中文摘要：

非线性振动系统中由于复杂动力学行为而导致安全盆边界出现分形的情况比较普遍。研究随机激励对软弹簧杜芬振子的安全盆的影响。通过对系统安全盆的描绘，利用龙格-库塔和蒙特-卡罗方法，给出了系统有正阻尼和无阻尼时高斯白噪声和有界噪声作用下安全盆的侵蚀状况。结果表明，类似于确定的谐和激励情形，安全盆的面积将随着随机外激励强度的增大而减少，无阻尼系统的安全盆的边界经高斯白噪声或有界噪声外激励的作用同样可出现分形形状。

英文摘要：

It is common for many dynamical systems to have complex incursive fractal basin boundaries under external excitations. The purpose of this paper is to discuss the effect of noises on erosion of safe basin in the softening Duffing oscillator. By introducing the portrait of the system＇s safe basin briefly and employing the Runge-Kutta and the Monte-Carlo methods, the eroded safe basins are presented when the softening Duffing oscillator with or without damping item is excited by the Gaussian white or the bounded noise. It is shown that, similar to the case of determinate harmonic excitation, the area of the safe basin decreases with the increase of the strength of the stochastic excitation. The boundary of the safe basin also becomes incursive fractal when the damping item is absent and the system is excited only by the Gaussian white or the bounded noise.