Machine Learning Insights into the Dynamics of Cusp CatastropheInstability Region

Abstract

This study investigates a novel application of Random Forest regression for analyzing the unstable set of the cusp catastrophe, a mathematical model describing abrupt, nonlinear transitions in dynamical systems. The cusp catastrophe effectively captures critical phenomena such as bifurcation, hysteresis, and multistability; however, modeling its unstable region remains challenging due to noise, sparsity, and the localized nature of transitions in real world data.

Random Forest’s ability to approximate complex, nonlinear relationships was evaluated across varying noise levels and data availabilities. The results demonstrate that the model excels in low noise scenarios, accurately capturing the critical features of the unstable set. However, performance declines with increasing noise and limited data, highlighting the need for noise tolerant strategies. This work provides new insights into leveraging machine learning for robust modeling and prediction of unstable regions in non-linear dynamical systems, offering a foundation for future advancements in catastrophe theory applications.