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August  2022, 21(8): 2831-2856. doi: 10.3934/cpaa.2022077

Emergent dynamics of the fractional Cucker-Smale model under general network topologies

Jinwook Jung 1, and  Peter Kuchling 2,,

1. 

Research Institute of Basic Sciences, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Republic of Korea
2. 

Department of Mathematics and Informatics, University of Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany

*Corresponding author

Received  January 2022 Revised  March 2022 Published  August 2022 Early access  April 2022

Fund Project: The first author is supported by NRF grant No. 2019R1A6A1A10073437

We study the fractional Cucker-Smale (in short, CS) model under general network topologies. In [15], the authors introduced the fractional CS model to see the interplay of memory effect and the flocking dynamics in the all-to-all network topology. As an extension of the previous work, we investigate under which network topologies flocking still emerges. Specifically, we first consider the symmetric network case and show that the existence of a leader guarantees the emergence of flocking. Furthermore, we present a framework for the non-symmetric network case where we can observe the flocking. We also conduct numerical simulations to support our theoretical results and see whether our framework gives necessary and sufficient conditions for the emergence of flocking.

Keywords: Caputo fractional derivative, collective motion, fractional calculus, fractional Cucker-Smale model, flocking, network topology.
Mathematics Subject Classification: Primary: 70F99, 92B25.
Citation: Jinwook Jung, Peter Kuchling. Emergent dynamics of the fractional Cucker-Smale model under general network topologies. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2831-2856. doi: 10.3934/cpaa.2022077
References:
[1]

S. Ahn and S. Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.

[2]

G. Albi, N. Bellomo, L. Fermo, S. Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[3]

R. Almeida, R. Kamocki, A. B. Malinowska and T. Odzijewicz, On the necessary optimality conditions for the fractional Cucker-Smale optimal control problem, Commun. Nonlinear Sci. Numer. Simul., 96 (2021), 105678, 22pp. doi: 10.1016/j.cnsns.2020.105678.

[4]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci., 105 (2008), 1232-1237. 

[5]

B. Bonilla, M. Rivero and J. J. Trujillo, On systems of linear fractional differential equations with constant coeffients, Appl. Math. Comput., 187 (2007), 68-78.  doi: 10.1016/j.amc.2006.08.104.

[6]

M. Caputo, Linear model of dissipation whose $Q$ is almost frequency independent-II, Geophys. J R. Astr. Soc., 13 (1967), 529-539. 

[7]

Y. P. Choi, S. Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol.I-Theory, Models, Applications, Birkhauser-Springer, 2017.

[8]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[9]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-14574-2.

[10]

J. G. Dong, S. Y. Ha and D. Kim, Emergent behaviors of continuous and discrete thermomechanical Cucker-Smale models on general digraphs, Math. Models Meth. Appl. Sci., 29 (2019), 589-632.  doi: 10.1142/S0218202519400013.

[11]

J. G. Dong and L. Qiu, Flocking of the Cucker-Smale model on general digraphs, IEEE Trans. Automat. Control, 62 (2017), 5234-5239.  doi: 10.1109/TAC.2016.2631608.

[12]

M. Eckert, K. Nagatou, F. Rey, O. Stark and S. Hohmann, Solution of time-variant fractional differential equations with a generalized Peano-Baker series, IEEE Control Syst. Lett., 3 (2019), 79-84. 

[13]

E. Girejko, D. Mozyrska and M. Wyrwas, Numerical analysis of behaviour of the Cucker-Smale type models with fractional operators, J. Comput. Appl. Math., 339 (2018), 111-123.  doi: 10.1016/j.cam.2017.12.013.

[14]

S. Y. Ha and J. Jung, Remarks on the slow relaxation for the fractional Kuramoto model for synchronization, J. Math. Phys., 59 (2018), 032703, 18pp. doi: 10.1063/1.5005865.

[15]

S. Y. Ha, J. Jung and P. Kuchling, Emergence of anomalous flocking in the fractional Cucker-Smale model, Discrete Contin. Dyn. Syst., 39 (2019), 5465-5489.  doi: 10.3934/dcds.2019223.

[16]

S. Y. Ha and J. G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325. 

[17]

S. Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[18]

C. Li, A. Chen and J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys., 230 (2011), 3352-3368.  doi: 10.1016/j.jcp.2011.01.030.

[19]

A. B. Malinowska, T. Odzijewicz and E. Schmeidel, On the existence of optimal controls for the fractional continuous-time Cucker-Smale model, in Theory and Applications of Non-integer Order Systems (eds. A. Babiarz, A. Czornik, J. Klamka, M. Niezabitowski), Springer International Publishing, (2017), 227–240.

[20]

M. Merkle, Completely monotone functions: a digest, Anal. Number Theor., Approx. Theor., Special Funct., (2014), 347–364.

[21]

S. Motsch and E. Tadmor, Heterophilious dynamics: Enhanced consensus, SIAM Rev., 56 (2014), 577-621.  doi: 10.1137/120901866.

[22]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guid. Control Dynam., 32 (2009) 527–537.

[23]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic press, 1998.

[24]

K. Sayevand, Fractional dynamical systems: A fresh view on the local qualitative theorems, Int. J. Nonlinear Anal. Appl., 7 (2016), 303-318. 

[25]

W. R. Schneider, Completely monotone generalized Mittag-Leffler functions, Expo. Math., 14 (1996), 3-16. 

[26]

E. D. Sontag, Mathematical Control Theory, $2^nd$ edition, Texts Appl. Math. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[27]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E., 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

[28]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

show all references

References:
[1]

S. Ahn and S. Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.

[2]

G. Albi, N. Bellomo, L. Fermo, S. Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[3]

R. Almeida, R. Kamocki, A. B. Malinowska and T. Odzijewicz, On the necessary optimality conditions for the fractional Cucker-Smale optimal control problem, Commun. Nonlinear Sci. Numer. Simul., 96 (2021), 105678, 22pp. doi: 10.1016/j.cnsns.2020.105678.

[4]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci., 105 (2008), 1232-1237. 

[5]

B. Bonilla, M. Rivero and J. J. Trujillo, On systems of linear fractional differential equations with constant coeffients, Appl. Math. Comput., 187 (2007), 68-78.  doi: 10.1016/j.amc.2006.08.104.

[6]

M. Caputo, Linear model of dissipation whose $Q$ is almost frequency independent-II, Geophys. J R. Astr. Soc., 13 (1967), 529-539. 

[7]

Y. P. Choi, S. Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol.I-Theory, Models, Applications, Birkhauser-Springer, 2017.

[8]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[9]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-14574-2.

[10]

J. G. Dong, S. Y. Ha and D. Kim, Emergent behaviors of continuous and discrete thermomechanical Cucker-Smale models on general digraphs, Math. Models Meth. Appl. Sci., 29 (2019), 589-632.  doi: 10.1142/S0218202519400013.

[11]

J. G. Dong and L. Qiu, Flocking of the Cucker-Smale model on general digraphs, IEEE Trans. Automat. Control, 62 (2017), 5234-5239.  doi: 10.1109/TAC.2016.2631608.

[12]

M. Eckert, K. Nagatou, F. Rey, O. Stark and S. Hohmann, Solution of time-variant fractional differential equations with a generalized Peano-Baker series, IEEE Control Syst. Lett., 3 (2019), 79-84. 

[13]

E. Girejko, D. Mozyrska and M. Wyrwas, Numerical analysis of behaviour of the Cucker-Smale type models with fractional operators, J. Comput. Appl. Math., 339 (2018), 111-123.  doi: 10.1016/j.cam.2017.12.013.

[14]

S. Y. Ha and J. Jung, Remarks on the slow relaxation for the fractional Kuramoto model for synchronization, J. Math. Phys., 59 (2018), 032703, 18pp. doi: 10.1063/1.5005865.

[15]

S. Y. Ha, J. Jung and P. Kuchling, Emergence of anomalous flocking in the fractional Cucker-Smale model, Discrete Contin. Dyn. Syst., 39 (2019), 5465-5489.  doi: 10.3934/dcds.2019223.

[16]

S. Y. Ha and J. G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325. 

[17]

S. Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.

[18]

C. Li, A. Chen and J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys., 230 (2011), 3352-3368.  doi: 10.1016/j.jcp.2011.01.030.

[19]

A. B. Malinowska, T. Odzijewicz and E. Schmeidel, On the existence of optimal controls for the fractional continuous-time Cucker-Smale model, in Theory and Applications of Non-integer Order Systems (eds. A. Babiarz, A. Czornik, J. Klamka, M. Niezabitowski), Springer International Publishing, (2017), 227–240.

[20]

M. Merkle, Completely monotone functions: a digest, Anal. Number Theor., Approx. Theor., Special Funct., (2014), 347–364.

[21]

S. Motsch and E. Tadmor, Heterophilious dynamics: Enhanced consensus, SIAM Rev., 56 (2014), 577-621.  doi: 10.1137/120901866.

[22]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guid. Control Dynam., 32 (2009) 527–537.

[23]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic press, 1998.

[24]

K. Sayevand, Fractional dynamical systems: A fresh view on the local qualitative theorems, Int. J. Nonlinear Anal. Appl., 7 (2016), 303-318. 

[25]

W. R. Schneider, Completely monotone generalized Mittag-Leffler functions, Expo. Math., 14 (1996), 3-16. 

[26]

E. D. Sontag, Mathematical Control Theory, $2^nd$ edition, Texts Appl. Math. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[27]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E., 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

[28]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

Figure 1.  The case of symmetric network topology
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Figure 2.  The case of nonsymmetric network topology
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