    March  2021, 16(1): 139-153. doi: 10.3934/nhm.2021002

## Periodic consensus in network systems with general distributed processing delays

 1 College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, P. R. China 2 College of Mathematics and Statistics, Changsha University of Science Technology, Changsha, 410076, P. R. China

* Corresponding author: Yicheng Liu

Received  September 2020 Revised  November 2020 Published  December 2020

Fund Project: This work was partially supported by National Natural Science Foundation of China (11671011 and 11428101)

How to understand the dynamical consensus patterns in network systems is of particular significance in both theories and applications. In this paper, we are interested in investigating the influences of distributed processing delay on the consensus patterns in a network model. As new observations, we show that the desired network model undergoes both weak consensus and periodic consensus behaviors when the parameters reach a threshold value and the connectedness of the network system may be absent. In results, some criterions of weak consensus and periodic consensus with exponential convergent rate are established by the standard functional differential equations analysis. An analytic formula is given to calculate the asymptotic periodic consensus in terms of model parameters and the initial time interval. Also, we post the threshold values for some typical distributions included uniform distribution and Gamma distribution. Finally, we give the numerical simulation and analyse the influences of different delays on the consensus.

Citation: Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks & Heterogeneous Media, 2021, 16 (1) : 139-153. doi: 10.3934/nhm.2021002
##### References:
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##### References:
  M. H. DeGroot, Reaching a consensus, J. Am. Stat. Assoc., 69 (1974), 118-121.   Google Scholar  J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Autom. Control, 49 (2004), 1465-1476.  doi: 10.1109/TAC.2004.834433.  Google Scholar  Q. Feng, S. K. Nguang and A. Seuret, Stability analysis of linear coupled differential–difference systems with general distributed delays, IEEE Transactions on Automatic Control, 65 (2020), 1356-1363.  doi: 10.1109/TAC.2019.2928145.  Google Scholar  J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar  J. Jost, F. M. Atay and W. Lu, Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays, Networks and Heterogeneous Media, 6 (2011), 329-349.  doi: 10.3934/nhm.2011.6.329.  Google Scholar  F. A. Khasawneh and B. P. Mann, A spectral element approach for the stability of delay systems, Int. J. Numer. Meth. Engng., 87 (2011), 566-592. Google Scholar  N. A. Lynch, Distributed Algorithms, San Francisco, CA: Morgan Kaufmann, 1996. doi: 10.1108/IMDS-01-2014-0013.  Google Scholar  F. Mazenc, M. Malisoff and H. $\ddot{O}$zbay, Stability and robustness analysis for switched systems with time-varying delays, SIAM J. Control Optim., 56 (2018), 158-182.  doi: 10.1137/16M1104895.  Google Scholar  M. Michiels, I. C. Morărescu and S.-I. Niculescu, Consensus problems with distributed delays, with application to traffic flow models, SIAM J. Control Optim., 48 (2009), 77-101.  doi: 10.1137/060671425.  Google Scholar  I. C. Morarescu, W. Michiels and M. Jungers, Synchronization of coupled nonlinear oscillatiors with gamma-distributed delays, in American Control Conference, ACC 2013, (2013), Washington, United States. Google Scholar  R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533.  doi: 10.1109/TAC.2004.834113.  Google Scholar  A. V. Proskurnikov, Average consensus in networks with nonlinearly delayed couplings and switching topology, Automatica, 49 (2013) 2928–2932. doi: 10.1016/j.automatica.2013.06.007.  Google Scholar  D. Serre, Matrices, Graduate Texts in Mathematics, 216 (2010), Springer. doi: 10.1007/978-1-4419-7683-3.  Google Scholar  J. W. H. So, X. Tang and X. Zou, Stability in a linear delay system without instantaneous negative feedback, SIAM J. Math. Anal., 33 (2002), 1297-1304.  doi: 10.1137/S0036141001389263.  Google Scholar  O. Solomon and E. Fridman, New stability conditions for systems with distributed delays, Automatica, 49 (2013), 3467-3475.  doi: 10.1016/j.automatica.2013.08.025.  Google Scholar  T. Vicsek, A Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar ). According to Theorem 2.1, if $\tilde{\lambda}\tau(1-\frac{8}{9})<\frac{\pi^2}{2}$, the system achieves a consensus(left:$\tilde{\lambda} = 9\pi^2$ and $\tau = 0.3$). When $\tilde{\lambda}\tau(1-\frac{8}{9}) = \frac{\pi^2}{2}$, the system achieves a periodic consensus(right: $\tilde{\lambda} = 9\pi^2$ and $\tau = 0.5$)">Figure 1.  Consensus and periodic consensus with a uniform distribution delay. $\varphi(s) = \frac{1}{\tau}$, $k^* = \frac{\pi^2}{2}$(Tab. 1). According to Theorem 2.1, if $\tilde{\lambda}\tau(1-\frac{8}{9})<\frac{\pi^2}{2}$, the system achieves a consensus(left:$\tilde{\lambda} = 9\pi^2$ and $\tau = 0.3$). When $\tilde{\lambda}\tau(1-\frac{8}{9}) = \frac{\pi^2}{2}$, the system achieves a periodic consensus(right: $\tilde{\lambda} = 9\pi^2$ and $\tau = 0.5$) Consensus and periodic consensus with an exponential distribution delay. $\varphi(s) = \frac{\alpha e^{\alpha\tau}}{e^{\alpha\tau}-1}e^{\alpha s}(\alpha = 2,\tau = 1)$, $k^* = 116.7278$. The critical condition is that $\tilde{\lambda}\tau(1-\frac{8}{9})<116.7278$. Thus, the left one is a consensus($\tilde{\lambda} = 270$ and $\tau = 1$) and the right one is a periodic consensus($\tilde{\lambda} = 1050.5502$ and $\tau = 1$) Consensus and periodic consensus with a Gamma distribution delay. $\varphi(s) = \frac{\alpha^2e^{\alpha\tau}}{e^{\alpha\tau}-\alpha\tau-1}|s|e^{\alpha s} (\alpha = 2,\tau = 1)$, $k^* = 3.8152$. Similarly, the left one is a consensus($\tilde{\lambda} = 13.5$ and $\tau = 1$) and the right one is a periodic consensus($\tilde{\lambda} = 34.3368$ and $\tau = 1$) Consensus and periodic consensus with a Gamma distribution delay. $\varphi(s) = \frac{\alpha^3e^{\alpha\tau}}{2e^{\alpha\tau}-(\alpha\tau+1)^2-1}s^2e^{\alpha s} (\alpha = 2,\tau = 1)$, $k^* = 2.7019$. The left one is the case $\tilde{\lambda} = 9$ and $\tau = 1$. The right one is the case $\tilde{\lambda} = 24.3171$ and $\tau = 1$ Consensus and periodic consensus with a Bernoulli distribution delay. $\varphi(s) = 0$ for $s\in (-\tau,0]$ and $\varphi(s) = 1$ for $s = -\tau$, $k^* = \frac{\pi}{2}$. The left one is the case $\tilde{\lambda} = 9\pi$ and $\tau = 0.3$. The right one is the case $\tilde{\lambda} = 9\pi$ and $\tau = 0.5$ Clustering consensus with a uniform distribution delay. $\varphi(s) = \frac{1}{\tau}$, $k^* = \frac{\pi^2}{2}$. According to Theorem 2.1, if $\tilde{\lambda}\tau(1-\frac{5}{6}) = \frac{\pi^2}{2}$, the nodes in Group 1(blue line) achieve a consensus and the ones in Group 2(red line) achieve a periodic consensus(left: $\tilde{\lambda} = 6\pi^2$ and $\tau = 0.5$). When $\tilde{\lambda}\tau(1-\frac{13}{15}) = \frac{\pi^2}{2}$, the nodes in Group 1(blue line)achieve a periodic consensus and the others in Group 2(red line) are divergence(right: $\tilde{\lambda} = \frac{15\pi^2}{2}$ and $\tau = 0.5$) Clustering consensus with an exponential distribution delay. $\varphi(s) = \frac{\alpha e^{\alpha\tau}}{e^{\alpha\tau}-1}e^{\alpha s}(\alpha = 2,\tau = 1)$, $k^* = 116.7278$. Similarly, the left one is the case $\tilde{\lambda} = 700.3668$ and $\tau = 1$, which is that Group 1(blue line) achieves a consensus and Group 2(red line) achieves a periodic consensus. The right one is the case $\tilde{\lambda} = 875.4585$ and $\tau = 1$, which is that Group 1(blue line)achieves a periodic consensus and Group 2(red line) is divergence Clustering consensus with a Gamma distribution delay. $\varphi(s) = \frac{\alpha^2e^{\alpha\tau}}{e^{\alpha\tau}-\alpha\tau-1}|s|e^{\alpha s} (\alpha = 2,\tau = 1)$, $k^* = 3.8152$. In the case of $\tilde{\lambda} = 22.8912$ and $\tau = 1$, the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case $\tilde{\lambda} = 28.614$ and $\tau = 1$, the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right) Clustering consensus with a Gamma distribution delay. $\varphi(s) = \frac{\alpha^3e^{\alpha\tau}}{2e^{\alpha\tau}-(\alpha\tau+1)^2-1}s^2e^{\alpha s}$ $(\alpha = 2,\tau = 1)$, $k^* = 2.7019$. In the case of $\tilde{\lambda} = 16.2114$ and $\tau = 1$, the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case $\tilde{\lambda} = 20.2643$ and $\tau = 1$, the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right) Clustering consensus with a Bernoulli distribution delay. $\varphi(s) = 0$ for $s\in (-\tau,0]$ and $\varphi(s) = 1$ for $s = -\tau$, $k^* = \frac{\pi}{2}$. In the case of $\tilde{\lambda} = 6\pi$ and $\tau = 0.5$, the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) achieve a periodic consensus(left). In the case $\tilde{\lambda} = \frac{15\pi}{2}$ and $\tau = 0.5$, the nodes in Group 1(blue line)achieve a consensus and the others in Group 2(red line) are divergence(right)
The values of $k^*$ and $y_{im}$ for some special cases
 Cases $k^*$ $y_{im}$ Descriptions $\varphi(s)=\frac{1}{\tau}$ $\frac{\pi^2}{2}$ $\frac{\pi}{\tau}$ Uniform distribution $\varphi(s)=\frac{2e^{2}}{e^{2}-1}e^{2s}$ 116.7278 16.8680 Exponential distribution $\varphi(s)=\frac{4e^{2}}{e^{2}-3}|s|e^{2s}$ 3.8152 2.8801 Special $\gamma$-distribution $\varphi(s)=\frac{4e^{2}}{e^{2}-5}s^2e^{2s}$ 2.7019 2.3530 Special $\gamma$-distribution $\varphi(s)=\left\{\begin{array}{ll} 0, & s \in(-\tau, 0] \\ 1, & s=-\tau \end{array}\right.$ $\frac{\pi}{2}$ $\frac{\pi}{2\tau}$ Bernoulli distribution
 Cases $k^*$ $y_{im}$ Descriptions $\varphi(s)=\frac{1}{\tau}$ $\frac{\pi^2}{2}$ $\frac{\pi}{\tau}$ Uniform distribution $\varphi(s)=\frac{2e^{2}}{e^{2}-1}e^{2s}$ 116.7278 16.8680 Exponential distribution $\varphi(s)=\frac{4e^{2}}{e^{2}-3}|s|e^{2s}$ 3.8152 2.8801 Special $\gamma$-distribution $\varphi(s)=\frac{4e^{2}}{e^{2}-5}s^2e^{2s}$ 2.7019 2.3530 Special $\gamma$-distribution $\varphi(s)=\left\{\begin{array}{ll} 0, & s \in(-\tau, 0] \\ 1, & s=-\tau \end{array}\right.$ $\frac{\pi}{2}$ $\frac{\pi}{2\tau}$ Bernoulli distribution
Initial values $x_i(\theta)(i = 1,2,...,N)$, $\theta\in[-\tau,0]$
 $x_1(\theta)$ $x_2(\theta)$ $x_3(\theta)$ $x_4(\theta)$ $x_5(\theta)$ 7.0605 0.3183 2.7692 0.4617 0.9713 $x_6(\theta)$ $x_7(\theta)$ $x_8(\theta)$ $x_9(\theta)$ $x_{10}(\theta)$ 8.2346 6.9483 3.1710 9.5022 0.3445 where the numbers are randomly selected in interval (0, 10).
 $x_1(\theta)$ $x_2(\theta)$ $x_3(\theta)$ $x_4(\theta)$ $x_5(\theta)$ 7.0605 0.3183 2.7692 0.4617 0.9713 $x_6(\theta)$ $x_7(\theta)$ $x_8(\theta)$ $x_9(\theta)$ $x_{10}(\theta)$ 8.2346 6.9483 3.1710 9.5022 0.3445 where the numbers are randomly selected in interval (0, 10).
The numerical simulations for Case Ⅰ
 Cases $\tilde{\lambda}$ $\tau$ Results Uniform distribution (Fig. 1) $9\pi^2$ $0.3$ consensus $9\pi^2$ $0.5$ periodic consensus Exponential distribution(Fig. 2) $270$ $1$ consensus $1050.5502$ $1$ periodic consensus Special $\gamma$-distribution 1(Fig. 3) $13.5$ $1$ consensus $34.3368$ $1$ periodic consensus Special $\gamma$-distribution 2(Fig. 4) $9$ $1$ consensus $24.3171$ $1$ periodic consensus Bernoulli distribution(Fig. 5) $9\pi$ $0.3$ consensus $9\pi$ $0.5$ periodic consensus
 Cases $\tilde{\lambda}$ $\tau$ Results Uniform distribution (Fig. 1) $9\pi^2$ $0.3$ consensus $9\pi^2$ $0.5$ periodic consensus Exponential distribution(Fig. 2) $270$ $1$ consensus $1050.5502$ $1$ periodic consensus Special $\gamma$-distribution 1(Fig. 3) $13.5$ $1$ consensus $34.3368$ $1$ periodic consensus Special $\gamma$-distribution 2(Fig. 4) $9$ $1$ consensus $24.3171$ $1$ periodic consensus Bernoulli distribution(Fig. 5) $9\pi$ $0.3$ consensus $9\pi$ $0.5$ periodic consensus
The numerical simulations for Case Ⅱ
 Distribution cases $\tilde{\lambda}$ $\tau$ Group 1(blue) Group 2(red) Uniform (Fig. 6) $6\pi^2$ $0.5$ consensus periodic consensus $\frac{15\pi^2}{2}$ $0.5$ periodic consensus divergence Exponential (Fig. 7) $700.3668$ $1$ consensus periodic consensus $875.4585$ $1$ periodic consensus divergence Gamma 1(Fig. 8) $22.8912$ $1$ consensus periodic consensus $28.614$ $1$ periodic consensus divergence Gamma 2(Fig. 9) $16.2114$ $1$ consensus periodic consensus $20.2643$ $1$ periodic consensus divergence Bernoulli(Fig. 10) $6\pi$ $0.5$ consensus periodic consensus $\frac{15\pi}{2}$ $0.5$ periodic consensus divergence
 Distribution cases $\tilde{\lambda}$ $\tau$ Group 1(blue) Group 2(red) Uniform (Fig. 6) $6\pi^2$ $0.5$ consensus periodic consensus $\frac{15\pi^2}{2}$ $0.5$ periodic consensus divergence Exponential (Fig. 7) $700.3668$ $1$ consensus periodic consensus $875.4585$ $1$ periodic consensus divergence Gamma 1(Fig. 8) $22.8912$ $1$ consensus periodic consensus $28.614$ $1$ periodic consensus divergence Gamma 2(Fig. 9) $16.2114$ $1$ consensus periodic consensus $20.2643$ $1$ periodic consensus divergence Bernoulli(Fig. 10) $6\pi$ $0.5$ consensus periodic consensus $\frac{15\pi}{2}$ $0.5$ periodic consensus divergence
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