This paper proposes a novel concept of the statistical basin of attraction to analyse the multiple stability in nonlinear time-delayed dynamical systems and shows how they can be computed. This concept has been applied to the cutting dynamics, which has been extensively investigated by the authors. Due to the nonlinearity and non-smoothness of tool-workpiece interactions, the cutting dynamics always exhibit large-amplitude chatter entering a linearly stable zone, making the area below stability boundaries unsafe for high material removal rates. Meanwhile, a thorough investigation of the multiple stability in the cutting dynamics is hampered by infinite-many dimensions introduced by time delays, which induce difficulties in computation and visualization of the conventional basin of attraction. To address this issue, infinite-many dimensional time-delayed states are approximated by a Fourier series aligned on a straight line, and the coefficients of the basis functions and the cutting process are used to construct the statistical basin of attraction. Inside the statistical basin of attraction, a safe basin with no probability of chatter occurrence exists. These findings are instrumental in designing a new state-dependent intermittent control to guide the cutting dynamics towards the safe basins. It is also seen that the state-dependent intermittent control is efficient in improving the cutting safety and shrinking the unsafe zones, even when the targeted basin for the control is larger than the real safe basin.

本文提出了一个统计吸引域的新概念，以分析非线性时滞动力系统的多重稳定性，并说明如何计算它们。该概念已应用于切割动力学，作者对此进行了广泛研究。由于工具-工件相互作用的非线性和非光滑性，切削动力学总是表现出进入线性稳定区域的大振幅震颤，使得稳定边界以下的区域对于高材料去除率而言是不安全的。同时，对切削动力学多重稳定性的彻底研究受到时间延迟引入的无数个维度的阻碍，这在传统吸引盆的计算和可视化方面造成了困难。为了解决这个问题，通过沿直线排列的傅立叶级数近似无数维的时滞状态，并使用基函数的系数和切割过程来构造统计吸引力。在统计吸引盆地内，存在一个没有发生of振可能性的安全盆地。这些发现有助于设计一种新的，取决于状态的间歇控制，以指导切割动态走向安全盆地。还可以看到，即使控制的目标盆大于实际的安全盆，基于状态的间歇控制也能有效提高切削安全性并缩小不安全区域。利用基函数系数和截割过程来构建统计吸引度。在统计吸引盆地内，存在一个没有发生of振可能性的安全盆地。这些发现有助于设计一种新的，取决于状态的间歇控制，以指导切割动态走向安全盆地。还可以看到，即使控制的目标盆大于实际的安全盆，基于状态的间歇控制也能有效提高切割安全性并缩小不安全区域。利用基函数系数和截割过程来构建统计吸引度。在统计吸引盆地内，存在一个没有发生of振可能性的安全盆地。这些发现有助于设计一种新的，取决于状态的间歇控制，以指导切割动态走向安全盆地。还可以看到，即使控制的目标盆大于实际的安全盆，基于状态的间歇控制也能有效提高切削安全性并缩小不安全区域。这些发现有助于设计一种新的，取决于状态的间歇控制，以指导切割动态走向安全盆地。还可以看到，即使控制的目标盆大于实际的安全盆，基于状态的间歇控制也能有效提高切削安全性并缩小不安全区域。这些发现有助于设计一种新的，取决于状态的间歇控制，以指导切割动态走向安全盆地。还可以看到，即使控制的目标盆大于实际的安全盆，基于状态的间歇控制也能有效提高切削安全性并缩小不安全区域。