Detecting chimeras by eigenvalue decomposition of the bivariate local order parameter

¹ Department of Biomedical Engineering, Amirkabir University of Technology - No. 350, Hafez Ave, Valiasr Square, Tehran 159163-4311, Iran
² Health Technology Research Institute, Amirkabir University of Technology - No. 350, Hafez Ave, Valiasr Square, Tehran 159163-4311, Iran
³ Faculty of Natural Sciences and Mathematics, University of Maribor - Koroška cesta 160, 2000 Maribor, Slovenia
⁴ Department of Medical Research, China Medical University Hospital, China Medical University - Taichung, Taiwan
⁵ Complexity Science Hub Vienna - Josefstädterstraße 39, 1080 Vienna, Austria

^(a) sajadjafari83@gmail.com

Received: 16 March 2020
Accepted: 4 May 2020

Abstract

It has been shown that the eigenvalue decomposition of the matrix of the bivariate phase synchronization measure can be used for the detection of cluster synchronization. It has also been shown that other measures, such as the strength of incoherence and various local order parameters, can be used to quantitatively characterize chimeras, or chimera states. Here we bridge these two domains by showing that the eigenvalue decomposition method can also be used for the detection of chimeras. We compute the local order parameter for all oscillator pairs and apply the eigenvalue decomposition on the bivariate matrix. We show that, in contrast to cluster synchronization, there are more eigenvalues above one than the number of synchronized clusters in the network. The corresponding eigenvectors correspond to synchronized groups, while the oscillators that are not represented by the eigenvectors form the chimeras. We demonstrate our approach on coupled Liénard equations and FitzHugh-Nagumo neurons.