American Institute of Mathematical Sciences

Advanced
September  2009, 8(5): 1493-1501. doi: 10.3934/cpaa.2009.8.1493

The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign

Amelia Álvarez 1, , José-Luis Bravo 1, and  Manuel Fernández 1,

1. 

Departamento de Matemáticas, Universidad de Extremadura, Badajoz, 06071, Spain, Spain, Spain

Received  June 2008 Revised  February 2009 Published  April 2009

We study the number of limit cycles (isolated periodic solutions in the set of all periodic solutions) for the generalized Abel equation $x'=a(t)x^{n_a}+b(t)x^{n_b}+c(t)x^{n_c}+d(t)x$, where $n_a > n_b > n_c > 1$, $a(t),b(t),c(t), d(t)$ are $2\pi$-periodic continuous functions, and two of $a(t),b(t),c(t)$ have definite sign.
    We obtain examples with at least seven limit cycles, and some sufficient conditions for the equation to have at most one or at most two positive limit cycles.
Keywords: limit cycle., Abel equation, periodic solution.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34A34, 37C27, 37G1.
Citation: 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
[1]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[2]

T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277
[3]

José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873
[4]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[5]

Josep M. Olm, Xavier Ros-Oton. Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1603-1614. doi: 10.3934/dcds.2013.33.1603
[6]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439
[7]

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
[8]

Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure & Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779
[9]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1
[10]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123
[11]

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
[12]

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081
[13]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893
[14]

Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic & Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57
[15]

Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41
[16]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[17]

Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268
[18]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003
[19]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[20]

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

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences