Ab initio analysis of extreme events in dynamical systems with Rényi entropy production rate

This article presents a novel framework, based on Rényi entropy production, for studying and anticipating extreme events in dynamical systems, which aims to quantify information creation, phase-space contraction, and instability across both stochastic (Markovian) and deterministic systems. We derive a generalized entropy balance equation that decomposes the rate of change of Rényi entropy into internal production and external flux, thereby establishing its consistency with the Shannon entropy limit. We further establish a connection between Rényi entropy production and Lyapunov exponents through a generalized Pesin identity, which directly links dynamical instability with information generation. Additionally, we propose an effective ‘early-warning index’ for extreme events in dynamical systems by integrating Rényi entropy production with Lyapunov exponents. This framework is demonstrated through three explicit examples: the Lorenz system, the FitzHugh–Nagumo model, and the Ikeda map, where extreme events are clearly detectable. The findings indicate that varying the Rényi entropy order parameter q can highlight either regular or rare trajectories, thereby providing a systematic, first-principles approach to analyzing extreme events in complex dynamical systems.