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

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

Ewa Girejko 1, , Luís Machado 2, , Agnieszka B. Malinowska 1,, and  Natália Martins 3,

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.

Keywords: Consensus formation, opinion dynamics, Cucker–Smale model, time scale.
Mathematics Subject Classification: Primary: 39A12; Secondary: 34N05.
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
References:
[1]

F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics, Math. Comput. Modelling, 43 (2006), 718-726.  doi: 10.1016/j.mcm.2005.08.014.  Google Scholar

[2]

B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in Qualitative Theory of Differential Equations (Szeged, 1988), Colloq. Math. Soc. J. Bolyai, 53, North-Holland, Amsterdam, 1990, 37–56.  Google Scholar

[3]

Z. Bartosiewicz, N. Martins and D. F. M. Torres, The second Euler–Lagrange equation of variational calculus on time scales, European Journal of Control, 17 (2011), 9-18.  doi: 10.3166/ejc.17.9-18.  Google Scholar

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Z. Bartosiewicz, Linear positive control systems on time scales: Controllability, Mathematics of Control, Signals and Systems, 25 (2013), 327-343.  doi: 10.1007/s00498-013-0106-6.  Google Scholar

[5]

J. Belikov, U. Kotta and M. Tonso, Realization of nonlinear MIMO system on homogeneous time scales, European Journal of Control, 23 (2015), 48-54.  doi: 10.1016/j.ejcon.2015.01.006.  Google Scholar

[6]

V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, On Krause's multi-agent consensus model with state-dependent connectivity, IEEE Transactions on Automatics Control, 54 (2009), 2586-2597.  doi: 10.1109/TAC.2009.2031211.  Google Scholar

[7]

V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, Continuous-time average-preserving opinion dynamics with opinion-dependent communications, SIAM J. Control Optim., 48 (2010), 5214-5240.  doi: 10.1137/090766188.  Google Scholar

[8]

M. Bohner, M. Fan and J. Zhang, Periodicity of scalar dynamic equations on time scales and applications to population models, J. Math. Anal. Appl., 330 (2007), 1-9.  doi: 10.1016/j.jmaa.2006.04.084.  Google Scholar

[9]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales Birkhäuser Boston, Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[10]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales Birkhäuser Boston, Boston, MA, 2003. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[11]

M. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015.  doi: 10.1080/00036810701474140.  Google Scholar

[12]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and optimal control of the Cucker-Smale model, Mathematics of Control and Related Fields, 3 (2013), 447-466.  doi: 10.3934/mcrf.2013.3.447.  Google Scholar

[13]

D. Casagrande, U. Kotta, M. Tonso and M. Wyrwas, Transfer equivalence and realization of nonlinear input-output delta-differential equations on homogeneous time scales, IEEE Transactions on Automatic Control, 55 (2010), 2601-2606.  doi: 10.1109/TAC.2010.2060251.  Google Scholar

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S. Chatterjee and E. Seneta, Towards consensus: Some convergence theorems on repeated averaging, J. Appl. Prob., 14 (1977), 89-97.  doi: 10.1017/S0021900200104681.  Google Scholar

[15]

Cucker Smale and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[16]

Cucker Smale and S. Smale, On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[17]

M. H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121.  doi: 10.1080/01621459.1974.10480137.  Google Scholar

[18]

J. French, A formal theory of social power, Social Networks, A Developing Paradigm, (1977), 35-48.  doi: 10.1016/B978-0-12-442450-0.50010-9.  Google Scholar

[19]

G. Fu, W. Zhang and Z. Li, Opinion dynamics of modified Hegselmann–Krause model in a group-based population with heterogeneous bounded confidence, Physica A, 419 (2015), 558-565.  doi: 10.1016/j.physa.2014.10.045.  Google Scholar

[20]

E. Girejko, L. Machado, A. B. Malinowska and N. Martins, Krause's model of opinion dynamics on isolated time scales, Mathematical Methods in the Applied Sciences, 39 (2016), 5302-5314.  doi: 10.1002/mma.3916.  Google Scholar

[21]

E. Girejko, A. B. Malinowska and D. F. M. Torres, The contingent epiderivative and the calculus of variations on time scales, Optimization Letters, 61 (2012), 251-264.  doi: 10.1080/02331934.2010.506615.  Google Scholar

[22]

E. Girejko and D. F. M. Torres, The existence of solutions for dynamic inclusions on time scales via duality, Applied Mathematic Letters, 25 (2012), 1632-1637.  doi: 10.1016/j.aml.2012.01.026.  Google Scholar

[23]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis, and simulation, J. Artificial Societies and Social Simulations, 5 (2002), 1-33.   Google Scholar

[24]

S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten Ph. D thesis, Universität Würzburg, 1988. Google Scholar

[25]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, Comunications in Difference Equations (eds. S. Elaydi, G. Ladas, J. Popenda and J. Rakowski), Gordon and Breach Publ. , Amsterdam, 2000,227–236.  Google Scholar

[26]

K. Lehrer and C. Wagner, Rational Consensus in Science and Society D. Reidel Publishing Company, Dordrecht, Holland, 1981.  Google Scholar

[27]

A. B. Malinowska and D. F. M. Torres, Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci., 33 (2010), 1712-1722.  doi: 10.1002/mma.1289.  Google Scholar

[28]

A. B. Malinowska, N. Martins and D. F. M. Torres, Transversality conditions for infinite horizon variational problems on time scales, Optim. Lett., 5 (2011), 41-53.  doi: 10.1007/s11590-010-0189-7.  Google Scholar

[29]

N. Martins and D. F. M. Torres, Calculus of variations on time scales with nabla derivatives, Nonlinear Anal., 71 (2009), e763-e773.  doi: 10.1016/j.na.2008.11.035.  Google Scholar

[30]

T. Vicsek, A. Czirók, E. Ben-Jacob and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Letters, 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[31]

H. Wang and L. Shang, Opinion dynamics in networks with common-neighbors-based connections, Physica A, 421 (2015), 180-186.  doi: 10.1016/j.physa.2014.10.090.  Google Scholar

show all references

References:
[1]

F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics, Math. Comput. Modelling, 43 (2006), 718-726.  doi: 10.1016/j.mcm.2005.08.014.  Google Scholar

[2]

B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in Qualitative Theory of Differential Equations (Szeged, 1988), Colloq. Math. Soc. J. Bolyai, 53, North-Holland, Amsterdam, 1990, 37–56.  Google Scholar

[3]

Z. Bartosiewicz, N. Martins and D. F. M. Torres, The second Euler–Lagrange equation of variational calculus on time scales, European Journal of Control, 17 (2011), 9-18.  doi: 10.3166/ejc.17.9-18.  Google Scholar

[4]

Z. Bartosiewicz, Linear positive control systems on time scales: Controllability, Mathematics of Control, Signals and Systems, 25 (2013), 327-343.  doi: 10.1007/s00498-013-0106-6.  Google Scholar

[5]

J. Belikov, U. Kotta and M. Tonso, Realization of nonlinear MIMO system on homogeneous time scales, European Journal of Control, 23 (2015), 48-54.  doi: 10.1016/j.ejcon.2015.01.006.  Google Scholar

[6]

V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, On Krause's multi-agent consensus model with state-dependent connectivity, IEEE Transactions on Automatics Control, 54 (2009), 2586-2597.  doi: 10.1109/TAC.2009.2031211.  Google Scholar

[7]

V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, Continuous-time average-preserving opinion dynamics with opinion-dependent communications, SIAM J. Control Optim., 48 (2010), 5214-5240.  doi: 10.1137/090766188.  Google Scholar

[8]

M. Bohner, M. Fan and J. Zhang, Periodicity of scalar dynamic equations on time scales and applications to population models, J. Math. Anal. Appl., 330 (2007), 1-9.  doi: 10.1016/j.jmaa.2006.04.084.  Google Scholar

[9]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales Birkhäuser Boston, Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[10]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales Birkhäuser Boston, Boston, MA, 2003. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[11]

M. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015.  doi: 10.1080/00036810701474140.  Google Scholar

[12]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and optimal control of the Cucker-Smale model, Mathematics of Control and Related Fields, 3 (2013), 447-466.  doi: 10.3934/mcrf.2013.3.447.  Google Scholar

[13]

D. Casagrande, U. Kotta, M. Tonso and M. Wyrwas, Transfer equivalence and realization of nonlinear input-output delta-differential equations on homogeneous time scales, IEEE Transactions on Automatic Control, 55 (2010), 2601-2606.  doi: 10.1109/TAC.2010.2060251.  Google Scholar

[14]

S. Chatterjee and E. Seneta, Towards consensus: Some convergence theorems on repeated averaging, J. Appl. Prob., 14 (1977), 89-97.  doi: 10.1017/S0021900200104681.  Google Scholar

[15]

Cucker Smale and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[16]

Cucker Smale and S. Smale, On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[17]

M. H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121.  doi: 10.1080/01621459.1974.10480137.  Google Scholar

[18]

J. French, A formal theory of social power, Social Networks, A Developing Paradigm, (1977), 35-48.  doi: 10.1016/B978-0-12-442450-0.50010-9.  Google Scholar

[19]

G. Fu, W. Zhang and Z. Li, Opinion dynamics of modified Hegselmann–Krause model in a group-based population with heterogeneous bounded confidence, Physica A, 419 (2015), 558-565.  doi: 10.1016/j.physa.2014.10.045.  Google Scholar

[20]

E. Girejko, L. Machado, A. B. Malinowska and N. Martins, Krause's model of opinion dynamics on isolated time scales, Mathematical Methods in the Applied Sciences, 39 (2016), 5302-5314.  doi: 10.1002/mma.3916.  Google Scholar

[21]

E. Girejko, A. B. Malinowska and D. F. M. Torres, The contingent epiderivative and the calculus of variations on time scales, Optimization Letters, 61 (2012), 251-264.  doi: 10.1080/02331934.2010.506615.  Google Scholar

[22]

E. Girejko and D. F. M. Torres, The existence of solutions for dynamic inclusions on time scales via duality, Applied Mathematic Letters, 25 (2012), 1632-1637.  doi: 10.1016/j.aml.2012.01.026.  Google Scholar

[23]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis, and simulation, J. Artificial Societies and Social Simulations, 5 (2002), 1-33.   Google Scholar

[24]

S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten Ph. D thesis, Universität Würzburg, 1988. Google Scholar

[25]

U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, Comunications in Difference Equations (eds. S. Elaydi, G. Ladas, J. Popenda and J. Rakowski), Gordon and Breach Publ. , Amsterdam, 2000,227–236.  Google Scholar

[26]

K. Lehrer and C. Wagner, Rational Consensus in Science and Society D. Reidel Publishing Company, Dordrecht, Holland, 1981.  Google Scholar

[27]

A. B. Malinowska and D. F. M. Torres, Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci., 33 (2010), 1712-1722.  doi: 10.1002/mma.1289.  Google Scholar

[28]

A. B. Malinowska, N. Martins and D. F. M. Torres, Transversality conditions for infinite horizon variational problems on time scales, Optim. Lett., 5 (2011), 41-53.  doi: 10.1007/s11590-010-0189-7.  Google Scholar

[29]

N. Martins and D. F. M. Torres, Calculus of variations on time scales with nabla derivatives, Nonlinear Anal., 71 (2009), e763-e773.  doi: 10.1016/j.na.2008.11.035.  Google Scholar

[30]

T. Vicsek, A. Czirók, E. Ben-Jacob and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Letters, 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[31]

H. Wang and L. Shang, Opinion dynamics in networks with common-neighbors-based connections, Physica A, 421 (2015), 180-186.  doi: 10.1016/j.physa.2014.10.090.  Google Scholar

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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