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Measuring dynamical phase transitions in time series

Bulcsú Sándor*, András Rusu, Károly Dénes, Mária Ercsey-Ravasz†, and Zsolt I. Lázár‡

  • Department of Physics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania and Transylvanian Institute of Neuroscience, 400191 Cluj-Napoca, Romania

  • *Contact author: bulcsu.sandor@ubbcluj.ro
  • †Contact author: maria.ercsey@ubbcluj.ro
  • ‡Contact author: zsolt.lazar@ubbcluj.ro
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Phys. Rev. Research 7, 033241 – Published 10 September, 2025

DOI: https://doi.org/10.1103/98vk-ws9s

Abstract

There is a growing interest in methods for detecting and interpreting changes in experimental time-evolution data. Based on measured time series, the quantitative characterization of dynamical phase transitions at bifurcation points of the underlying chaotic systems is a notoriously difficult task. Building on prior theoretical studies that focus on the discontinuities at q=1 in the order-q dynamical Rényi entropy, we measure the derivative of its spectrum given by the variance of the logarithmic probability of trajectories. To overcome the difficulties inherent in trajectory-space descriptions of finite-length time series, we derive—within the general framework of Markov processes—a computationally efficient closed-form expression for this measure, while maintaining control over the trade-off between statistical and truncation errors. We investigate the properties of the measure and assess the efficacy of the associated method in estimating characteristic quantities of well-known dynamical systems. Furthermore, we examine its scope and limitations, including its potential utility in predicting dynamical phase transitions from time series and in characterizing the complexity of the underlying behavior. Grounded in the thermodynamic formalism of chaotic systems, the developed framework bridges dynamical systems theory, network theory, and time-series analysis, extending classical concepts such as bit-number variance or varentropy in phase-space statistics, permutation entropy in time-series analysis, and path-length statistics in network science.

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Phys. Rev. Research 7, 033241– Published 10 September, 2025
  • Received 19 July 2024
  • Revised 6 March 2025
  • Accepted 1 August 2025
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