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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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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. 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.

[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.

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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