Deep Learning for Prediction and Classifying the Dynamical behaviour of Piecewise Smooth Maps

This paper explores the prediction of the dynamics of piecewise smooth maps using various deep learning models. We have shown various novel ways of predicting the dynamics of piecewise smooth maps using 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. 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. 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). Finally, deep learning models such as Long Short-Term Memory (LSTM) and Recurrent Neural Network (RNN) are used for reconstructing the two parametric charts of 2D border collision bifurcation normal form map.

Comparative Analysis of Predicting Subsequent Steps in Hénon Map

This paper explores the prediction of subsequent steps in H\'enon Map using various machine learning techniques. The H\'enon map, well known for its chaotic behaviour, finds applications in various fields including cryptography, image encryption, and pattern recognition. Machine learning methods, particularly deep learning, are increasingly essential for understanding and predicting chaotic phenomena. This study evaluates the performance of different machine learning models including Random Forest, Recurrent Neural Network (RNN), Long Short-Term Memory (LSTM) networks, Support Vector Machines (SVM), and Feed Forward Neural Networks (FNN) in predicting the evolution of the H\'enon map. Results indicate that LSTM network demonstrate superior predictive accuracy, particularly in extreme event prediction. Furthermore, a comparison between LSTM and FNN models reveals the LSTM's advantage, especially for longer prediction horizons and larger datasets. This research underscores the significance of machine learning in elucidating chaotic dynamics and highlights the importance of model selection and dataset size in forecasting subsequent steps in chaotic systems.