一类时滞位移反馈参数激励系统的复杂动力学行为

 考虑一个具有二次方和三次方非线性的单自由度参数激励系统，对系统引入一个主动控制即线性时滞位移反馈，定性地研究系统中时滞反馈对系统动力学行为的影响。首先运用规范型方法，给出由分岔产生的周期解的解析形式。进而解析地预测了由时滞导致的系统周期解的个数及其稳定性随时滞量的变化规律。发现时滞能够引起系统平衡点失稳，出现多吸引子共存现象。最后采用4阶Runge-Kutta法和点映射方法给出数值结果。并对多吸引子的吸引域进行了划分，给出了时滞导致的系统的概周期吸引子。数值结果与理论预测的一致性验证了理论分析结果的有效性。研究发现时滞可使系统出现复杂的动力学行为。本文结果对控制系统的镇定和系统同步有潜在的应用价值。

Complex Dynamics of a Parametrically Excited System with Delayed Position Feedbacks

A single-degree-of-freedom system with quadratic, cubic, and parametrically excited terms is considered. We introduce an active control, i.e. a linear delayed-position feedback control, to the system, and the effects of the delayed feedback on the dynamics of the system are studied qualitatively. First, the normal form method is proposed to investigate the dynamics of the system with varying delay. Then the second-order approximation of the periodic solution is obtained, and is used to predict the stability of the bifurcation branches and the variation of the number of solutions with varying time delay. It is found that time delay can make the trivial equilibrium lose its stability, and induce the multiple attractors coexisting in the system. Finally, the numerical results are obtained through the 4th-order Runge-Kutta and point-to-point mapping methods. The basins of attraction are classified. In addition, the quasi-periodic attractor time delay induces is also obtained. The agreement of the numerical and theoretical results verifies the validity of the theoretical predictions. It is found that varying the delay can induce the complex dynamical behaviors in the system. This paper provides potential applications of these findings for the study of stabilization of controlled systems and chaotic motions.