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Generalized synchronization in strong-mismatched, bidirectional-coupled systems: Empirical detection and experimental analysis

Ludovico Minati1,2,3,*, Léandre Kamdjeu Kengne1,4, Michele Castelluzzo5, Chiara Masci6, Pedro A. Valdes-Sosa1,7, Stefano Boccaletti8,9,10, Mattia Frasca11, and Leonardo Ricci3,5

  • *Contact author: lminati@uestc.edu.cn, ludovico.minati@unitn.it, lminati@ieee.org
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Phys. Rev. Research 7, 033259 – Published 17 September, 2025

DOI: https://doi.org/10.1103/m4rr-td7r

Abstract

Generalized synchronization, in which a nontrivial interdependence is established between the trajectories of two or more coupled units, has been observed in a wide range of simulated and physical systems. Although several numerical methods are available for its detection, visualization techniques have not been reported, and bidirectional coupling remains challenging, especially for experimental data. This paper introduces a simple empirical method for approximating the functional relation, based on expressing the current value of one system variable as a function of the state variable of the other system, reconstructed by time-delay embedding, along with suitable nonlinearities. Two strongly parametrically mismatched Rössler systems with weak bidirectional coupling have been considered. The proposed method allows the computation of a time series, namely, a “proxy signal,” which shows significantly stronger phase synchronization compared to the original system variables. The underlying approximated functional is visualized through regression coefficients, and its relationship with the Lyapunov exponents is demonstrated by simulated parameter sweeps in two dimensions. The results are validated via tunable analog electronic realization of the coupled systems. The effects of nonidealities in the physical apparatus are also investigated, demonstrating the robustness of the proposed method.

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Phys. Rev. Research 7, 033259– Published 17 September, 2025
  • Received 14 May 2025
  • Accepted 20 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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