An Application of Convolutional Neural Networks to Chaotic Systems

We propose a novel method of analyzing dynamic systems for chaotic behavior using machine learning techniques. The study of chaos in dynamic systems often involves analyzing visual representations of data points generated by the systems. One method includes embedding one-dimensional time-series data into a two-dimensional space (a "pq diagram") as an interim step in the test for chaotic behavior. We study the application of a machine learning visual recognition model—a Convolutional Neural Network—on the pq diagrams generated in various systems and determine that they can be used to reliably differentiate chaotic and periodic systems from one another, and possibly to differentiate between different types of chaotic systems themselves.

I. Introduction

Because of their non-linear (and thus non-separable) nature, systems of ordinary differential equations (ODEs) that give rise to chaotic behavior are often initially investigated via visualization. Visual representations illustrate relationships and patterns of distributions that are not immediately obvious from the equations themselves. This can be seen in visualizations of systems such as the Lorenz attractor and the Rössler attractor. Formal examinations of chaotic systems rely on more rigorous techniques. Of particular note is the method presented in [1] that expands upon the 0-1 test for chaos detection described in [2]. The technique in [1] takes the projection of the signal into a "p-q" or "pq" space. This is visualized as pq diagrams; the authors then take the data in the pq space and apply topological data analysis techniques to make reliable inferences on the presence of chaos based on the shape of the projected data. Moreover, these inferences are robust to the presence of noise even when the signal to noise ratio (SNR) falls to 30 dB [1]. Since real-world systems are expected to be noisy, this is a critical feature for practical applications.