Abstract

This paper introduces a novel hybrid approach for time series forecasting of chaotic systems, integrating the strengths of fractional calculus and deep learning. Chaotic systems, characterized by their sensitive dependence on initial conditions, pose significant challenges for accurate prediction. While deep learning models, particularly Long Short-Term Memory (LSTM) networks, have shown promise in capturing complex temporal dependencies, they often struggle with long-range dependencies and noise inherent in chaotic data. We propose a hybrid model that leverages fractional derivatives to enhance the representation of past states, thereby improving the LSTM network's ability to learn and forecast chaotic time series. The fractional derivative captures non-local dependencies more effectively than traditional integer-order derivatives, providing richer information for the deep learning component. We evaluate the performance of our proposed model on benchmark chaotic systems, including the Lorenz attractor and the Rossler system. The results demonstrate that our hybrid approach significantly outperforms traditional LSTM networks and other established forecasting methods in terms of prediction accuracy, especially over longer forecasting horizons. This work provides a valuable contribution to the field of time series forecasting for chaotic systems, offering a powerful tool for modeling and predicting complex dynamical behaviors.