Periodic consensus in network systems with general distributed processing delays

Yicheng Liu; Yipeng Chen; Jun Wu; Xiao Wang

doi:10.3934/nhm.2021002

Article Contents

2021, Volume 16, Issue 1: 139-153. Doi: 10.3934/nhm.2021002

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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
^* Corresponding author: Yicheng Liu

Received: September 2020

Revised: November 2020

Early access: December 2020

Published: March 2021

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 93C23, 93D09, 93C95.

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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 $)

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Figure 2. 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 $)

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Figure 3. 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 $)

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Figure 4. 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 $

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Figure 5. 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 $

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Figure 6. 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 $)

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Figure 7. 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

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Figure 8. 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)

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Figure 9. 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)

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Figure 10. 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)

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Table 1. 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

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Table 2. 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).

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Table 3. The numerical simulations for Case Ⅰ

Cases	$ \tilde{\lambda} $	$ \tau $	Results
Uniform distribution (Fig. 1)	$ 9\pi^2 $	$ 0.3 $	consensus
Uniform distribution (Fig. 1)	$ 9\pi^2 $	$ 0.5 $	periodic consensus
Exponential distribution(Fig. 2)	$ 270 $	$ 1 $	consensus
Exponential distribution(Fig. 2)	$ 1050.5502 $	$ 1 $	periodic consensus
Special $ \gamma $-distribution 1(Fig. 3)	$ 13.5 $	$ 1 $	consensus
Special $ \gamma $-distribution 1(Fig. 3)	$ 34.3368 $	$ 1 $	periodic consensus
Special $ \gamma $-distribution 2(Fig. 4)	$ 9 $	$ 1 $	consensus
Special $ \gamma $-distribution 2(Fig. 4)	$ 24.3171 $	$ 1 $	periodic consensus
Bernoulli distribution(Fig. 5)	$ 9\pi $	$ 0.3 $	consensus
Bernoulli distribution(Fig. 5)	$ 9\pi $	$ 0.5 $	periodic consensus

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Table 4. 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
Uniform (Fig. 6)	$ \frac{15\pi^2}{2} $	$ 0.5 $	periodic consensus	divergence
Exponential (Fig. 7)	$ 700.3668 $	$ 1 $	consensus	periodic consensus
Exponential (Fig. 7)	$ 875.4585 $	$ 1 $	periodic consensus	divergence
Gamma 1(Fig. 8)	$ 22.8912 $	$ 1 $	consensus	periodic consensus
Gamma 1(Fig. 8)	$ 28.614 $	$ 1 $	periodic consensus	divergence
Gamma 2(Fig. 9)	$ 16.2114 $	$ 1 $	consensus	periodic consensus
Gamma 2(Fig. 9)	$ 20.2643 $	$ 1 $	periodic consensus	divergence
Bernoulli(Fig. 10)	$ 6\pi $	$ 0.5 $	consensus	periodic consensus
Bernoulli(Fig. 10)	$ \frac{15\pi}{2} $	$ 0.5 $	periodic consensus	divergence

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