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

TL;DR

This paper demonstrates how various deep learning models can predict, classify, and reconstruct the complex dynamical behaviors of piecewise smooth maps, including bifurcations and chaos, across multiple dimensions.

Contribution

It introduces novel applications of deep learning techniques for predicting and classifying the dynamics of piecewise smooth maps, including bifurcations and chaotic regimes.

Findings

Deep learning models accurately predict border collision bifurcations.

CNN, ResNet50, and ConvLSTM classify regular and chaotic behaviors.

LSTM and RNN reconstruct parametric charts of bifurcation maps.

Abstract

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.