The framework Topology of Sustainable Management by Heitzig et al. (Earth Syst Dyn 7:21. https://doi.org/10.5194/esd-7-21-2016, 2016) distinguishes qualitatively different regions in state space of dynamical models representing manageable systems with default dynamics. In this paper, we connect the framework to viability theory by defining its main components based on viability kernels and capture basins. This enables us to use the Saint-Pierre algorithm to visualize the shape and calculate the volume of the main partition of the Topology of Sustainable Management. We present an extension of the algorithm to compute implicitly defined capture basins. To demonstrate the applicability of our approach, we introduce a low-complexity model coupling environmental and socioeconomic dynamics. With this example, we also address two common estimation problems: an unbounded state space and highly varying time scales. We show that appropriate coordinate transformations can solve these problems. It is thus demonstrated how algorithmic approaches from viability theory can be used to get a better understanding of the state space of manageable dynamical systems.

Heitzig 等人的可持续管理框架拓扑。（Earth Syst Dyn 7:21。https://doi.org/10.5194/esd-7-21-2016, 2016）在代表具有默认动态的可管理系统的动态模型的状态空间中，在质量上区分了不同的区域。在本文中，我们通过基于生存力内核和捕获池定义其主要组成部分，将框架与生存力理论联系起来。这使我们能够使用圣皮埃尔算法来可视化形状并计算可持续管理拓扑的主分区的体积. 我们提出了算法的扩展来计算隐式定义的捕获盆地。为了证明我们方法的适用性，我们引入了一个耦合环境和社会经济动态的低复杂性模型。在这个例子中，我们还解决了两个常见的估计问题：无界状态空间和高度变化的时间尺度。我们表明适当的坐标变换可以解决这些问题。因此，演示了如何使用可行性理论中的算法方法来更好地理解可管理动态系统的状态空间。