# American Institute of Mathematical Sciences

December  2021, 41(12): 5765-5787. doi: 10.3934/dcds.2021095

## Grassmannian reduction of cucker-smale systems and dynamical opinion games

 Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607, USA

* Corresponding author: lear@uic.edu

Received  November 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

Fund Project: This work was supported in part by NSF grant DMS-1813351

In this note we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a new method which allows us to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than $\pi$.

In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions.

Citation: Daniel Lear, David N. Reynolds, Roman Shvydkoy. Grassmannian reduction of cucker-smale systems and dynamical opinion games. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5765-5787. doi: 10.3934/dcds.2021095
##### References:
 [1] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374. [2] J. Carrillo, Y.-P. Choi and S. Perez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, in Active Particles. Advances in Theory, Models, and Application, 1, Birkhäuser, 2017. [3] J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, Birkhäuser, 2010,297–336. doi: 10.1007/978-0-8176-4946-3_12. [4] J. A. Carrillo, Y.-P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017. [5] Y.-l. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.  doi: 10.1016/j.physd.2007.05.007. [6] J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, Mathematical Surveys, 11, American Mathematical Society, Providence, R.I., 1964. [7] F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355. [8] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [9] F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [10] M. H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 121-132. [11] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. [12] H. Dietert and R. Shvydkoy, On cucker-smale dynamical systems with degenerate communication, Analysis and Appl., accepted. doi: 10.1142/S0219530520500050. [13] T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, Arch. Ration. Mech. Anal., 228 (2018), 1-37.  doi: 10.1007/s00205-017-1184-2. [14] S.-Y. Ha, T. Ha and J.-H. Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A, 43 (2010), 315201. doi: 10.1088/1751-8113/43/31/315201. [15] R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis, and simulations, Journal of Artificial Societies and Social Simulation, 5 (2002). [16] S.-Z. Huang, Gradient inequalities, Mathematical Surveys and Monographs, 126, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/126. [17] J. Kim and J. Peszek, Cucker-smale model with a bonding force and a singular interaction kernel, 2018. [18] Z. Li, X. Xue and D. Yu, On the Lojasiewicz exponent of Kuramoto model, J. Math. Phys., 56 (2015), 022704. doi: 10.1063/1.4908104. [19] S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in Les Équations aux Dérivées Partielles (Paris, 1962), Éditions du Centre National de la Recherche Scientifique, Paris, 1963, 87–89. [20] Z. Mao, Z. Li and G. E. Karniadakis, Nonlocal flocking dynamics: learning the fractional order of PDEs from particle simulations, Commun. Appl. Math. Comput., 1 (2019), 597-619.  doi: 10.1007/s42967-019-00031-y. [21] P. Minakowski, P. B. Mucha, J. Peszek and E. Zatorska, Singular Cucker-Smale dynamics, in Active Particles, Vol. 2, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2019,201–243. [22] J. Nash, Non-cooperative games, Ann. of Math. (2), 54 (1951), 286-295.  doi: 10.2307/1969529. [23] J. Palis Jr. and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York-Berlin, 1982. [24] C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, ACM SIGGRAPH Computer Graphics, 21 (1987), 25–34. [25] R. Shu and E. Tadmor, Anticipation breeds alignment. doi: 10.1007/s00205-021-01609-8. [26] R. Shu and E. Tadmor, Flocking hydrodynamics with external potentials. doi: 10.1007/s00205-020-01544-0. [27] R. Shvydkoy, Dynamics and analysis of alignment models of collective behavior., Available at: https://shvydkoy.people.uic.edu/alignment.pdf. [28] R. Shvydkoy and E. Tadmor, Multi-flocks: Emergent dynamics in systems with multi-scale collective behavior, preprint. [29] R. Shvydkoy and E. Tadmor, Topologically-based fractional diffusion and emergent dynamics with short-range interactions, to appear in SIMA. doi: 10.1137/19M1292412. [30] R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing, Transactions of Mathematics and Its Applications, 1 (2017). doi: 10.1093/imatrm/tnx001. [31] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118 (1983), 525-571.  doi: 10.2307/2006981.

show all references

##### References:
 [1] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374. [2] J. Carrillo, Y.-P. Choi and S. Perez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, in Active Particles. Advances in Theory, Models, and Application, 1, Birkhäuser, 2017. [3] J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, Birkhäuser, 2010,297–336. doi: 10.1007/978-0-8176-4946-3_12. [4] J. A. Carrillo, Y.-P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017. [5] Y.-l. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.  doi: 10.1016/j.physd.2007.05.007. [6] J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, Mathematical Surveys, 11, American Mathematical Society, Providence, R.I., 1964. [7] F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355. [8] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [9] F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [10] M. H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 121-132. [11] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. [12] H. Dietert and R. Shvydkoy, On cucker-smale dynamical systems with degenerate communication, Analysis and Appl., accepted. doi: 10.1142/S0219530520500050. [13] T. Do, A. Kiselev, L. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, Arch. Ration. Mech. Anal., 228 (2018), 1-37.  doi: 10.1007/s00205-017-1184-2. [14] S.-Y. Ha, T. Ha and J.-H. Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A, 43 (2010), 315201. doi: 10.1088/1751-8113/43/31/315201. [15] R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis, and simulations, Journal of Artificial Societies and Social Simulation, 5 (2002). [16] S.-Z. Huang, Gradient inequalities, Mathematical Surveys and Monographs, 126, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/126. [17] J. Kim and J. Peszek, Cucker-smale model with a bonding force and a singular interaction kernel, 2018. [18] Z. Li, X. Xue and D. Yu, On the Lojasiewicz exponent of Kuramoto model, J. Math. Phys., 56 (2015), 022704. doi: 10.1063/1.4908104. [19] S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in Les Équations aux Dérivées Partielles (Paris, 1962), Éditions du Centre National de la Recherche Scientifique, Paris, 1963, 87–89. [20] Z. Mao, Z. Li and G. E. Karniadakis, Nonlocal flocking dynamics: learning the fractional order of PDEs from particle simulations, Commun. Appl. Math. Comput., 1 (2019), 597-619.  doi: 10.1007/s42967-019-00031-y. [21] P. Minakowski, P. B. Mucha, J. Peszek and E. Zatorska, Singular Cucker-Smale dynamics, in Active Particles, Vol. 2, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2019,201–243. [22] J. Nash, Non-cooperative games, Ann. of Math. (2), 54 (1951), 286-295.  doi: 10.2307/1969529. [23] J. Palis Jr. and W. de Melo, Geometric Theory of Dynamical Systems, Springer-Verlag, New York-Berlin, 1982. [24] C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, ACM SIGGRAPH Computer Graphics, 21 (1987), 25–34. [25] R. Shu and E. Tadmor, Anticipation breeds alignment. doi: 10.1007/s00205-021-01609-8. [26] R. Shu and E. Tadmor, Flocking hydrodynamics with external potentials. doi: 10.1007/s00205-020-01544-0. [27] R. Shvydkoy, Dynamics and analysis of alignment models of collective behavior., Available at: https://shvydkoy.people.uic.edu/alignment.pdf. [28] R. Shvydkoy and E. Tadmor, Multi-flocks: Emergent dynamics in systems with multi-scale collective behavior, preprint. [29] R. Shvydkoy and E. Tadmor, Topologically-based fractional diffusion and emergent dynamics with short-range interactions, to appear in SIMA. doi: 10.1137/19M1292412. [30] R. Shvydkoy and E. Tadmor, Eulerian dynamics with a commutator forcing, Transactions of Mathematics and Its Applications, 1 (2017). doi: 10.1093/imatrm/tnx001. [31] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118 (1983), 525-571.  doi: 10.2307/2006981.
Isosceles triangles in the proof of Lemma 4.3
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