# American Institute of Mathematical Sciences

February  2018, 11(1): 77-89. doi: 10.3934/dcdss.2018005

## 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

Received  September 2016 Revised  April 2017 Published  January 2018

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.

Citation: Ewa Girejko, Luís Machado, Agnieszka B. Malinowska, Natália Martins. On consensus in the Cucker–Smale type model on isolated time scales. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 77-89. doi: 10.3934/dcdss.2018005
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##### References:
Time evolution of 5 consensus parameters with 30 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=\left\{1-\frac{1}{k}\, :\, k=1, \ldots, 10\right\}$ $\cup$ $\bigl\{t_k=1+2.5\sum_{i=0}^k|\sin(i)|\, :\, k\in \mathbb{N}_0\bigr\}.$
Time evolution of 5 consensus parameters with 20 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=\left\{1-\frac{1}{k}\, :\, k=1, \ldots, 10\right\}$ $\cup$ $\bigl\{t_k=1+6\sum_{i=0}^k|\sin(i)|\, :\, k\in \mathbb{N}_0\bigr\}.$
Time evolution of 5 consensus parameters with 50 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=\left\{0;0.5; 0.75; 0.875; 1.375; 1.625;\ldots\right\}$
Time evolution of 5 consensus parameters with 150 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=\left\{1-\frac{1}{k}~:~k=1, \ldots, 50\right\} \cup \left\{1+1.2771 k~:~k\in \mathbb{N}_0\right\}$
Time evolution of 5 consensus parameters with 200 iterations (left) and their states in the last 16 iterations (right) on $\mathbb{T}=1.2871\mathbb{N}_0$
Time evolution of 5 consensus parameters with 300 iterations (left) and their states in the last 16 iterations (right) on the time scale when $\mu=1.2771, 1.2871, 1.2771, 1.2871, \ldots.$
Time evolution of 30 consensus parameters with 40 iterations (left) and their states (right) on $\mathbb{T}=\bigl\{t_n=\sum_{k=1}^n\frac{1}{k}\, :\, n\in\mathbb{N} \bigr\}.$
Time evolution of 30 consensus parameters with 50 iterations (left) and their states (right) on the time scale when $t_0=0$ and $\mu=\frac{1}{4}, \frac{5}{2}, 2, \frac{1}{4}, \frac{5}{2}, 2, \ldots.$
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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