Deep learning for prediction and classifying the dynamical behaviour of piecewise-smooth maps

Abstract

This paper explores novel ways of predicting and classifying the dynamics of piecewise smooth maps using various deep learning models. Moreover, we have used machine learning models such as Decision Tree Classifier, Logistic Regression, K-Nearest Neighbor, Random Forest, and Support Vector Machine for predicting the border collision bifurcation in the 1D normal form map and the 1D tent map. The decision tree classifier best predicts the border collision bifurcation for the 1D normal form map, the random forest, and the 1D tent map. This study introduces a novel application of deep learning models to cobweb diagrams and phase portraits, which provides a new perspective for classifying regular and chaotic behaviour. Further, we classified the regular and chaotic behaviour of the 1D tent map and the 2D Lozi map using deep learning models like Convolutional Neural Network (CNN), ResNet50, and ConvLSTM via cobweb diagram and phase portraits, where CNN exhibits better performance than other models. We also classified the chaotic and hyperchaotic behaviour of the 3D piecewise smooth map using deep learning models such as the Feed Forward Neural Network (FNN), Long Short-Term Memory (LSTM), and Recurrent Neural Network (RNN). We have shown that LSTM performs best for classifying chaotic and hyperchaotic behaviour. Additionally, LSTM outperforms other models in accuracy and computational efficiency, making it highly effective for real-time analysis. Finally, deep learning models such as Long Short Term Memory (LSTM) and Recurrent Neural Network (RNN) are used for reconstructing the two-parameter bifurcation charts of 2D normal form map, in which LSTM is more precise than RNN in reconstructing the two-parametric charts.

Keywords

• Cobweb diagram
• Phase portrait
• Machine learning
• Deep learning
• Hyperchaos
• Piecewise -smooth map