Bifurcation of Limit Cycles for Some Liénard Systems with a Nilpotent Singular Point

In this paper, we first present some general theorems on bifurcation of limit cycles in near-Hamiltonian systems with a nilpotent saddle or a nilpotent cusp. Then we apply the theorems to study the number of limit cycles for some polynomial Lienard systems with a nilpotent saddle or a nilpotent cusp, and obtain some new estimations on the number of limit cycles of these systems.

[1]  Maoan Han,et al.  On Hopf Cyclicity of Planar Systems , 2000 .

[2]  Yulin Zhao,et al.  Abelian integrals for cubic vector fields , 1999 .

[3]  Valery G. Romanovski,et al.  Limit cycle bifurcations of some Liénard systems , 2010 .

[4]  Maoan Han,et al.  Melnikov function and limit cycle bifurcation from a nilpotent center , 2008 .

[5]  Huaiping Zhu,et al.  Limit Cycle bifurcations in Near-Hamiltonian Systems by Perturbing a Nilpotent Center , 2008, Int. J. Bifurc. Chaos.

[6]  V. Romanovski,et al.  On the number of limit cycles of polynomial Liénard systems , 2011, 1109.6470.

[7]  Robert Roussarie,et al.  On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields , 1986 .

[8]  Lubomir Gavrilov,et al.  Petrov modules and zeros of Abelian integrals , 1998 .

[9]  Global wellposedness and limit behavior for the generalized finite-depth-fluid equation with small data in critical Besov spaces B˙2,1s , 2008 .

[10]  Dongmei Xiao,et al.  Bifurcations on a Five-Parameter Family of Planar Vector Field , 2008 .

[11]  Junmin Yang,et al.  Computation of expansion coefficients of Melnikov functions near a nilpotent center , 2012, Comput. Math. Appl..

[12]  N. G. Lloyd,et al.  The number of small-amplitude limit cycles of Liénard equations , 1984, Mathematical Proceedings of the Cambridge Philosophical Society.

[13]  Abraham C.-L. Chian,et al.  Polynomial Hamiltonian systems with a nilpotent critical point , 2010 .

[14]  Pei Yu,et al.  Limit cycles in generalized Liénard systems , 2006 .

[15]  Pei Yu,et al.  Hopf bifurcations for Near-Hamiltonian Systems , 2009, Int. J. Bifurc. Chaos.

[16]  Stephen Lynch,et al.  Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces , 1999 .

[17]  Maoan Han,et al.  On Melnikov functions of a homoclinic loop through a nilpotent saddle for planar near-Hamiltonian systems , 2008 .

[18]  Noel G. Lloyd,et al.  Symmetry in Planar Dynamical Systems , 2002, J. Symb. Comput..

[19]  Maoan Han,et al.  Limit cycle bifurcations of some Liénard systems with a cuspidal loop and a homoclinic loop , 2011 .

[20]  Freddy Dumortier,et al.  Perturbations from an Elliptic Hamiltonian of Degree Four , 2001 .

[21]  Chengzhi Li,et al.  Perturbation from an elliptic Hamiltonian of degree four—IV figure eight-loop , 2003 .

[22]  Maoan Han,et al.  Limit Cycles Near Homoclinic and Heteroclinic Loops , 2008 .