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April  2021, 14(4): 1359-1373. doi: 10.3934/dcdss.2020371

## 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  April 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (4) : 1359-1373. doi: 10.3934/dcdss.2020371
##### References:
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[16] D. Y. Kenett and S. Havlin, Network science: A useful tool in economics and finance, Mind & Society, 14 (2015), 155-167.  doi: 10.1007/s11299-015-0167-y. [17] D. Y. Kenett, M. Perc and S. Boccaletti, Networks of networks–an introduction, Chaos, Solitons & Fractals, 80 (2015), 1-6. [18] J. Ma and J. Tang, A review for dynamics of collective behaviors of network of neurons, Science China Technological Sciences, 58 (2015), 2038-2045.  doi: 10.1007/s11431-015-5961-6. [19] S. Majhi, B. K. Bera, D. Ghosh and M. Perc, Chimera states in neuronal networks: A review, Physics of Life Reviews, 28 (2019), 100-121. [20] S. Majhi, D. Ghosh and J. Kurths, Emergence of synchronization in multiplex networks of mobile rössler oscillators, Physical Review E, 99 (2019), 012308, 13pp. doi: 10.1103/physreve.99.012308. [21] A. Y. Mutlu and S. Aviyente, Multivariate empirical mode decomposition for quantifying multivariate phase synchronization, EURASIP Journal on Advances in Signal Processing, 2011 (2011), 615717. doi: 10.1155/2011/615717. [22] V. Patidar and K. Sud, Identical synchronization in chaotic jerk dynamical systems, Electronic Journal of Theoretical Physics, 3 (2006), 33-70. [23] L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Physical Review Letters, 80 (1998), 2109. [24] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743. [25] F. A. Rodrigues, T. K. D. Peron, P. Ji and J. Kurths, The kuramoto model in complex networks, Physics Reports, 610 (2016), 1-98.  doi: 10.1016/j.physrep.2015.10.008. [26] M. Rosenblum, A. Pikovsky, J. Kurths, C. Schäfer and P. A. Tass, Phase synchronization: From theory to data analysis, Handbook of Biological Physics, 4 (2001), 279-321. [27] M. G. Rosenblum, A. S. Pikovsky and J. Kurths, Phase synchronization of chaotic oscillators, Physical Review Letters, 76 (1996), 1804. [28] I. Stamova, T. Stamov and X. Li, Global exponential stability of a class of impulsive cellular neural networks with supremums, International Journal of Adaptive Control and Signal Processing, 28 (2014), 1227-1239.  doi: 10.1002/acs.2440. [29] S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276.  doi: 10.1038/35065725. [30] X. Sun, J. Lei, M. Perc, J. Kurths and G. Chen, Burst synchronization transitions in a neuronal network of subnetworks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 21 (2011), 016110. doi: 10.1063/1.3559136. [31] X. Sun, M. Perc, J. Kurths and Q. Lu, Fast regular firings induced by intra-and inter-time delays in two clustered neuronal networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 106310, 10pp. doi: 10.1063/1.5037142. [32] U. K. Verma, A. Sharma, N. K. Kamal, J. Kurths and M. D. Shrimali, Explosive death induced by mean–field diffusion in identical oscillators, Scientific Reports, 7 (2017), 7936. doi: 10.1038/s41598-017-07926-x. [33] C. Wang and J. Ma, A review and guidance for pattern selection in spatiotemporal system, International Journal of Modern Physics B, 32 (2018), 1830003, 15pp. doi: 10.1142/S0217979218300037. [34] Q. Wang, G. Chen and M. Perc, Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling, PLoS One, 6 (2011), e15851. doi: 10.1371/journal.pone.0015851. [35] D. Wen, Y. Zhou and X. Li, A critical review: Coupling and synchronization analysis methods of eeg signal with mild cognitive impairment, Frontiers in Aging Neuroscience, 7 (2015), 54. doi: 10.3389/fnagi.2015.00054. [36] C. W. Wu and L. O. Chua, On a conjecture regarding the synchronization in an array of linearly coupled dynamical systems, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 43 (1996), 161-165. [37] J. Xiang and G. Chen, On the v-stability of complex dynamical networks, Automatica, 43 (2007), 1049-1057.  doi: 10.1016/j.automatica.2006.11.014. [38] X. Yang, X. Li, Q. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences & Engineering, 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069. [39] X. Zhang, X. Li, J. Cao and F. Miaadi, Design of memory controllers for finite-time stabilization of delayed neural networks with uncertainty, Journal of the Franklin Institute, 355 (2018), 5394-5413.  doi: 10.1016/j.jfranklin.2018.05.037.

show all references

##### References:
 [1] G. R. Ávila, J. Kurths, J.-L. Guisset and J.-L. Deneubourg, How do small differences in nonidentical pulse-coupled oscillators induce great changes in their synchronous behavior?, The European Physical Journal Special Topics, 223 (2014), 2759-2773. [2] D. S. Bassett and O. Sporns, Network neuroscience, Nature Neuroscience, 20 (2017), 353-364.  doi: 10.1038/nn.4502. [3] V. N. Belykh, I. V. Belykh and M. Hasler, Connection graph stability method for synchronized coupled chaotic systems, Physica D: Nonlinear Phenomena, 195 (2004), 159-187.  doi: 10.1016/j.physd.2004.03.012. [4] S. Boccaletti, J. Almendral, S. Guan, I. Leyva, Z. Liu, I. Sendiña-Nadal, Z. Wang and Y. Zou, Explosive transitions in complex networks' structure and dynamics: Percolation and synchronization, Physics Reports, 660 (2016), 1-94.  doi: 10.1016/j.physrep.2016.10.004. [5] S. Boccaletti, J. Bragard, F. Arecchi and H. Mancini, Synchronization in nonidentical extended systems, Physical Review Letters, 83 (1999), 536. doi: 10.1103/PhysRevLett.83.536. [6] S. Boccaletti, J. Kurths, G. Osipov, D. Valladares and C. Zhou, The synchronization of chaotic systems, Physics Reports, 366 (2002), 1-101.  doi: 10.1016/S0370-1573(02)00137-0. [7] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Physics Reports, 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009. [8] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, The Bulletin of Mathematical Biophysics, 17 (1955), 257-278.  doi: 10.1007/BF02477753. [9] L. C. Freeman, Research Methods in Social Network Analysis, Routledge, 2017. doi: 10.4324/9781315128511. [10] H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Progress of Theoretical Physics, 69 (1983), 32-47.  doi: 10.1143/PTP.69.32. [11] D. Gabor, Theory of communication. part 1: The analysis of information, Journal of the Institution of Electrical Engineers-Part Ⅲ: Radio and Communication Engineering, 93 (1946), 429-441.  doi: 10.1049/ji-3-2.1946.0074. [12] J. Gao, B. Barzel and A.-L. Barabási, Universal resilience patterns in complex networks, Nature, 530 (2016), 307-312.  doi: 10.1038/nature16948. [13] M. Gosak, R. Markovič, J. Dolenšek, M. S. Rupnik, M. Marhl, A. Stožer and M. Perc, Network science of biological systems at different scales: A review, Physics of Life Reviews, 24 (2018), 118-135.  doi: 10.1016/j.plrev.2017.11.003. [14] D. J. Hill and J. Zhao, Global synchronization of complex dynamical networks with non-identical nodes, in 2008 47th IEEE Conference on Decision and Control, IEEE, 2008,817–822. [15] J. L. Hindmarsh and R. Rose, A model of neuronal bursting using three coupled first order differential equations, Proceedings of the Royal society of London. Series B. Biological sciences, 221 (1984), 87-102. [16] D. Y. Kenett and S. Havlin, Network science: A useful tool in economics and finance, Mind & Society, 14 (2015), 155-167.  doi: 10.1007/s11299-015-0167-y. [17] D. Y. Kenett, M. Perc and S. Boccaletti, Networks of networks–an introduction, Chaos, Solitons & Fractals, 80 (2015), 1-6. [18] J. Ma and J. Tang, A review for dynamics of collective behaviors of network of neurons, Science China Technological Sciences, 58 (2015), 2038-2045.  doi: 10.1007/s11431-015-5961-6. [19] S. Majhi, B. K. Bera, D. Ghosh and M. Perc, Chimera states in neuronal networks: A review, Physics of Life Reviews, 28 (2019), 100-121. [20] S. Majhi, D. Ghosh and J. Kurths, Emergence of synchronization in multiplex networks of mobile rössler oscillators, Physical Review E, 99 (2019), 012308, 13pp. doi: 10.1103/physreve.99.012308. [21] A. Y. Mutlu and S. Aviyente, Multivariate empirical mode decomposition for quantifying multivariate phase synchronization, EURASIP Journal on Advances in Signal Processing, 2011 (2011), 615717. doi: 10.1155/2011/615717. [22] V. Patidar and K. Sud, Identical synchronization in chaotic jerk dynamical systems, Electronic Journal of Theoretical Physics, 3 (2006), 33-70. [23] L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Physical Review Letters, 80 (1998), 2109. [24] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743. [25] F. A. Rodrigues, T. K. D. Peron, P. Ji and J. Kurths, The kuramoto model in complex networks, Physics Reports, 610 (2016), 1-98.  doi: 10.1016/j.physrep.2015.10.008. [26] M. Rosenblum, A. Pikovsky, J. Kurths, C. Schäfer and P. A. Tass, Phase synchronization: From theory to data analysis, Handbook of Biological Physics, 4 (2001), 279-321. [27] M. G. Rosenblum, A. S. Pikovsky and J. Kurths, Phase synchronization of chaotic oscillators, Physical Review Letters, 76 (1996), 1804. [28] I. Stamova, T. Stamov and X. Li, Global exponential stability of a class of impulsive cellular neural networks with supremums, International Journal of Adaptive Control and Signal Processing, 28 (2014), 1227-1239.  doi: 10.1002/acs.2440. [29] S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276.  doi: 10.1038/35065725. [30] X. Sun, J. Lei, M. Perc, J. Kurths and G. Chen, Burst synchronization transitions in a neuronal network of subnetworks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 21 (2011), 016110. doi: 10.1063/1.3559136. [31] X. Sun, M. Perc, J. Kurths and Q. Lu, Fast regular firings induced by intra-and inter-time delays in two clustered neuronal networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 106310, 10pp. doi: 10.1063/1.5037142. [32] U. K. Verma, A. Sharma, N. K. Kamal, J. Kurths and M. D. Shrimali, Explosive death induced by mean–field diffusion in identical oscillators, Scientific Reports, 7 (2017), 7936. doi: 10.1038/s41598-017-07926-x. [33] C. Wang and J. Ma, A review and guidance for pattern selection in spatiotemporal system, International Journal of Modern Physics B, 32 (2018), 1830003, 15pp. doi: 10.1142/S0217979218300037. [34] Q. Wang, G. Chen and M. Perc, Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling, PLoS One, 6 (2011), e15851. doi: 10.1371/journal.pone.0015851. [35] D. Wen, Y. Zhou and X. Li, A critical review: Coupling and synchronization analysis methods of eeg signal with mild cognitive impairment, Frontiers in Aging Neuroscience, 7 (2015), 54. doi: 10.3389/fnagi.2015.00054. [36] C. W. Wu and L. O. Chua, On a conjecture regarding the synchronization in an array of linearly coupled dynamical systems, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 43 (1996), 161-165. [37] J. Xiang and G. Chen, On the v-stability of complex dynamical networks, Automatica, 43 (2007), 1049-1057.  doi: 10.1016/j.automatica.2006.11.014. [38] X. Yang, X. Li, Q. Xi and P. Duan, Review of stability and stabilization for impulsive delayed systems, Mathematical Biosciences & Engineering, 15 (2018), 1495-1515.  doi: 10.3934/mbe.2018069. [39] X. Zhang, X. Li, J. Cao and F. Miaadi, Design of memory controllers for finite-time stabilization of delayed neural networks with uncertainty, Journal of the Franklin Institute, 355 (2018), 5394-5413.  doi: 10.1016/j.jfranklin.2018.05.037.
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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