一类非线性自治Liu系统的Hopf分岔分析

Hopf bifurcation analysis in a nonlinear autonomic Liu system

针对新提出的三维自治Liu系统进行研究,求得该系统的平衡点,并分析平衡点的稳定性.对平衡点进行了Hopf分岔分析,得出Hopf分岔的参数条件.通过对系统的第一李雅普诺夫系数的分析,推导出系统发生超临界、亚临界以及余维二退化Hopf分岔的参数条件.对Liu系统进行数值仿真,验证了理论推导的正确性.

A three-dimensional autonomous Liu system proposed recently is studied. The equilibrium of the system is obtained through the simple calculation, and then its stability is analyzed. Through the analysis of Hopf bifurcation of the equilibrium, the parameter condition of Hopf bifurcation is derived. Parameter conditions for the supercritical, the subcritical and the codimension two degenerating Hopf bifurcation are presented by analyzing Lyapunov＇ s first coefficient of the system. Numerical simulation is given to illustrate the theoretical analysis with the aid of the computer. It is shown that the numerical results agree quite well with our theoretical analysis.