Characterization of Fractal Basins Using Deep Convolutional Neural Networks

Abstract

Neural network models have recently demonstrated impressive prediction performance in complex systems where chaos and unpredictability appear. In spite of the research efforts carried out on predicting future trajectories or improving their accuracy compared to numerical methods, not sufficient work has been done by using deep learning techniques in which they characterize the unpredictability of chaotic systems or give a general view of the global unpredictability of a system. In this work we propose a novel approach based on deep learning techniques to measure the fractal dimension of the basins of attraction of the Duffing oscillator for a variety of parameters. As a consequence, we provide an algorithm capable of predicting fractal dimension measures as accurately as the conventional algorithm, but with a computation speed about ten times faster.