• Previous Article
    Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics
  • DCDS Home
  • This Issue
  • Next Article
    $ L^1 $ estimates for oscillating integrals and their applications to semi-linear models with $ \sigma $-evolution like structural damping
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. 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
[1]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168

[2]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[3]

Jinwook Jung, Peter Kuchling. Emergent dynamics of the fractional Cucker-Smale model under general network topologies. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022077

[4]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[5]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062

[6]

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 and Continuous Dynamical Systems - S, 2020, 13 (3) : 975-993. doi: 10.3934/dcdss.2020057

[7]

Piotr Grabowski. On analytic semigroup generators involving Caputo fractional derivative. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022014

[8]

Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic and Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027

[9]

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 and Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

[10]

Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic and Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008

[11]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419

[12]

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 and Continuous Dynamical Systems - B, 2019, 24 (7) : 3227-3247. doi: 10.3934/dcdsb.2018317

[13]

Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195

[14]

Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021026

[15]

Iman Malmir. Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 395-426. doi: 10.3934/naco.2021013

[16]

Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155

[17]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023

[18]

Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic and Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026

[19]

Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253

[20]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic and Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (341)
  • HTML views (197)
  • Cited by (1)

Other articles
by authors

[Back to Top]