Lü系统的Hopf分岔反控制研究

Anti-control of Hopf Bifurcation for Lü System

分析了Lü系统平衡点的非线性动力学性质,根据Hopf分岔产生的条件,设计控制器,使原系统不稳定的零平衡点产生极限环.对原系统的非零平衡点,该控制器也使其在一个更大的参数区域,在所期望的位置产生Hopf分岔.基于中心流形定理和规范型理论求得的稳定性指标保证了分岔解的稳定性.因此,该控制器成功地实现了Lü系统平衡点的Hopf分岔反控制,并且原系统的平衡点并未改变.最后,通过数值模拟来验证理论分析的结果.

The nonlinear dynamic property of the equilibrium of Lü system is studied. According to the conditions of Hopf bifurcation, a certain limit cycle is created from the zero steady state by appropriate control. For nonzero steady state, the controlled system can exhibit Hopf bifurcation in a much larger parameter region at the desired location. Based on the center manifold theory and normal form reduction, the stability index of bifurcation solution is given. The anti-control strategy used keeps the equilibrium structure of the system. Finally, numerical simulation results are presented to illustrate analytical results found.