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## New synchronization index of non-identical networks

 Department of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 15875-4413, Iran

Received  September 2019 Revised  January 2020 Published  May 2020

Recently, quantifying the level of the synchrony in non-identical networks has got considerable attention. In the first part of this paper, a new synchronization index for non-identical networks is proposed. Non-identical networks can be categorized into two main types. The first group consists of similar oscillators with miss-match in their parameters, and the second group is organized from completely different oscillators. The synchronizability of the second group of the non-identical networks is more challenging since the amplitude and frequencies of the different oscillators may not be matched. Thus, one way to investigate the limitation of the synchronizability of these networks is to explore the parameter space of their amplitude and frequency. In the second part of this research, the amplitude and frequency of each individual system of the non-identical network are considered as varying parameters and the effect of these parameters on the synchronizability of the network is measured with the propsed index. The results are compared with the conventional indexes, such as the root-mean-square error and phase synchrony with the help of Hilbert transform. The outcomes show that the new proposed synchronization index not only is simple and accurate, but also fast with short computational time. It is not affected by amplitude, phase, or polarity. It can detect the similarity in the fluctuations which is a sign of synchrony in the non-identical networks.

Citation: Shirin Panahi, Sajad Jafari. New synchronization index of non-identical networks. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020371
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Phase space and time series of the Rössler and Lorenz oscillators, respectively
(a) Time series of the first state of each oscillator of the network in different coupling strength. (b) the first state of each oscillator of network with respect to each other
The flowchart of the proposed method to estimate the synchronization degree on nonidentical networks
Compare the result of the pattern synchrony with new proposed method on Eq. 2 with the Error (RMSE) and PS with respect to Hilbert transform approaches concerning to changing the coupling strength (d). The blue, red and green lines are corresponding to the pattern synchrony, Error and PS methods, respectively
Phase space and time series of the HR and FHN, respectively
(a) Time series of the first state of each oscillator of the Eq. 6 in different coupling strength. (b) the first state of each oscillator of network with respect to each other
Compare the result of the pattern synchrony with the new proposed method on Eq. 6 with the Error (RMSE) and PS with respect to Hilbert transform approaches concerning to changing the coupling strength $d$. The blue, red and green lines are corresponding to the pattern synchrony, Error and PS methods, respectively
Compare the result of the pattern synchrony with the new proposed method on Eq. 6 in three different values of coupling strength which is (a) $d = 0.6$ (b) $d = 0.7$ and (c) $d = 0.8$
Time series of the Eq. 7 with $d = 0$ when the other parameters are set to (a) $A = 1 f = 1$. (b) $A = 1 f = 5$. (c) $A = 0.5 f = 1$. (d) $A = 0.8 f = 6$
Synchronization degree in the parameter space of Eq. 7 with respect to changing the coupling strength with the help of the pattern synchrony (Sec. 2.1). The synchronization of the network is constant in each column, while the amplitude is changing. Since the important point in new proposed method is to consider the pattern similarity, not the similarity in the amplitudes. The light yellow and dark blue represent the most and the least synchronized pattern, respectively. The unstable state of the network is shown in the white color
Synchronization degree of the Eq. 7 with respect to changing the parameter F and coupling strength by the help of the proposed algorithm. The unbounded states of the network are shown in white color
Synchronization degree of the Eq. 7 in the parameter space of A & F with respect to changing the coupling strength by the help of the RMSE synchronization index. The dark blue is responsible for the minimum error and best synchronization error. The unstable state of the network is shown at the white color
Synchronization degree of the Eq. 7 in the parameter space of A & F with respect to changing the coupling strength by the help of the PS. The dark blue is responsible for the minimum phase error and best PS. The unstable state of the network is shown at the white color
Comparison the time series of the Eq. 7 with the optimum parameters of the new proposed method, E & PS with Hilbert transform synchronization index when the coupling strength is set to (a) $d = 0.5$ and (b) $d = 1$
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