Deep Learning for Bifurcation Detection: Extending Early Warning Signals to Dynamical Systems with Coloured Noise

Deep learning models have demonstrated remarkable success in recognising tipping points and providing early warning signals. However, there has been limited exploration of their application to dynamical systems governed by coloured noise, which characterizes many real-world systems. In this study, we demonstrated that, using the normal form theorem, it is possible to leverage the normal forms of three primary types of bifurcations (Fold, Transcritical, and Hopf) to construct a training set that enables deep learning architectures to perform effectively. Furthermore, we showed that this approach could accommodate coloured noise by replacing white noise with red noise during the training process. To evaluate the classifier trained on red noise compared to one trained on white noise, we tested their performance on mathematical models using ROC curves and AUC scores. Our findings reveal that the DL architecture can be effectively trained on colored noise inputs, as evidenced by high validation accuracy and minimal sensitivity to redness (ranging from 0.83 to 0.85). However, classifiers trained on white noise also demonstrate impressive performance in identifying tipping points in colored time series. This is further supported by high AUC scores (ranging from 0.9 to 1) for both classifiers across different colored stochastic time series.