Tipping elements occur in various systems such as in socio-economics, ecology and the climate system. In many cases, the individual tipping elements are not independent of each other, but they interact across scales in time and space. To model systems of interacting tipping elements, we here introduce the PyCascades open source software package for studying interacting tipping elements (https://doi.org/10.5281/zenodo.4153102). PyCascades is an object-oriented and easily extendable package written in the programming language Python. It allows for investigating under which conditions potentially dangerous cascades can emerge between interacting dynamical systems, with a focus on tipping elements. With PyCascades it is possible to use different types of tipping elements such as double-fold and Hopf types and interactions between them. PyCascades can be applied to arbitrary complex network structures and has recently been extended to stochastic dynamical systems. This paper provides an overview of the functionality of PyCascades by introducing the basic concepts and the methodology behind it. In the end, three examples are discussed, showing three different applications of the software package. First, the moisture recycling network of the Amazon rainforest is investigated. Second, a model of interacting Earth system tipping elements is discussed. And third, the PyCascades modelling framework is applied to a global trade network.

小费因素出现在各种系统中，例如社会经济、生态和气候系统。在许多情况下，各个倾翻元素并不是相互独立的，而是在时间和空间上跨尺度相互作用。为了对交互小费元素系统建模，我们在此介绍了用于研究交互小费元素的PyCascades开源软件包 (https://doi.org/10.5281/zenodo.4153102)。PyCascades是一个用 Python 编程语言编写的面向对象且易于扩展的包。它允许调查在相互作用的动力系统之间可能出现潜在危险级联的条件下，重点是倾斜元素。使用PyCascades可以使用不同类型的倾斜元素，例如双折和 Hopf 类型以及它们之间的相互作用。PyCascades可以应用于任意复杂的网络结构，最近已经扩展到随机动态系统。本文通过介绍基本概念及其背后的方法来概述PyCascades的功能。最后讨论了三个例子，展示了该软件包的三种不同应用。首先，对亚马逊雨林的水分循环网络进行了调查。其次，讨论了相互作用的地球系统倾覆要素的模型。第三，将PyCascades建模框架应用于全球贸易网络。