Nonlinear Dynamics

, Volume 97, Issue 4, pp 2399–2411 | Cite as

Phase synchronization analysis of bridge oscillators between clustered networks

  • Arthur N. MontanariEmail author
  • Leandro Freitas
  • Leonardo A. B. Torres
  • Luis A. Aguirre
Original paper


Recent works aim to establish necessary and sufficient conditions to guarantee phase synchronization between clusters of oscillators, usually assuming knowledge of the intra-cluster connections, that is, connections among oscillators of the same cluster. In this context, this paper takes a different approach in studying the stability of the synchronous manifold between clusters. By focusing on the inter-cluster relations between the bridge oscillators, a simplified problem is considered where intra-cluster effects are described as perturbations. Based on Lyapunov’s direct method, a framework is put forward to derive sufficient conditions for the ultimately boundedness of the phase difference between the bridge oscillators. This analysis does not rely on full information on the adjacency matrix describing the specific connections among oscillators within each cluster, an information that is not always available. The established theoretical conditions are compared to numerical simulations in two examples: (i) two interconnected clusters of Kuramoto oscillators, and (ii) a benchmark model of a power grid. Results indicate that the method is effective and that its conservativeness depends on the available network information. This framework can be generalized to different networks and oscillators.


Phase synchronization Stability analysis Kuramoto oscillators Clustered networks Perturbed systems 



ANM acknowledges financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (finance code 001). LABT and LAA acknowledges Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grants Nos. 309268/2017-6 and 302079/2011-4). LF gratefully acknowledges IFMG Campus Betim for an academic leave.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Graduate Program in Electrical Engineering of the Universidade Federal de Minas Gerais (UFMG)Belo HorizonteBrazil
  2. 2.Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais - Campus BetimBetimBrazil
  3. 3.Departamento de Engenharia EletrônicaUFMGBelo HorizonteBrazil

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