Figure 1.
$ \lambda = 3 $ , $ \delta = 0 $ and $ r = 1 $ . In this case, only the neighbours attract each other. Because the value of $ r $ is small, the system does not achieve flocking
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On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping
June
2022, 11(3): 729-748.
doi: 10.3934/eect.2021023
Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance
College of Liberal Arts and Science, National University of Defense Technology, Changsha, 410073, China |
In this paper, a generalized Motsch-Tadmor model with piecewise interaction function is investigated, which can be viewed as a generalization of the model proposed in [
Mathematics Subject Classification:
93C15, 93D09, 93C95.
Citation:
Yipeng Chen, Yicheng Liu, Xiao Wang. Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance. Evolution Equations and Control Theory,
2022, 11
(3)
: 729-748.
doi: 10.3934/eect.2021023
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References:
| [1] |
S.-M. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, Journal of Mathematical Physics, 51 (2010), 103301, 17pp.
doi: 10.1063/1.3496895. |
| [2] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
| [3] |
F. Cucker and S. Smale,
On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
| [4] |
F. Cucker and E. Mordecki,
Flocking in noisy environments, Journal de Mathématiques Pures et Appliquées, 89 (2008), 278-296.
doi: 10.1016/j.matpur.2007.12.002. |
| [5] |
J.-G. Dong, S.-Y. Ha, J. Jung and D. Kim,
On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies, SIAM Journal on Control and Optimization, 58 (2020), 2332-2353.
doi: 10.1137/19M1279150. |
| [6] |
R. Erban, J. Haskovec and Y. Sun,
A Cucker-Smale model with noise and delay, SIAM Journal on Applied Mathematics, 76 (2016), 1535-1557.
doi: 10.1137/15M1030467. |
| [7] |
S.-Y. Ha and J. Liu,
A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Communications in Mathematical Sciences, 7 (2009), 297-325.
doi: 10.4310/CMS.2009.v7.n2.a2. |
| [8] |
S.-Y. Ha, K. Lee and D. Levy,
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Communications in Mathematical Sciences, 7 (2009), 453-469.
doi: 10.4310/CMS.2009.v7.n2.a9. |
| [9] |
Y. Jin,
Flocking of the Motsch-Tadmor model with a cut-off interaction function, Journal of Statistical Physics, 171 (2018), 345-360.
doi: 10.1007/s10955-018-2006-0. |
| [10] |
Z. Li and S.-Y. Ha,
On the Cucker-Smale flocking with alternating leaders, Quarterly of Applied Mathematics, 73 (2015), 693-709.
doi: 10.1090/qam/1401. |
| [11] |
Y. Liu and J. Wu,
Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, Journal Mathematical Analysis and Applications, 415 (2014), 53-61.
doi: 10.1016/j.jmaa.2014.01.036. |
| [12] |
Y. Liu and J. Wu,
Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling, Journal Mathematical Analysis and Applications, 444 (2016), 947-956.
doi: 10.1016/j.jmaa.2016.06.049. |
| [13] |
Y. Liu and J. Wu,
Opinion consensus with delay when the zero eigenvalue of the connection matrix is semi-simple, Journal Mathematical Analysis and Applications, 29 (2017), 1539-1551.
doi: 10.1007/s10884-016-9548-0. |
| [14] |
X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
| [15] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, Journal of Statistical Physics, 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
| [16] |
L. Pedeches,
Asymptotic properties of various stochastic Cucker-Smale dynamics, Discrete and Continuous Dynamical Systems, 38 (2018), 2731-2762.
doi: 10.3934/dcds.2018115. |
| [17] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM Journal on Applied Mathematics, 68 (2007), 694-719.
doi: 10.1137/060673254. |
| [18] |
T. V. Ton, N. T. Lihn and A. yagi,
Flocking and non-flocking behaviour in a stochastic Cucker-Smale system, Analysis and Application, 12 (2014), 63-73.
doi: 10.1142/S0219530513500255. |
| [19] |
X. Wang, L. Wang and J. Wu,
Impacts of time delay on flocking dynamics of a two-agent flock model, Communications in Nonlinear Science and Numerical Simulation, 70 (2019), 80-88.
doi: 10.1016/j.cnsns.2018.10.017. |
show all references
References:
| [1] |
S.-M. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, Journal of Mathematical Physics, 51 (2010), 103301, 17pp.
doi: 10.1063/1.3496895. |
| [2] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
| [3] |
F. Cucker and S. Smale,
On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
| [4] |
F. Cucker and E. Mordecki,
Flocking in noisy environments, Journal de Mathématiques Pures et Appliquées, 89 (2008), 278-296.
doi: 10.1016/j.matpur.2007.12.002. |
| [5] |
J.-G. Dong, S.-Y. Ha, J. Jung and D. Kim,
On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies, SIAM Journal on Control and Optimization, 58 (2020), 2332-2353.
doi: 10.1137/19M1279150. |
| [6] |
R. Erban, J. Haskovec and Y. Sun,
A Cucker-Smale model with noise and delay, SIAM Journal on Applied Mathematics, 76 (2016), 1535-1557.
doi: 10.1137/15M1030467. |
| [7] |
S.-Y. Ha and J. Liu,
A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Communications in Mathematical Sciences, 7 (2009), 297-325.
doi: 10.4310/CMS.2009.v7.n2.a2. |
| [8] |
S.-Y. Ha, K. Lee and D. Levy,
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Communications in Mathematical Sciences, 7 (2009), 453-469.
doi: 10.4310/CMS.2009.v7.n2.a9. |
| [9] |
Y. Jin,
Flocking of the Motsch-Tadmor model with a cut-off interaction function, Journal of Statistical Physics, 171 (2018), 345-360.
doi: 10.1007/s10955-018-2006-0. |
| [10] |
Z. Li and S.-Y. Ha,
On the Cucker-Smale flocking with alternating leaders, Quarterly of Applied Mathematics, 73 (2015), 693-709.
doi: 10.1090/qam/1401. |
| [11] |
Y. Liu and J. Wu,
Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, Journal Mathematical Analysis and Applications, 415 (2014), 53-61.
doi: 10.1016/j.jmaa.2014.01.036. |
| [12] |
Y. Liu and J. Wu,
Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling, Journal Mathematical Analysis and Applications, 444 (2016), 947-956.
doi: 10.1016/j.jmaa.2016.06.049. |
| [13] |
Y. Liu and J. Wu,
Opinion consensus with delay when the zero eigenvalue of the connection matrix is semi-simple, Journal Mathematical Analysis and Applications, 29 (2017), 1539-1551.
doi: 10.1007/s10884-016-9548-0. |
| [14] |
X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
| [15] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, Journal of Statistical Physics, 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
| [16] |
L. Pedeches,
Asymptotic properties of various stochastic Cucker-Smale dynamics, Discrete and Continuous Dynamical Systems, 38 (2018), 2731-2762.
doi: 10.3934/dcds.2018115. |
| [17] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM Journal on Applied Mathematics, 68 (2007), 694-719.
doi: 10.1137/060673254. |
| [18] |
T. V. Ton, N. T. Lihn and A. yagi,
Flocking and non-flocking behaviour in a stochastic Cucker-Smale system, Analysis and Application, 12 (2014), 63-73.
doi: 10.1142/S0219530513500255. |
| [19] |
X. Wang, L. Wang and J. Wu,
Impacts of time delay on flocking dynamics of a two-agent flock model, Communications in Nonlinear Science and Numerical Simulation, 70 (2019), 80-88.
doi: 10.1016/j.cnsns.2018.10.017. |
Figure 2.
$ \lambda = 3 $ , $ \delta = 0.1 $ and $ r = 1 $ . In this case, not only the neighbours attract each other but also the distant relatives attract each other in a weak force. Hence unconditional flocking occurs
Figure 4.
$ \lambda = 3 $ , $ \delta = -0.1 $ and $ r = 2 $ . In this case, the neighbours attract each other however the distant relatives repel each other in a weak force. The system also achieves flocking
Figure 5.
$ \lambda = 3 $ , $ \delta = -0.3 $ and $ r = 3 $ . By increasing the repulsive force between distant relatives, the system does not achieve flocking
Figure 6.
$ \lambda = 3 $ , $ \delta = -0.5 $ and $ r = 3 $ . Continue to increase the repulsive force between distant relatives, the system still does not achieve flocking
Figure 7.
$ \lambda = 3 $ , $ \delta = -0.1 $ , $ \sigma_i = 2(i = 1, 2, ..n) $ and $ r = 2 $ . The system achieves flocking under a moderate stochastic disturbance
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