Full Lyapunov Exponents spectrum with Deep Learning from single-variable time series

TL;DR

This paper demonstrates that a convolutional neural network can accurately approximate the full Lyapunov exponents spectrum of dynamical systems using only single-variable time series, enabling faster analysis of complex behaviors.

Contribution

It introduces a deep learning approach to estimate the entire Lyapunov spectrum from minimal data, advancing the analysis of chaotic and hyperchaotic systems.

Findings

Accurately approximates Lyapunov spectra from single-variable data

Enables faster analysis of chaotic systems

Provides insights into hyperchaotic dynamics

Abstract

In this article we study if a Deep Learning technique can be used to obtain an approximated value of the Lyapunov exponents of a dynamical system. Moreover, we want to know if Machine Learning techniques are able, once trained, to provide the complete Lyapunov exponents spectrum with just single-variable time series. We train a Convolutional Neural Network and we use the resulting network to approximate the complete spectrum using the time series of just one variable from the studied systems (Lorenz system and coupled Lorenz system). The results are quite stunning as all the values are well approximated with only partial data. This strategy permits to speed up the complete analysis of the systems and also to study the hyperchaotic dynamics in the coupled Lorenz system.