时滞Duffing方程的多周期解

研究了时滞线性位移反馈对一类单自由度非线性的自激振动系统动力学行为的影响规律。所考虑的数学模型为时滞Duffing方程,是由原Van der Pol-Duffing振子系统加入线性时滞位置反馈而得到。定性地研究时滞和反馈增益联合作用对Van der Pol-Duffing系统周期解的影响规律,发现时滞可使该系统出现多个周期解共存的现象。通过本文构造的解析方法,从理论上预测了由时滞导致的系统周期解个数及其稳定性随着时滞反馈增益和时滞量的变化规律,得到了不同周期解的频率和振幅。从数值上采用Runge-Kutta法,验证了理论分析结果的有效性,并划分不同周期解所对应的吸引域。结果对进一步研究镇定系统和混沌运动机理有着潜在的应用价值。

Multiple Periodic Solutions in Delayed Duffing Equation

The main purpose of this paper is to investigate the effects of time delayed position feedback on the dynamics of a sort of single degree-of-freedom self exciting vibration system.The original mathematical model under consideration was a Van der Pol-Duffing oscillator.A delayed system was obtained by adding linear time delayed position feedback to the original system.The effects of combined action of time delay and feedback-gain on the periodic solutions of Van der Pol-Duffing system was studied qualitatively.It was found that there coexisted multiple solutions derived from the delayed feedback.A method was proposed analytically to predict the number and the stability of periodic solutions with the feedback-gain and the delay varying,and the frequency and amplitude of the distinct periodic solutions are obtained.The analytical results were in agreement with the numerical ones from Runge-Kutta approach,which verified the validity of the analytical results.And the attraction domain of different periodic solutions was also classified by Runge-Kutta approach.The paper provided some potential applications for the study of stabilization of controlled systems and chaotic motions.