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Matryoshka and disjoint cluster synchronization of networks
Amirhossein Nazerian1, Shirin Panahi1, Ian Leifer2
1Department of Mechanical Engineering, University of New Mexico, Albuquerque, New Mexico 87131, USA.
Chaos (Woodbury, N.Y.)
|April 30, 2022
Summary
This study introduces a new definition for cluster synchronization (CS) in complex networks, analyzing its complexity and synchronization ranges. It categorizes network structures and synchronization behaviors for better understanding of network dynamics.
Area of Science:
- Complex Networks
- Nonlinear Dynamics
- Systems Biology
Background:
- Complete synchronization in complex networks has been well-characterized.
- Cluster synchronization presents unique challenges due to inter-cluster dynamics.
- Existing methods for complete synchronization do not directly apply to cluster synchronization.
Purpose of the Study:
- To characterize network synchronizability for cluster synchronization (CS).
- To extend the master stability function approach to analyze CS.
- To define and categorize different types of cluster synchronization based on stability ranges.
Main Methods:
- Analysis of network structures, distinguishing between intertwined and non-intertwined clusters.
- Extension of the master stability function (MSF) to assess synchronizability.
- Classification of synchronization behaviors based on the MSF negativity range (bounded vs. unbounded).
Main Results:
- Cluster synchronization is significantly more complex than complete synchronization.
- Introduced a definition of cluster synchronizability based on individual cluster stability.
- Identified three distinct cases of pairwise cluster synchronization: Matryoshka, partially disjoint, and completely disjoint.
Conclusions:
- The stability of synchronous solutions within individual clusters can be independent.
- Network topology and the nature of the master stability function significantly impact CS.
- The proposed framework provides a detailed characterization of cluster synchronization in complex networks.
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