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doi: 10.3934/dcdss.2022033
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## Bifurcation of limit cycles from a Li$\acute{E}$nard system with asymmetric figure eight-loop case

 Department of Mathematics, Jiujiang University, Jiujiang 332005, China

*Corresponding author: Hong Li

Dedicated to the 80th birthday of Professor Jibin Li

Received  December 2021 Early access February 2022

In this paper, we consider the number of limit cycles of the Li$\acute{e}$nard system of the form $\dot{x} = y, \ \dot{y} = -x(x^2+bx-1)+\varepsilon f_{m}(x)y$, where $b>0$, $f_{m}(x) = \sum_{i = 0}^m a_{i}x^{i}$ is a polynomial of $x$ with degree not greater than $m$ and $0<\varepsilon \ll 1$. By studying the number of isolated zeros of the corresponding Abelian integral $I(h) = \oint_{L_{h}}f_{m}(x)ydx,$ we obtain the upper bound of the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for $\varepsilon = 0$.

Citation: Hong Li. Bifurcation of limit cycles from a Li$\acute{E}$nard system with asymmetric figure eight-loop case. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022033
##### References:
 [1] V. I. Arnol'd, Loss of stability of self-induced oscillations near resonance, and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11 (1977), 1-10. [2] T. R. Blows and N. G. Lloyd, The number of small-amplitude limit cycles of li$\acute{e}$nard equations, Math. Proc. Camb. Phil. Soc., 95 (1984), 359-366.  doi: 10.1017/S0305004100061636. [3] R. Bogdanov, Versal deformation of a singularity of a vector feld on the plane in the case of zero eigenvalues, Sel. Math. Sov., 1 (1981), 389-421. [4] C. Christopher and S. Lynch, Small-amplitude limit cycle bifurcations for li$\acute{e}$nard systems with quadratic or cubic damping or restoring forces, Nonlinearity, 12 (1999), 1099-1112.  doi: 10.1088/0951-7715/12/4/321. [5] F. Dumortier and C. Li, Perturbation from an elliptic Hamiltonian of degree four-IV Figure eight-loop, J. Differ. Equ., 188 (2003), 512-554.  doi: 10.1016/S0022-0396(02)00111-0. [6] E. Horozov and I. Iliev, Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians, Nonlinearity, 11 (1998), 1521-1537.  doi: 10.1088/0951-7715/11/6/006. [7] C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z. [8] J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurcat. Chaos, 13 (2003), 47-106.  doi: 10.1142/S0218127403006352. [9] W. Li, Y. Zhao, C. Li and Z. Zhang, Abelian integrals for quadratic centers having almost all their orbits formed by quartics, Nonlinearity, 15 (2002), 863-885.  doi: 10.1088/0951-7715/15/3/321. [10] J. Llibre, A. C. Mereu and M. A. Teixeira, Limit cycles of the generalized polynomial li$\acute{e}$nard differential equations, Math. Proc. Camb. Phil. Soc., 148 (2010), 363-383.  doi: 10.1017/S0305004109990193. [11] J. Yang, M. Han and V. G. Romanovski, Limit cycle bifurcations of some li$\acute{e}$nard systems, J. Math. Anal. Appl., 366 (2010), 242-255.  doi: 10.1016/j.jmaa.2009.12.035. [12] Y. Zhao and Z. Zhang, Linear estimate of the number of zeros of abelian integrals for a kind of quartic Hamiltonians, J. Differ. Equ., 155 (1999), 73-88.  doi: 10.1006/jdeq.1998.3581.

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##### References:
 [1] V. I. Arnol'd, Loss of stability of self-induced oscillations near resonance, and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11 (1977), 1-10. [2] T. R. Blows and N. G. Lloyd, The number of small-amplitude limit cycles of li$\acute{e}$nard equations, Math. Proc. Camb. Phil. Soc., 95 (1984), 359-366.  doi: 10.1017/S0305004100061636. [3] R. Bogdanov, Versal deformation of a singularity of a vector feld on the plane in the case of zero eigenvalues, Sel. Math. Sov., 1 (1981), 389-421. [4] C. Christopher and S. Lynch, Small-amplitude limit cycle bifurcations for li$\acute{e}$nard systems with quadratic or cubic damping or restoring forces, Nonlinearity, 12 (1999), 1099-1112.  doi: 10.1088/0951-7715/12/4/321. [5] F. Dumortier and C. Li, Perturbation from an elliptic Hamiltonian of degree four-IV Figure eight-loop, J. Differ. Equ., 188 (2003), 512-554.  doi: 10.1016/S0022-0396(02)00111-0. [6] E. Horozov and I. Iliev, Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians, Nonlinearity, 11 (1998), 1521-1537.  doi: 10.1088/0951-7715/11/6/006. [7] C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z. [8] J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurcat. Chaos, 13 (2003), 47-106.  doi: 10.1142/S0218127403006352. [9] W. Li, Y. Zhao, C. Li and Z. Zhang, Abelian integrals for quadratic centers having almost all their orbits formed by quartics, Nonlinearity, 15 (2002), 863-885.  doi: 10.1088/0951-7715/15/3/321. [10] J. Llibre, A. C. Mereu and M. A. Teixeira, Limit cycles of the generalized polynomial li$\acute{e}$nard differential equations, Math. Proc. Camb. Phil. Soc., 148 (2010), 363-383.  doi: 10.1017/S0305004109990193. [11] J. Yang, M. Han and V. G. Romanovski, Limit cycle bifurcations of some li$\acute{e}$nard systems, J. Math. Anal. Appl., 366 (2010), 242-255.  doi: 10.1016/j.jmaa.2009.12.035. [12] Y. Zhao and Z. Zhang, Linear estimate of the number of zeros of abelian integrals for a kind of quartic Hamiltonians, J. Differ. Equ., 155 (1999), 73-88.  doi: 10.1006/jdeq.1998.3581.
Phase portraits of Eq. (3) on the $(x, y)$ plane
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