# From lakes and glades to viability algorithms: Automatic classification of system states according to the Topology of Sustainable Management

## TL;DR

This paper integrates viability theory with the Topology of Sustainable Management framework to visualize and analyze system states, introducing algorithms for better understanding complex environmental and socioeconomic models.

## Contribution

It connects the Topology of Sustainable Management to viability theory, extending algorithms to compute capture basins and applying them to a coupled environmental-socioeconomic model.

## Key findings

- Viability algorithms effectively visualize system state partitions.
- Coordinate transformations address unbounded state spaces.
- The approach enhances understanding of manageable dynamical systems.

## Abstract

The framework Topology of Sustainable Management by Heitzig et al. (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.