中文摘要：

对一个Helmoholtz振子系统引入线性时滞位移反馈，研究时滞位移反馈对系统的安全盆侵蚀的控制作用．将Melnikov函数法推广到时滞系统，从而给出时滞受控系统的安全盆的边界分形条件，在此基础上得出了安全盆分形的激励振幅阈值．再以时滞量为变参数，运用四阶Rung-Kutta方法和点映射方法从定量上研究了时滞对受控系统安全盆的影响规律．结果表明在弱反馈下，时滞量的增大能够提高安全盆边界分形的阈值，进而有效地抑制安全盆的分形侵蚀．说明时滞位移反馈是控制系统的安全盆侵蚀的良好方法．

英文摘要：

A linear delayed position feedback is introduced into a Helmhohz oscillator system, and its effects on controlling the erosion of safe basins are investigated. The condition of fractal erosion of the basin boundary of the delayed feedback controlled system is acquired, there by the threshold of the excitation of the fractal erosion is obtained. Then considering time delay as a variable parameter and employing the 4th-order Runge-Kutta method and the point-to-point mapping method, evolutions of safe basins with time delay are presented. It is found that the increasing of time delay can enhance the threshold of the fractal erosion and reduce the erosion of safe basin effectively under a weak feedback gain. All the results imply that delayed position feedback can be used as a good strategy to control the erosion of safe basins.