具带限反馈的时滞系统的Hopf分支分析

Hopf bifurcations in time-delay systems with band-limited feedback

具反馈的非线性装置中不可避免地带有时滞,时滞和反馈控制参数的变化对系统的动力学性质会产生一定影响.研究了具带限反馈时滞系统中滞量和控制参数对稳定性和Hopf分支性质所起的作用.通过分析系统线性部分相应特征方程,发现当控制参数和滞量变化时,系统的拓扑结构会发生变化,并且当穿过一系列临界值时会发生Hopf分支.应用中心流形定理和Hassard规范型理论,得到了判断Hopf分支方向和分支周期解稳定性的计算公式.最后,给出了几个算例,其数值模拟结果与理论分析结果一致.可见,通过调整时滞和反馈参数的大小可以实现对系统动力学行为的控制.

To discuss the effects of parameter change of time-delay and feedback control on the dynamics of a nonlinear device with feedback, the stability and Hopf bifurcation analyses on the role played by delay and control parameter in a time-delay system with band-limited feedback were investigated. By analyzing the associated characteristic equations, it is found that the control parameter and delay can qualitatively change the dynamics, and the Hopf bifurcation occurs when the parameters pass through a sequence of critical values. The stability and direction of Hopf bifurcation are determined by applying the Hassard normal form theory and the center manifold theorem. Results of some numerical simulations on Hopf bifurcation agree well with the theoretical results, which means that the system control can be realized by adjusting the magnitude of time delay and feedback gain.