# American Institute of Mathematical Sciences

• Previous Article
$L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping
• DCDS Home
• This Issue
• Next Article
Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics
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 & Continuous Dynamical Systems - A, 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
 [1] Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023 [2] Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 [3] Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062 [4] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 975-993. doi: 10.3934/dcdss.2020057 [5] Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028 [6] Pierre Aime Feulefack, Jean Daniel Djida, Atangana Abdon. A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3227-3247. doi: 10.3934/dcdsb.2018317 [7] Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419 [8] Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019033 [9] Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023 [10] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [11] Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447 [12] Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic & Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040 [13] Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116 [14] Hayat Zouiten, Ali Boutoulout, Delfim F. M. Torres. Regional enlarged observability of Caputo fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1017-1029. doi: 10.3934/dcdss.2020060 [15] Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic & Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028 [16] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 937-956. doi: 10.3934/dcdss.2020055 [17] Ioannis Markou. Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5245-5260. doi: 10.3934/dcds.2018232 [18] Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045 [19] Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks & Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017 [20] Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2019072

2017 Impact Factor: 1.179