KUZNETSOV Y A.Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations[J].International Journal of Bifurcation and Chaos,2005,15:3535-3546.
[2]
KOKUBU K,OKA H,WANG Duo.Linear grading function and further reduction of normal form[J].Journal of Differential Equations,1996,132:293-318.
[3]
WANG Duo,LI Jing,HUANG Ming-hai,et al.Unique normal form of Bogdanov-Takens singularities[J].Journal of Differential Equations,2000,163:223-238.
[4]
LI Jing,WANG Duo,ZHANG Wei.General forms of the simplest normal form of Bogdanov-Takens singularities[J].Dynamics of Continuous,Discrete and Impulsive Systems,Series A:Mathematical Analysis,2001,8:519-530.
[5]
PENG Jian-ping,WANG Duo.A sufficient condition for the uniqueness of normal forms and unique normal forms of generalized Hopf singularities[J].International Journal of Bifurcation and Chaos,2004,14(9):3337-3345.
[6]
WANG Duo,ZHENG Ming,PENG Jian-ping.Further reduction of normal forms and unique normal forms of smooth maps[J].International Journal of Bifurcation and Chaos,2008,18(3):803-825.
[7]
BAIDER A,SANDERS J.Further reduction of the Takens-Bogdanov normal form[J].Journal of Differential Equations,1992,99:205-244.
[8]
ALGABA A,FREIRE E,GAMERO E,et al.On the Takens-Bogdanov bifurcation in the Chua's equation[J].IEICE Trans,1999,82:1722-1782.
[9]
CHEN Guo-ting,DORA J.An algorithm for computing a new normal form for dynamical systems[J].Journal of Symbolic Computation,2000,29(3):393-418.
[10]
GAO Bo,ZHANG Wei-nian.Parametric normal forms of vector fields and their further simplification[J].Nonlinearity,2010,23:2539-2557.
[11]
ZHANG Qi-chang,HE Xue-jun,HAO Shu-ying.Computation of the simplest normal forms for resonant double Hopf bifurcations system based on Lie transforms[J].Transactions of Tianjin University,2006,12(3):180-185.
[12]
PENG Guo-jun,JIANG Yao-lin.Practical computation of normal forms of the Bogdanov-Takens bifurcation[J].Nonlinear Dynamics,2011,66:99-132.