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Revealing compactness of basins of attraction of multi-DoF dynamical systems

机译：揭示多自由度动力系统吸引盆的紧凑性

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摘要

Global properties of Multi-Degrees-of-Freedom (M-DoF) systems, in particular phase space organization, are largely unexplored due to the computational challenge requested to build basins of attraction. To overcome this problem, various techniques have been developed, some trying to improve algorithms and to exploit high speed computing, others giving up to possibility of having the exact phase space organization and trying to extract major information on a probability base. Following the last approach, this work exploits the method of “basin stability” (Menck et al., 2013) in order to drastically reduce the numerical effort. The probability of reaching the attractors is evaluated using a reasonable number of trials with random initial conditions. Then we investigate how this probability depends on particular generalized coordinate or a pair of coordinates. The method allows to obtain information about the basins compactness and reveals particular features of the phase space topology. We focus the study on a 2-DoF multistable paradigmatic system represented by a parametric pendulum on a moving support and model of a Church Bell. The trustworthiness of the proposed approach is enhanced through the comparison with the classical computation of basins of attraction performed in the full range of initial conditions. The proposed approach can be effectively utilized to investigate the phase space in multidimensional nonlinear dynamical systems by providing additional insights over traditional methods.
机译：由于建立吸引盆地所需的计算挑战，多自由度（M-DoF）系统的全局属性（尤其是相空间组织）在很大程度上尚未得到开发。为了克服这个问题，已经开发了各种技术，一些试图改进算法并利用高速计算，而另一些则放弃了具有精确的相空间组织的可能性并试图以概率为基础提取主要信息。按照最后一种方法，这项工作采用了“盆地稳定性”方法（Menck等，2013），以大大减少数值工作量。使用合理数量的随机初始条件试验评估达到吸引子的概率。然后，我们研究此概率如何取决于特定的广义坐标或一对坐标。该方法可以获取有关盆地密实度的信息，并揭示相空间拓扑的特定特征。我们将研究重点放在以教堂钟的移动支撑和模型为代表的参数摆所表示的2自由度多稳态范式系统上。通过与在整个初始条件范围内执行的吸引盆地的经典计算进行比较，提高了所提出方法的可信赖性。通过提供对传统方法的更多了解，可以有效地利用所提出的方法来研究多维非线性动力学系统中的相空间。

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