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

TL;DR

This paper explores the prediction of the dynamics of piecewise smooth maps using various deep learning models and 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.

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.

Figures (21)

• Figure 1: One-parameter border collision bifurcation diagram of normal form map \ref{['normaleq']} where $x$-axis presents the parameter $\mu$ and $y$-axis represents the state variable $x$, $\mu=(-0.1,2)$.
• Figure 2: One-parameter Period vs $\mu$ diagram of normal form map \ref{['normaleq']} where $x$-axis represents $\mu$ values and $y$-axis represents periods, $\mu$ varies from $-0.10$ to $0.20$ and period varies from $1$ to $9$.
• Figure 3: One-parameter border collision bifurcation diagram of normal form map \ref{['normaleq']} showing the $\mu$ values at which border collision bifurcation occur i.e., $\mu=1.3877787807814457e-17$ and $\mu=0.10000000000000003$ denoted by grey vertical lines.
• Figure 4: One-parameter border collision bifurcation diagram of the $1D$ tent map where $x$-axis represent the parameter $\mu$ and $y$-axis represent the state variable $x$, $\mu=(-1.5,1.5)$.
• Figure 5: One parameter period vs $\mu$ diagram of tent map where where $x$-axis represents the parameter $\mu$ values and $y$-axis represents the periods, $\mu$ varies from $-1.0$ to $1.0$ and period varies from $2$ to $16$.