Identifying and describing the dynamics of complex systems is a central challenge in various areas of science, such as physics, finance, or climatology. While machine learning algorithms are increasingly overtaking traditional approaches, their inner workings and, thus, the drivers of causality remain elusive. In this paper, we analyze the causal structure of chaotic systems using Fourier transform surrogates and three different inference techniques: While we confirm that Granger causality is exclusively able to detect linear causality, transfer entropy and convergent cross-mapping indicate that causality is determined to a significant extent by nonlinear properties. For the Lorenz and Halvorsen systems, we find that their contribution is independent of the strength of the nonlinear coupling. Furthermore, we show that a simple rationale and calibration algorithm are sufficient to extract the governing equations directly from the causal structure of the data. Finally, we illustrate the applicability of the framework to real-world dynamical systems using financial data before and after the COVID-19 outbreak. It turns out that the pandemic triggered a fundamental rupture in the world economy, which is reflected in the causal structure and the resulting equations.

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