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Global Phase Portraits of Symmetrical CubicHamiltonian Systems with a Nilpotent SingularPoint
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KeyWord:Hamiltonian systems  nilpotent singular point  global phase por- traits  Poincar´ e transformation
Author NameAffiliation
Huiyang Zhang School of Mathematics and Computing Science, Guilin University of Elec- tronic Technology, Guilin, Guangxi 541004, China 
Aiyong Chen Department of Mathematics, Hunan First Normal University, Changsha, Hu- nan 410205, China 
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      Han et al. [Han et al., Polynomial Hamiltonian systems with a nilpotent critical point, J. Adv. Space Res. 2010, 46, 521–525] successfully studied local behavior of an isolated nilpotent critical point for polynomial Hamiltonian systems. In this paper, we extend the previous result by analyzing the global phase portraits of polynomial Hamiltonian systems. We provide 12 non-topological equivalent classes of global phase portraits in the Poincar´ e disk of cubic polynomial Hamiltonian systems with a nilpotent center or saddle at origin under some conditions of symmetry.