Emergent Dynamics of an Orientation Flocking Model With Distance-Dependent Delay
Zhengyang Qiao
Department of Mathematics, National University of Defense Technology, Changsha, China
Contribution: Conceptualization, Methodology, Writing - original draft
Search for more papers by this authorCorresponding Author
Yicheng Liu
Department of Mathematics, National University of Defense Technology, Changsha, China
Correspondence:
Yicheng Liu ([email protected])
Contribution: Writing - review & editing, Supervision, Project administration
Search for more papers by this authorXiao Wang
Department of Mathematics, National University of Defense Technology, Changsha, China
Contribution: Writing - review & editing, Supervision
Search for more papers by this authorMaoli Chen
Beijing Institute of Control and Electronic Technology, Beijing, China
Contribution: Writing - review & editing
Search for more papers by this authorZhengyang Qiao
Department of Mathematics, National University of Defense Technology, Changsha, China
Contribution: Conceptualization, Methodology, Writing - original draft
Search for more papers by this authorCorresponding Author
Yicheng Liu
Department of Mathematics, National University of Defense Technology, Changsha, China
Correspondence:
Yicheng Liu ([email protected])
Contribution: Writing - review & editing, Supervision, Project administration
Search for more papers by this authorXiao Wang
Department of Mathematics, National University of Defense Technology, Changsha, China
Contribution: Writing - review & editing, Supervision
Search for more papers by this authorMaoli Chen
Beijing Institute of Control and Electronic Technology, Beijing, China
Contribution: Writing - review & editing
Search for more papers by this authorFunding: This work was partially supported by the National Natural Science Foundation of China (12371180) and Hunan Provincial Graduate Student Innovation Program (CX20230004).
ABSTRACT
In this paper, we investigate the dynamical behaviors of an orientation flocking model that incorporates state-dependent delay. The research unfolds in three principal phases: First, we establish the global existence and uniqueness of solutions for our system. Subsequently, we develop a comprehensive theoretical framework that provides sufficient conditions for the emergence of orientation flocking phenomena. In particular, this framework remains valid regardless of the number of particles . In the final phase of our analysis, we perform a systematic derivation of the mean-field limit for the discrete system, accompanied by a proof of global well-posedness for the resulting mean-field equations. Furthermore, we demonstrate that our theoretical results and analytical framework naturally extend to and encompass the case of orientation flocking models with constant time delays.
Conflicts of Interest
The authors declare no conflicts of interest.
References
- 1T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, “Novel Type of Phase Transition in a System of Self-Driven Particles,” Physical Review Letters 75, no. 6 (1995): 1226.
- 2Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, 1984).
10.1007/978-3-642-69689-3 Google Scholar
- 3F. Cucker and S. Smale, “Emergent Behavior in Flocks,” IEEE Transactions on Automatic Control 52, no. 5 (2007): 852–862.
- 4G. Albi, N. Bellomo, L. Fermo, et al., “Vehicular Traffic, Crowds, and Swarms: From Kinetic Theory and Multiscale Methods to Applications and Research Perspectives,” Mathematical Models and Methods in Applied Sciences 29, no. 10 (2019): 1901–2005.
- 5S. Motsch and E. Tadmor, “Heterophilious Dynamics Enhances Consensus,” SIAM Review 56, no. 4 (2014): 577–621.
- 6J. R. Lawton and R. W. Beard, “Synchronized Multiple Spacecraft Rotations,” Automatica 38, no. 8 (2002): 1359–1364.
- 7R. Tron, R. Vidal, and A. Terzis, “ Distributed Pose Averaging in Camera Networks via Consensus on SE(3),” in 2008 Second ACM/IEEE International Conference on Distributed Smart Cameras (IEEE, 2008), 1–10.
10.1109/ICDSC.2008.4635701 Google Scholar
- 8P. Degond, A. Frouvelle, and S. Merino-Aceituno, “A New Flocking Model Through Body Attitude Coordination,” Mathematical Models and Methods in Applied Sciences 27, no. 06 (2017): 1005–1049.
10.1142/S0218202517400085 Google Scholar
- 9S. Y. Ha, D. Kim, J. Lee, and S. E. Noh, “Emergent Dynamics of an Orientation Flocking Model for Multi-Agent System,” Discrete and Continuous Dynamical Systems 40, no. 4 (2020): 2037–2060.
- 10R. C. Fetecau, S. Y. Ha, and H. Park, “Emergent Behaviors of Rotation Matrix Flocks,” SIAM Journal on Applied Dynamical Systems 21, no. 2 (2022): 1382–1425.
- 11R. C. Fetecau, S. Y. Ha, and H. Park, “An Intrinsic Aggregation Model on the Special Orthogonal Group SO(3): Well-Posedness and Collective Behaviours,” Journal of Nonlinear Science 31 (2021): 1–61.
- 12P Degond, A Diez, and A Frouvelle. “ Body-Attitude Coordination in Arbitrary Dimension.” arXiv preprint arXiv:2111.05614, 2021.
- 13H. Smith, An Introduction to Delay Differential Equations With Applications to the Life Sciences (Springer, 2011).
- 14R. Erban, J. Haskovec, and Y. Z. Sun, “A Cucker–Smale Model With Noise and Delay,” SIAM Journal on Applied Mathematics 76, no. 4 (2016): 1535–1557.
- 15Y. P. Choi and J. Haskovec, “Cucker–Smale Model With Normalized Communication Weights and Time Delay,” Kinetic and Related Models 10, no. 4 (2017): 1011–1033.
- 16J. Haskovec, “A Simple Proof of Asymptotic Consensus in the Hegselmann–Krause and Cucker–Smale Models With Normalization and Delay,” SIAM Journal on Applied Dynamical Systems 20, no. 1 (2021): 130–148.
- 17Y. C. Liu and J. H. Wu, “Flocking and Asymptotic Velocity of the Cucker–Smale Model With Processing Delay,” Journal of Mathematical Analysis and Applications 415, no. 1 (2014): 53–61.
- 18Z. Y. Qiao, Y. C. Liu, and X. Wang, “Multi-Cluster Flocking Behavior Analysis for a Delayed Cucker–Smale Model With Short-Range Communication Weight,” Journal of Systems Science and Complexity 35, no. 1 (2022): 137–158.
- 19S. Y. Ha, D. Kim, D. Kim, H. Park, and W. Shim, “Emergent Dynamics of the Lohe Matrix Ensemble on a Network Under Time-Delayed Interactions,” Journal of Mathematical Physics 61, no. 1 (2020): 012702.
- 20S. H. Choi and S. Y. Ha, “Interplay of the Unit-Speed Constraint and Time-Delay in Cucker–Smale Flocking,” Journal of Mathematical Physics 59, no. 8 (2018): 082701.
- 21J. G. Dong, S. Y. Ha, D. Kim, and J. Kim, “Time-Delay Effect on the Flocking in an Ensemble of Thermomechanical Cucker–Smale Particles,” Journal of Differential Equations 266, no. 5 (2019): 2373–2407.
- 22Y. P. Choi and Z. C. Li, “Emergent Behavior of Cucker–Smale Flocking Particles With Time Delays,” Applied Mathematics Letters 86 (2018): 49–56.
10.1016/j.aml.2018.06.018 Google Scholar
- 23R. C. Mauro, “Cucker-Smale Model With Time Delay,” Discrete and Continuous Dynamical Systems 42 (2022): 2409–2432.
10.3934/dcds.2021195 Google Scholar
- 24J. Haskovec and I. Markou, “Asymptotic Flocking in the Cucker–Smale Model With Reaction-Type Delays in the Non-Oscillatory Regime,” Kinetic and Related Models 13 (2020): 795–813.
- 25Z. S. Liu, Y. C. Liu and X. Wang, “Emergence of Time-Asymptotic Flocking for a General Cucker-Smale-Type Model with Distributed Time Delays,“ Mathematical Methods in the Applied Sciences 43, no. 15 (2020): 8657–8668.
- 26C. Pignotti and E. Trlat, “Convergence to Consensus of the General Finite-Dimensional Cucker–Smale Model With Time-Varying Delays,” Communications in Mathematical Sciences 16 (2018): 2053–2076.
- 27J. Haskovec, “Well Posedness and Asymptotic Consensus in the Hegselmann-Krause Model With Finite Speed of Information Propagation,” Proceedings of the American Mathematical Society 149, no. 8 (2021): 3425–3437.
- 28J. Haskovec, “Cucker–Smale Model With Finite Speed of Information Propagation: Well-Posedness, Flocking and Mean-Field Limit,” Kinetic and Related Models 16, no. 3 (2023): 394–422.
- 29S. H. Choi and H. Seo, “Unit Speed Flocking Model With Distance-Dependent Time Delay,” Applicable Analysis 102, no. 8 (2023): 2338–2364.
- 30E. Continelli and C. Pignotti, “Consensus for Hegselmann-Krause Type Models With Time Variable Time Delays,” Mathematical Methods in the Applied Sciences 46, no. 18 (2023): 18916–18934.
- 31Y. P. Choi and J. Haskovec, “Hydrodynamic Cucker–Smale Model With Normalized Communication Weights and Time Delay,” SIAM Journal on Mathematical Analysis 51, no. 3 (2019): 2660–2685.
- 32S. Y. Ha, E. Jeong, and M.-J. Kang, “Emergent Behaviour of a Generalized Viscek-Type Flocking Model,” Nonlinearity 23, no. 12 (2010): 3139–3156.
- 33E. W. Justh and P. S. Krishnaprasad, “ Steering Laws and Continuum Models for Planar Formations,” in 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475, 2003).
10.1109/CDC.2003.1271708 Google Scholar
- 34S. Y. Ha, D. Ko, W. Shim, and Y. Hui, “Emergent Behaviors of the Justh-Krishnaprasad Model With Uncertain Communications,” Journal of Differential Equations 322 (2022): 38–70.
- 35S. Y. Ha and E. Tadmor, “From Particle to Kinetic and Hydrodynamic Descriptions of Flocking,” Kinetic and Related Models 1, no. 3 (2017): 415–435.
10.3934/krm.2008.1.415 Google Scholar
- 36J. Canizo, J. Carrillo, and J. Rosado, “A Well-Posedness Theory in Measures for Some Kinetic Models of Collective Motion,” Mathematical Models and Methods in Applied Sciences 21, no. 03 (2011): 515–539.
10.1142/S0218202511005131 Google Scholar
- 37S. Y. Ha and J. G. Liu, “A Simple Proof of the Cucker–Smale Flocking Dynamics and Mean-Field Limit,” Communications in Mathematical Sciences 7, no. 2 (2009): 297–325.
- 38F. Golse and S. Y. Ha, “A Mean-Field Limit of the Lohe Matrix Model and Emergent Dynamics,” Archive for Rational Mechanics and Analysis 234, no. 3 (2019): 1445–1491.
- 39C. Villani, Optimal Transport: Old and New, vol. 338 (Springer, 2009).
