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Master stability functions for torus and stable equilibrium attractors

Sajad Jafari1,2,*, Fatemeh Parastesh3,4, Atiyeh Bayani3,4, Karthikeyan Rajagopal3,5, Qiang Lai6, and Jürgen Kurths7,8

  • *Contact author: sajadjafari83@gmail.com; sajadjafari@aut.ac.ir
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Phys. Rev. E 113, 024211 – Published 19 February, 2026

DOI: https://doi.org/10.1103/fh87-dntc

Abstract

Synchronization is a fundamental phenomenon in networked dynamical systems with applications ranging from power grids to biological networks. While much progress has been made in understanding synchronization in chaotic and periodic systems through the master stability function (MSF) framework, less attention has been given to systems exhibiting simpler attractors, such as fixed points or quasiperiodic (torus) behaviors. This paper addresses this gap by systematically analyzing the synchronization properties of coupled systems with point and torus attractors using the MSF approach. The results show that systems with point attractors can display an unexpected synchronization scenario in which stable synchrony can emerge in disjoint regions of the coupling parameter space. We propose a generalized classification scheme for synchronization types in such systems, drawing parallels to existing frameworks for chaotic and periodic oscillators. This study contributes to a more comprehensive understanding of synchronization behavior in complex networks.

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Phys. Rev. E 113, 024211– Published 19 February, 2026
  • Received 28 September 2025
  • Accepted 13 January 2026
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