# American Institute of Mathematical Sciences

October  2018, 23(8): 3461-3482. doi: 10.3934/dcdsb.2018234

## Synchronization in networks with strongly delayed couplings

 1 Associate Laboratory of Applied Computing and Mathematics - LAC, National Institute for Space Research - INPE, Av. dos Astronautas, 1758, São José dos Campos-SP, 12227-010, Brazil 2 Institute of Mathematics and Computer Sciences - ICMC, University of São Paulo - USP, Av. do Trabalhador São-Carlense 400, São Carlos-SP, 13566-590, Brazil 3 Institute of Mathematics, Technical University of Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany

1Current address: Department of Mathematics and Statistics, State University of Rio Grande do Norte - UERN, Mossoró-RN, 59610-210 Brazil

Received  September 2017 Revised  April 2018 Published  August 2018

We investigate the stability of synchronization in networks of dynamical systems with strongly delayed connections. We obtain strict conditions for synchronization of periodic and equilibrium solutions. In particular, we show the existence of a critical coupling strength $κ_{c}$, depending only on the network structure, isolated dynamics and coupling function, such that for large delay and coupling strength $κ<κ_{c}$, the network possesses stable synchronization. The critical coupling $κ_{c}$ can be chosen independently of the delay for the case of equilibria, while for the periodic solution, $κ_{c}$ depends essentially on the delay and vanishes as the delay increases. We observe that, for random networks, the synchronization interval is maximal when the network is close to the connectivity threshold. We also derive scaling of the coupling parameter that allows for a synchronization of large networks for different network topologies.

Citation: Daniel M. N. Maia, Elbert E. N. Macau, Tiago Pereira, Serhiy Yanchuk. Synchronization in networks with strongly delayed couplings. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3461-3482. doi: 10.3934/dcdsb.2018234
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Synchronization map in the $\sigma\times\tau$ parameter space. The color scale represents $\Re(\lambda)<0$ in which $\lambda$ is a solution of (6.7) with maximal real part. The white color stands for the instability region ($\Re(\lambda)>0$). The parameters used were $\alpha = 1$ and $\beta = \pi$.
Numerically-computed-spectrum}Numerically computed spectrum for the equilibrium of two Stuart-Landau oscillators (see Sec. 6) coupled as in (1.1) with identity coupling function and parameters $\alpha = 1$, $\beta = \pi$, $\tau = 20$, and $\kappa = 0.7$. The points approaching the curve on the left side of the figure belongs to the pseudo-continuous spectrum and the isolated points on the right belongs to the strongly unstable spectrum. Solid lines show the re-scaled asymptotic continuous spectrum $\Gamma_{A}$. The gray strip represents a break on the figure, which is necessary due to the different scales of the two parts of the spectrum.
A directed ring network with 4 nodes.
Time series of the synchronization error for Eq. (3.3) and network given in Fig. 6.1 where $\kappa = 0.49$ for the left figure and $\kappa = 0.51$ for the right. Other parameters are $\alpha = -1$, $\beta = \pi$, $\tau = 100$. The history functions were taken as constant and non-zero.
The asymptotic continuous spectrum (blue lines) and the pseudo-continuous spectrum (red dots) for the periodic solution of Stuart-Landau system given by Eqs. (6.6) and (6.7) respectively. The parameters are $\sigma = -0.08$ (with $\mu = \rho_{L} = 2$, the spectral radius of the Laplacian matrix of the network in Fig. 6.1, and $\kappa = 0.04$), $\alpha = 1$, $\beta = \pi$ and $\tau = 20$.
Characteristic time for the synchronization of two Stuart-Landau coupled oscillators. The red curve is $t_{\text{tr}}(\kappa) = 20\ln^{-1}(2\kappa)$. The blue dots were obtained by fixing $\kappa$ and computing $\eta$, which stands for the angular coefficient of Eq. (6.8) in log scale in which $||\xi(t)|| = ||x_{1}(t)-x_{2}(t)||$, and then taking $t_{\text{tr}} = 1/\eta$. The parameters used were $\alpha = -1$, $\beta = \pi$ and $\tau = 20$. The history functions were taken as constant and non-zero.
Illustrations of a BA network (left) and an ER network (right), both with $n = 100$. Some hubs in the BA network are highlighted with black color and bigger size.
Laplacian spectral radius $\rho_{L}$ and synchronization window for the coupling parameter $\kappa$ (for strong delay) of some regular graphs.
 Graph $\rho_{L}$ Synchronization window Complete $n$ $(0, r_{0}/n)$ Ring $\begin{array}{l} {\rm{4}}\;{\rm{if}}\;{n}\;{\rm{is}}\;{\rm{even}}\\ 2 + 2\cos \left( {2{\rm{\pi }}/n} \right)\;{\rm{if}}\;{n}\;{\rm{is}}\;{\rm{odd}} \end{array}$ $(0, r_{0}/4)$ or $\left(0, r_{0}/(2+2\cos\left({2\pi}/{n}\right))\right)$ Star $n$ $(0, r_{0}/n)$ Path $2+2\cos\left({\pi}/{n}\right)$ $\left(0, r_{0}/\left(2+2\cos\left({\pi}/{n}\right)\right)\right)$
 Graph $\rho_{L}$ Synchronization window Complete $n$ $(0, r_{0}/n)$ Ring $\begin{array}{l} {\rm{4}}\;{\rm{if}}\;{n}\;{\rm{is}}\;{\rm{even}}\\ 2 + 2\cos \left( {2{\rm{\pi }}/n} \right)\;{\rm{if}}\;{n}\;{\rm{is}}\;{\rm{odd}} \end{array}$ $(0, r_{0}/4)$ or $\left(0, r_{0}/(2+2\cos\left({2\pi}/{n}\right))\right)$ Star $n$ $(0, r_{0}/n)$ Path $2+2\cos\left({\pi}/{n}\right)$ $\left(0, r_{0}/\left(2+2\cos\left({\pi}/{n}\right)\right)\right)$
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