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Comparative Analysis of Predicting Subsequent Steps in Hénon Map

Vismaya V S, Alok Hareendran, Bharath V Nair, Sishu Shankar Muni, Martin Lellep

TL;DR

This study tackles the problem of predicting subsequent steps in chaotic systems by evaluating multiple machine learning approaches on the $Hénon$ map, a two-dimensional discrete dynamical system with chaotic behavior. Using a data-driven setup, the authors generate $10{,}000$ iterations with $a=1.4$, $b=0.3$, select the last $20\%$ of data for analysis, and split it into $80\%$ training and $20\%$ testing, then forecast the next step via RF, RNN, LSTM, SVM, and FNN, evaluating with mean squared error (MSE). The results show that LSTM achieves outstanding predictive accuracy, with MSE around $2.00\times 10^{-6}$, outperforming RF, RNN, SVM, and FNN across dataset sizes and horizons; FNN can excel on small datasets but its performance is surpassed by LSTM on larger datasets. The study highlights the importance of model selection and data volume in forecasting chaotic dynamics and points to future directions such as attention-based architectures and parameter studies to further improve predictability in chaotic systems.

Abstract

This paper explores the prediction of subsequent steps in Hénon Map using various machine learning techniques. The Hénon 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 Hénon 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.

Comparative Analysis of Predicting Subsequent Steps in Hénon Map

TL;DR

This study tackles the problem of predicting subsequent steps in chaotic systems by evaluating multiple machine learning approaches on the map, a two-dimensional discrete dynamical system with chaotic behavior. Using a data-driven setup, the authors generate iterations with , , select the last of data for analysis, and split it into training and testing, then forecast the next step via RF, RNN, LSTM, SVM, and FNN, evaluating with mean squared error (MSE). The results show that LSTM achieves outstanding predictive accuracy, with MSE around , outperforming RF, RNN, SVM, and FNN across dataset sizes and horizons; FNN can excel on small datasets but its performance is surpassed by LSTM on larger datasets. The study highlights the importance of model selection and data volume in forecasting chaotic dynamics and points to future directions such as attention-based architectures and parameter studies to further improve predictability in chaotic systems.

Abstract

This paper explores the prediction of subsequent steps in Hénon Map using various machine learning techniques. The Hénon 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 Hénon 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.
Paper Structure (9 sections, 2 equations, 12 figures)

This paper contains 9 sections, 2 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Hénon Map
  3. Predictions using Machine Learning Methods
  4. Random Forest
  5. Reccurent Neural Network(RNN)
  6. Long Short-Term Memory (LSTM)
  7. Support Vector Machine (SVM)
  8. Results
  9. Conclusion

Figures (12)

  • Figure 1: The Hénon-map chaotic attractor shown as blue dots and dashed red lines represent the threshold $\theta$
  • Figure 2: Hénon Attractor with time steps T = 4, 6, and 8 where black: Criterion not satisfied; Orange, Magenta, Cyan: Criterion satisfied (T = 4, 6, 8)
  • Figure 3: Schematic diagram showing the process of generating training data, training the prediction model, and finally using the trained model for prediction
  • Figure 5: Topology of the neural network. There are input nodes (blue box), hidden layers (black box), and output neurons (green box), where N is the number of previous time steps. (a) Recuuent Neural Network (RNN),(b) Long Short-Term Memory (LSTM),(c) Feedforward Neural Networks (FNN)
  • Figure 6: A point plot of Mean Squared Error (MSE) for five models: Random Forest, RNN, LSTM, SVM, and FNN.
  • ...and 7 more figures