Infinite dimensions always challenge the analysis of multiple stability in nonlinear time-delayed systems, as the computation and visualization of conventional basin of attraction are hampered by the increase in systems’ dimensions. To address this issue, this paper introduces an orthonormal basis to approximate the delayed states, uses their signal energy to represent them, and generalises the concept of basin of attraction into stochastic, where each pixel of the basin has the same energy level for each delayed state but corresponds to many initial conditions. Thus, the probabilities are estimated by Monte Carlo method, which is then extensively boosted by artificial neural networks including both classification and regression types. This procedure has been successively applied in the analysis of multiple stability in three typical time-delayed systems, which are a two-dimensional autonomous cutting process, a three-dimensional autonomous neural system, and a two-dimensional non-autonomous forced vibration isolator. They, respectively, have one, two, and two delayed states, with two, three, and five coexisting attractors. It is found that the energy distribution in the delayed state determines both the convergence of Monte Carlo simulation and sensitivity of the classification neural network. It is also seen that the performance of classification neural networks decreases with respect to the increase in the number of attractors, but the regression neural networks show a robuster performance. As a result, the stochastic basin of attraction can be accurately and efficiently computed to reveal the multiple stability in various time-delayed systems.

无限尺寸总是挑战非线性时滞系统的多重稳定性，因为传统吸引池的计算和可视化受到系统尺寸增加的阻碍。为了解决这个问题，本文介绍了一个正交基来近似延迟状态，使用它们的信号能量来表示它们，并概括了吸引池为随机的概念，其中吸引池的每个像素对于每个延迟都具有相同的能级状态，但对应许多初始条件。因此，概率是通过蒙特卡洛方法进行估计的，然后通过人工神经网络（包括分类和回归类型）来大大提高概率。该程序已成功应用于三个典型时滞系统的多重稳定性分析，这三个系统是二维自主切割过程，三维自主神经系统和二维非自主强制隔振器。它们分别具有一个，两个和两个延迟状态，以及两个，三个和五个共存吸引子。发现延迟状态下的能量分布决定了蒙特卡洛模拟的收敛性和分类神经网络的灵敏度。还可以看出，分类神经网络的性能随吸引子数量的增加而降低，但是回归神经网络的性能更强。结果是，