# American Institute of Mathematical Sciences

September  2015, 35(9): 4241-4268. doi: 10.3934/dcds.2015.35.4241

## A nonlinear model of opinion formation on the sphere

 1 Équipe M2N - EA 7340, Conservatoire National des Arts et Métiers, Paris, France 2 Dipartimento di Matematica e Fisica, Università degli studi di Roma Tre, Rome, Italy 3 Department of Mathematical Sciences & Center for Computational and Integrative Biology, Rutgers University, Camden, NJ

Received  June 2014 Revised  October 2014 Published  April 2015

In this paper we present a model for opinion dynamics on the $d$-dimensional sphere based on classical consensus algorithms. The choice of the model is motivated by the analysis of the comprehensive literature on the subject, both from the mathematical and the sociological point of views. The resulting dynamics is highly nonlinear and therefore presents a rich structure. Equilibria and asymptotic behavior are then analysed and sufficient condition for consensus are established. Finally we address global stabilization and controllability.
Citation: Marco Caponigro, Anna Chiara Lai, Benedetto Piccoli. A nonlinear model of opinion formation on the sphere. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4241-4268. doi: 10.3934/dcds.2015.35.4241
##### References:

show all references

##### References:
 [1] Marina Dolfin, Mirosław Lachowicz. Modeling opinion dynamics: How the network enhances consensus. Networks & Heterogeneous Media, 2015, 10 (4) : 877-896. doi: 10.3934/nhm.2015.10.877 [2] Sergei Yu. Pilyugin, M. C. Campi. Opinion formation in voting processes under bounded confidence. Networks & Heterogeneous Media, 2019, 14 (3) : 617-632. doi: 10.3934/nhm.2019024 [3] Pierre Degond, Gadi Fibich, Benedetto Piccoli, Eitan Tadmor. Special issue on modeling and control in social dynamics. Networks & Heterogeneous Media, 2015, 10 (3) : i-ii. doi: 10.3934/nhm.2015.10.3i [4] Masatoshi Shiino, Keiji Okumura. Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena. Conference Publications, 2013, 2013 (special) : 685-694. doi: 10.3934/proc.2013.2013.685 [5] Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021 [6] Robin Cohen, Alan Tsang, Krishna Vaidyanathan, Haotian Zhang. Analyzing opinion dynamics in online social networks. Big Data & Information Analytics, 2016, 1 (4) : 279-298. doi: 10.3934/bdia.2016011 [7] Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4665-4681. doi: 10.3934/dcds.2015.35.4665 [8] Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367 [9] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 [10] Giacomo Albi, Lorenzo Pareschi, Mattia Zanella. Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinetic & Related Models, 2017, 10 (1) : 1-32. doi: 10.3934/krm.2017001 [11] Domenica Borra, Tommaso Lorenzi. Asymptotic analysis of continuous opinion dynamics models under bounded confidence. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1487-1499. doi: 10.3934/cpaa.2013.12.1487 [12] Clinton Innes, Razvan C. Fetecau, Ralf W. Wittenberg. Modelling heterogeneity and an open-mindedness social norm in opinion dynamics. Networks & Heterogeneous Media, 2017, 12 (1) : 59-92. doi: 10.3934/nhm.2017003 [13] David L. Russell. Modeling and control of hybrid beam systems with rotating tip component. Evolution Equations & Control Theory, 2014, 3 (2) : 305-329. doi: 10.3934/eect.2014.3.305 [14] Silviu-Iulian Niculescu, Peter S. Kim, Keqin Gu, Peter P. Lee, Doron Levy. Stability crossing boundaries of delay systems modeling immune dynamics in leukemia. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 129-156. doi: 10.3934/dcdsb.2010.13.129 [15] Doron Levy, Tiago Requeijo. Modeling group dynamics of phototaxis: From particle systems to PDEs. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 103-128. doi: 10.3934/dcdsb.2008.9.103 [16] Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122 [17] Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002 [18] M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137 [19] Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063 [20] Chunjiang Qian, Wei Lin, Wenting Zha. Generalized homogeneous systems with applications to nonlinear control: A survey. Mathematical Control & Related Fields, 2015, 5 (3) : 585-611. doi: 10.3934/mcrf.2015.5.585

2018 Impact Factor: 1.143