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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 & Continuous Dynamical Systems - A, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223
References:
[1]

S. AhnH. ChoiS.-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. Google Scholar

[2]

B. BonillaM. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci., 4 (2010), 1021-1032. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

E. GirejkoD. 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. Google Scholar

[11]

E. GirejkoD. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Communications in Applied Analysis, 11 (2007), 395-402. Google Scholar

[16]

C. LiA. 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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[22]

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

[23]

E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, 23 (2002), 397-404. Google Scholar

[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. Google Scholar

[25]

V. K. Vladimir and L. L. Jose, Application of fractional calculus to fluid mechanics, J. Fluids Eng, 124 (2002), 803-806. Google Scholar

show all references

References:
[1]

S. AhnH. ChoiS.-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. Google Scholar

[2]

B. BonillaM. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci., 4 (2010), 1021-1032. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

E. GirejkoD. 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. Google Scholar

[11]

E. GirejkoD. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Communications in Applied Analysis, 11 (2007), 395-402. Google Scholar

[16]

C. LiA. 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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[22]

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

[23]

E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, 23 (2002), 397-404. Google Scholar

[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. Google Scholar

[25]

V. K. Vladimir and L. L. Jose, Application of fractional calculus to fluid mechanics, J. Fluids Eng, 124 (2002), 803-806. Google Scholar

Figure 1.  Initial configurations for $ \psi_m >0 $.
Figure 2.  Slow velocity alignment for $ \psi_m >0 $
Figure 3.  Relaxation rate toward velocity alignment for $ \psi_m>0 $
Figure 4.  Initial configurations for each case, when $ \psi $ is just nonnegative.
Figure 5.  Slow velocity alignment when $ \psi $ is just nonnegative
Figure 6.  Relaxation rate toward velocity alignment when $ \psi $ is just nonnegative
Figure 7.  Non-flocking result when $ \psi $ is just nonnegative
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