非自治时滞反馈控制系统的周期解分岔和混沌

研究时滞反馈控制对具有周期外激励非线性系统复杂性的影响机理，研究对应的线性平衡态失稳的临界边界，将时滞非线性控制方程化为泛函微分方程，给出由Hopf分岔产生的周期解的解析形式．通过分析周期解的稳定性得到周期解的失稳区域，使用数值分析观察到时滞在该区域可以导致系统出现倍周期运动、锁相运动、概周期运动和混沌运动以及两条通向混沌的道路：倍周期分岔和环面破裂．其结果表明，时滞在控制系统中可以作为控制和产生系统的复杂运动的控制“开关”．

BIFURCATIONS AND CHAOS DUE TO TIME DELAY IN A DELAYED CONTROL NON-AUTONOMOUS SYSTEM

The mechanism for the action of delayed feedback control in a nonlinear system with external periodic forcing is investigated in this paper. The system under consideration is a nonlinear controlled system with delayed position feedbacks and external periodic forcing. The time delay is chosen as a control parameter (or bifurcation parameter) in order to observe effect of time delay on the system. The critical stability conditions for a static equilibrium of the linearized system are studied. Moreover, functional analysis is used to change the delayed system into a functional differential equation (FDE) to obtain the periodic solution from Hopf bifurcation analytically in a closed form. The good agreements with comparing the analytical solution with the numerical solution demonstrate the validity and accuracy of the method provided in the present paper. The stability of the periodic solution is also analyzed to show the stable and unstable region in time delay. The numerical simulation is employed to observe effect of time delay on the dynamics of the system such as the stability and bifurcation of an equilibrium point, periodic solution, period-doubling, phase-locked, quasi-periodic motion and even chaos. Two routes to chaos are represented, namely period-doubling bifurcation and torus breaking. This suggests that as a control parameter, time delay may be used as a simple but efficient "switch" to control motions of a system: either from order motion to chaos or from chaotic motion to order for different applications.