annulus around the center. This bound is exact.
Keywords: period annulus, Bifurcation of limit cycles, elliptic phase curves., small perturbations, reversible quadratic system.
Mathematics Subject Classification: Primary: 34C07, 34C08; Secondary: 37G1.
Citation: Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583
[1]

Linping Peng, Yazhi Lei. The cyclicity of the period annulus of a quadratic reversible system with a hemicycle. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 873-890. doi: 10.3934/dcds.2011.30.873
[2]

Yi Shao, Yulin Zhao. The cyclicity of the period annulus of a class of quadratic reversible system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1269-1283. doi: 10.3934/cpaa.2012.11.1269
[3]

Ricardo M. Martins, Otávio M. L. Gomide. Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142
[4]

Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807
[5]

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
[6]

Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073
[7]

Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846
[8]

B. Coll, Chengzhi Li, Rafel Prohens. Quadratic perturbations of a class of quadratic reversible systems with two centers. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 699-729. doi: 10.3934/dcds.2009.24.699
[9]

Haihua Liang, Yulin Zhao. Quadratic perturbations of a class of quadratic reversible systems with one center. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 325-335. doi: 10.3934/dcds.2010.27.325
[10]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795
[11]

G. Chen, C. Li, C. Liu, Jaume Llibre. The cyclicity of period annuli of some classes of reversible quadratic systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 157-177. doi: 10.3934/dcds.2006.16.157
[12]

Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091
[13]

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
[14]

José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020
[15]

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
[16]

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
[17]

J. C. Artés, Jaume Llibre, J. C. Medrado. Nonexistence of limit cycles for a class of structurally stable quadratic vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 259-270. doi: 10.3934/dcds.2007.17.259
[18]

Yunming Zhou, Desheng Shang, Tonghua Zhang. Seventeen limit cycles bifurcations of a fifth system. Conference Publications, 2007, 2007 (Special) : 1070-1081. doi: 10.3934/proc.2007.2007.1070
[19]

Maoan Han, Yuhai Wu, Ping Bi. A new cubic system having eleven limit cycles. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 675-686. doi: 10.3934/dcds.2005.12.675
[20]

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

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences