On consensus in the Cucker–Smale type model on isolated time scales

Ewa Girejko; Luís Machado; Agnieszka B. Malinowska; Natália Martins

doi:10.3934/dcdss.2018005

Article Contents

2018, Volume 11, Issue 1: 77-89. Doi: 10.3934/dcdss.2018005

This issue Previous Article A necessary condition of Pontryagin type for fuzzy fractional optimal control problems Next Article Noether currents for higher-order variational problems of Herglotz type with time delay

On consensus in the Cucker–Smale type model on isolated time scales

1.
Faculty of Computer Science, Bialystok University of Technology, 15-351 Bia lystok, Poland
2.
Department of Mathematics, UTAD, 5001-801 Vila Real, Portugal
3.
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

* Corresponding author: A. B. Malinowska
* Corresponding author: A. B. Malinowska

Received: August 31, 2016

Revised: March 31, 2017

Published: February 2018

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Abstract

This article addresses a consensus phenomenon in a Cucker-Smale model where the magnitude of the step size is not necessarily a constant but it is a function of time. In the considered model the weights of mutual influences in the group of agents do not change. A sufficient condition under which the proposed model tends to a consensus is obtained. This condition strikingly demonstrates the importance of the graininess function in a consensus phenomenon. The results are illustrated by numerical simulations.

Keywords:

Mathematics Subject Classification: Primary: 39A12; Secondary: 34N05.

Citation:

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Figure 1. Time evolution of 5 consensus parameters with 30 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=\left\{1-\frac{1}{k}\, :\, k=1, \ldots, 10\right\}$ $\cup$ $\bigl\{t_k=1+2.5\sum_{i=0}^k|\sin(i)|\, :\, k\in \mathbb{N}_0\bigr\}.$

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Figure 2. Time evolution of 5 consensus parameters with 20 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=\left\{1-\frac{1}{k}\, :\, k=1, \ldots, 10\right\}$ $\cup $ $\bigl\{t_k=1+6\sum_{i=0}^k|\sin(i)|\, :\, k\in \mathbb{N}_0\bigr\}.$

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Figure 3. Time evolution of 5 consensus parameters with 50 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=\left\{0;0.5; 0.75; 0.875; 1.375; 1.625;\ldots\right\}$

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Figure 4. Time evolution of 5 consensus parameters with 150 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=\left\{1-\frac{1}{k}~:~k=1, \ldots, 50\right\} \cup \left\{1+1.2771 k~:~k\in \mathbb{N}_0\right\}$

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Figure 5. Time evolution of 5 consensus parameters with 200 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=1.2871\mathbb{N}_0$

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Figure 6. Time evolution of 5 consensus parameters with 300 iterations (left) and their states in the last 16 iterations (right) on the time scale when $\mu=1.2771, 1.2871, 1.2771, 1.2871, \ldots.$

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Figure 7. Time evolution of 30 consensus parameters with 40 iterations (left) and their states (right) on $\mathbb{T}=\bigl\{t_n=\sum_{k=1}^n\frac{1}{k}\, :\, n\in\mathbb{N} \bigr\}.$

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Figure 8. Time evolution of 30 consensus parameters with 50 iterations (left) and their states (right) on the time scale when $t_0=0$ and $\mu=\frac{1}{4}, \frac{5}{2}, 2, \frac{1}{4}, \frac{5}{2}, 2, \ldots.$

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Figure 9. Time evolution of 30 consensus parameters with 20 iterations (left) and their states (right) on the time scale when $t_0=0$ and $\mu=\frac{1}{4}, \frac{3}{4}, 2, \frac{1}{4}, \frac{3}{4}, 2, \ldots.$

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