A generalized method for estimating parameters of chaotic systems using synchronization with modern optimizers

Abstract

Deriving governing equations from time series data is an ongoing topic of research across different disciplines in science. While the terms of the governing equations can be reconstructed by combinations of the input coordinates or other more sophisticated methods, inferring the coefficients of each term is a complex task on its own. Here, we extend and discuss an algorithm for finding the correct coefficients of the governing equations of chaotic systems by introducing a unidirectional coupling. We achieve this by treating the data as a primary system and coupling a secondary system to it. Then by inducing synchronization, we can push the parameters of the secondary system in the direction minimizing a loss function. After the loss has reached its minimum, the found parameters are a good estimate of the real parameters producing the data. We apply our algorithm on numerous chaotic systems and we find that this method identifies the correct coefficients for all of them, while being robust to noise and incorrect terms in the governing equations. Additionally, we discover that the Lorenz equations are not the only ones producing the—or a—butterfly-shaped attractor.