# Climbing Escher's stairs: a way to approximate stability landscapes in multidimensional systems

## TL;DR

This paper introduces a numerical method and an R package to approximate stability landscapes in multidimensional systems by decomposing differential equations into gradient and non-gradient parts, enabling potential landscape estimation where it traditionally does not exist.

## Contribution

The authors present a novel numerical approach and software tool to approximate stability landscapes in complex systems lacking scalar potentials.

## Key findings

- The method effectively decomposes differential equations into gradient and non-gradient components.
- Approximate potential landscapes can be constructed in regions where non-gradient effects are small.
- An R package implementation makes the approach accessible for practical use.

## Abstract

Stability landscapes are useful for understanding the properties of dynamical systems. These landscapes can be calculated from the system's dynamical equations using the physical concept of scalar potential. Unfortunately, for most biological systems with two or more state variables such potentials do not exist. Here we use an analogy with art to provide an accessible explanation of why this happens. Additionally, we introduce a numerical method for decomposing differential equations into two terms: the gradient term that has an associated potential, and the non-gradient term that lacks it. In regions of the state space where the magnitude of the non-gradient term is small compared to the gradient part, we use the gradient term to approximate the potential as quasi-potential. The non-gradient to gradient ratio can be used to estimate the local error introduced by our approximation. Both the algorithm and a ready-to-use implementation in the form of an R package are provided.