Alessandro Maria Selvitella, Purdue University, Fort Wayne, IN
Elliot Allen, Purdue University, Fort Wayne, IN

Abstract: Traditionally, applied mathematics and nonlinear analysis have focused on creating models of real-world systems using first principles. However, in many contemporary scientific domains, first-principles approaches face increasing limitations, including but not limited to the challenges posed by complex, high-dimensional, or poorly characterized systems. In response, data science leverages the availability of large datasets and computational power to study problems where empirical understanding is incomplete or the underlying mechanisms are only partially known. A combination of both paradigms is crucial when some data is available, but our understanding of the phenomenon remains limited. In this context, methods such as sparse identification of nonlinear dynamics (SINDy), a data-driven technique designed to discover nonlinear dynamical systems from empirical data using regularization methods, have proved to be successful in the study of multivariate time series and nonlinear dynamics. Sparse identification capitalizes on the observation that many natural phenomena can be described by systems with only a few nonlinear terms, yielding interpretable models. In this paper, we will discuss the effectiveness of sparse identification in accurately determining the nonlinear dynamics of systems such as the van der Pol equation and a coupled system of van der Pol oscillators, two systems often used as a major benchmark examples for testing new data-driven methods on systems with rich dynamics, also when chaotic behavior of solutions and synchronization are possible.

Keywords: SINDy, van der Pol equation, bipolar disorder, pitfalls, misdiagnosis, data-driven dynamical systems