Two-dimensional master stability function analysis of synchronization in periodic and chaotic networks

Tayebeh Moalemi; Gopinath Barathi; Atiyeh Bayani; Karthikeyan Rajagopal; Sajad Jafari; Matjaž Perc; Tayebeh Moalemi; Gopinath Barathi; Atiyeh Bayani; Karthikeyan Rajagopal; Sajad Jafari; Matjaž Perc

doi:10.3934/math.2026269

AIMS Mathematics

2026, Volume 11, Issue 3: 6499-6524. doi: 10.3934/math.2026269

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Two-dimensional master stability function analysis of synchronization in periodic and chaotic networks

1.
Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Iran
2.
Center for Research, Easwari Engineering College, Chennai, India
3.
Center for Research, SRM TRP Engineering College, Trichy, India
4.
Center for Cognitive Science, Trichy SRM Medical College Hospital and Research Center, Trichy, India
5.
Health Technology Research Institute, Amirkabir University of Technology (Tehran Polytechnic), Iran
6.
Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
7.
Community Healthcare Center Dr. Adolf Drolc Maribor, Ulica talcev 9, 2000 Maribor, Slovenia
8.
Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea
9.
University College, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Republic of Korea

Received: 09 December 2025 Revised: 21 February 2026 Accepted: 04 March 2026 Published: 13 March 2026
MSC : 65P20, 34D06, 05C82

Here we used the master stability function (MSF) framework to develop a two-dimensional MSF representation that incorporates both real and complex eigenvalues, thereby capturing the dynamics of diagonalizable directed networks. Our analysis shows how synchronization stability depends on complex coupling strengths and reveals conditions under which bounded or unbounded regions of negative MSF arise, corresponding to stable synchronization. This two-dimensional approach unifies periodic and chaotic dynamics within a common framework, offering sharper insights into when and how diagonalizable directed networks can sustain coherent collective behavior.
- master stability function,
- synchronization,
- two-dimensional analysis,
- chaotic systems,
- periodic systems
Citation: Tayebeh Moalemi, Gopinath Barathi, Atiyeh Bayani, Karthikeyan Rajagopal, Sajad Jafari, Matjaž Perc. Two-dimensional master stability function analysis of synchronization in periodic and chaotic networks[J]. AIMS Mathematics, 2026, 11(3): 6499-6524. doi: 10.3934/math.2026269

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Abstract

Here we used the master stability function (MSF) framework to develop a two-dimensional MSF representation that incorporates both real and complex eigenvalues, thereby capturing the dynamics of diagonalizable directed networks. Our analysis shows how synchronization stability depends on complex coupling strengths and reveals conditions under which bounded or unbounded regions of negative MSF arise, corresponding to stable synchronization. This two-dimensional approach unifies periodic and chaotic dynamics within a common framework, offering sharper insights into when and how diagonalizable directed networks can sustain coherent collective behavior.

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