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September  2019, 39(9): 5465-5489. doi: 10.3934/dcds.2019223

## Emergence of anomalous flocking in the fractional Cucker-Smale model

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea 2 Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea 3 Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea 4 Faculty of Mathematics, Bielefeld University, Bielefeld 33501, Germany

* Corresponding author: Jinwook Jung

Received  November 2018 Revised  March 2019 Published  May 2019

In this paper, we study the emergent behaviors of the Cucker-Smale (C-S) ensemble under the interplay of memory effect and flocking dynamics. As a mathematical model incorporating aforementioned interplay, we introduce the fractional C-S model which can be obtained by replacing the usual time derivative by the Caputo fractional time derivative. For the proposed fractional C-S model, we provide a sufficient framework which admits the emergence of anomalous flocking with the algebraic decay and an $\ell^2$-stability estimate with respect to initial data. We also provide several numerical examples and compare them with our theoretical results.

Citation: Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223
##### References:
 [1] S. Ahn, H. Choi, S.-Y. Ha and H. Lee, On the collision avoiding initial-configurations to the Cucker-Smale type flocking models, Comm. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10. [2] B. Bonilla, M. Rivero and J. J. Trujillo, On systems of linear fractional differential equations with constant coeffients, Applied Mathematics and Computation, 187 (2007), 68-78.  doi: 10.1016/j.amc.2006.08.104. [3] M. Caputo, Linear model of dissipation whose $Q$ is almost frequency independent-II, Geophys. J R. Astr. Soc., 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [4] J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, in Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12. [5] 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, Series: Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhauser-Springer, (2017), 299–331. doi: 10.1007/978-3-319-49996-3_8. [6] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [7] M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci., 4 (2010), 1021-1032. [8] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition Using Differential Operators of Caputo Type, Lecture notes in mathematics, 2004, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-14574-2. [9] Z. E. A. Fellah and C. Depollier, Application of fractional calculus to the sound waves propagation in rigid porous materials: Validation via ultrasonic measurement, Acta Acustica, 88 (2002), 34-39. [10] E. Girejko, D. Mozyrska and M. Wyrwas, Numerical analysis of behaviour of the Cucker-Smale type models with fractional operators, Journal of Computational and Applied Mathematics, 339 (2018), 111-123.  doi: 10.1016/j.cam.2017.12.013. [11] E. Girejko, D. Mozyrska and M. Wyrwas, On the fractional variable order Cucker-Smale type model, IFAC-PapersOnLine, 51 (2018), 693-697.  doi: 10.1016/j.ifacol.2018.06.184. [12] S.-Y. Ha and J. Jung, Remarks on the slow relaxation for the fractional Kuramoto model for synchronization, J. Math. Phys., 59 (2018), 032702, 18pp. doi: 10.1063/1.5005865. [13] S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2. [14] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415. [15] V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Communications in Applied Analysis, 11 (2007), 395-402. [16] 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. [17] 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 and M. Niezabitowski), Springer International Publishing, (2017), 227–240. [18] M. Merkle, Completely monotone functions: A digest, in Analytic Number Theory, Approximation Theory, and Special Functions (eds. G. V. Milovanović and M. Th. Rassias), Springer New York, (2014), 347–364. [19] S. Motsch and E. Tadmor, Heterophilious dynamics: Enhanced consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866. [20] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198, Academic press, 1999. [21] K. Sayevand, Fractional dynamical systems: A fresh view on the local qualitative theorems, Int. J. Nonlinear Anal. Appl., 7 (2016), 303-318. [22] W. R. Schneider, Completely monotone generalized Mittag-Leffler functions, Expo. Math., 14 (1996), 3-16. [23] E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, 23 (2002), 397-404. [24] 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. [25] V. K. Vladimir and L. L. Jose, Application of fractional calculus to fluid mechanics, J. Fluids Eng, 124 (2002), 803-806.

show all references

##### References:
 [1] S. Ahn, H. Choi, S.-Y. Ha and H. Lee, On the collision avoiding initial-configurations to the Cucker-Smale type flocking models, Comm. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10. [2] B. Bonilla, M. Rivero and J. J. Trujillo, On systems of linear fractional differential equations with constant coeffients, Applied Mathematics and Computation, 187 (2007), 68-78.  doi: 10.1016/j.amc.2006.08.104. [3] M. Caputo, Linear model of dissipation whose $Q$ is almost frequency independent-II, Geophys. J R. Astr. Soc., 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [4] J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, in Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12. [5] 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, Series: Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhauser-Springer, (2017), 299–331. doi: 10.1007/978-3-319-49996-3_8. [6] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [7] M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci., 4 (2010), 1021-1032. [8] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition Using Differential Operators of Caputo Type, Lecture notes in mathematics, 2004, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-14574-2. [9] Z. E. A. Fellah and C. Depollier, Application of fractional calculus to the sound waves propagation in rigid porous materials: Validation via ultrasonic measurement, Acta Acustica, 88 (2002), 34-39. [10] E. Girejko, D. Mozyrska and M. Wyrwas, Numerical analysis of behaviour of the Cucker-Smale type models with fractional operators, Journal of Computational and Applied Mathematics, 339 (2018), 111-123.  doi: 10.1016/j.cam.2017.12.013. [11] E. Girejko, D. Mozyrska and M. Wyrwas, On the fractional variable order Cucker-Smale type model, IFAC-PapersOnLine, 51 (2018), 693-697.  doi: 10.1016/j.ifacol.2018.06.184. [12] S.-Y. Ha and J. Jung, Remarks on the slow relaxation for the fractional Kuramoto model for synchronization, J. Math. Phys., 59 (2018), 032702, 18pp. doi: 10.1063/1.5005865. [13] S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2. [14] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415. [15] V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Communications in Applied Analysis, 11 (2007), 395-402. [16] 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. [17] 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 and M. Niezabitowski), Springer International Publishing, (2017), 227–240. [18] M. Merkle, Completely monotone functions: A digest, in Analytic Number Theory, Approximation Theory, and Special Functions (eds. G. V. Milovanović and M. Th. Rassias), Springer New York, (2014), 347–364. [19] S. Motsch and E. Tadmor, Heterophilious dynamics: Enhanced consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866. [20] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198, Academic press, 1999. [21] K. Sayevand, Fractional dynamical systems: A fresh view on the local qualitative theorems, Int. J. Nonlinear Anal. Appl., 7 (2016), 303-318. [22] W. R. Schneider, Completely monotone generalized Mittag-Leffler functions, Expo. Math., 14 (1996), 3-16. [23] E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, 23 (2002), 397-404. [24] 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. [25] V. K. Vladimir and L. L. Jose, Application of fractional calculus to fluid mechanics, J. Fluids Eng, 124 (2002), 803-806.
Initial configurations for $\psi_m >0$.
Slow velocity alignment for $\psi_m >0$
Relaxation rate toward velocity alignment for $\psi_m>0$
Initial configurations for each case, when $\psi$ is just nonnegative.
Slow velocity alignment when $\psi$ is just nonnegative
Relaxation rate toward velocity alignment when $\psi$ is just nonnegative
Non-flocking result when $\psi$ is just nonnegative
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