Hopf bifurcation in an opinion model with state-dependent delay

• Published in: Chaos, solitons and fractals solitons and fractals, 2021-12, Vol.153, p.111511, Article 111511
• Main Author: Qesmi, Redouane
• Format: Article
• Language: English
• Subjects: Functional differential equation; Hopf bifurcation; Opinion; Periodic solution; State-dependent delay
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2021.111511

•We considered an opinion model of threshold-type delay differential equations.•We proved that the model with either a transcritical forward or a backward bifurcation, can exhibit Hopf bifurcation.•We showed that the obtained Hopf bifurcation appears only on upper bifurcated branches of the equilibria.•We performed numerical simulations to illustrate and support our theoretical results. In a recent paper [R. Qesmi, Dynamics of an opinion model with threshold-type delay, Chaos, Solitons & Fractals 98 (2020). (https://doi.org/10.1016/j.chaos.2020.110379)], we proposed a mathematical model of threshold-type delay differential equations describing the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. The study there showed the possibility of a transcritical forward and backward bifurcations of positive equilibria. In the present paper, we show that the opinion model undergoes a Hopf bifurcation through which one of the bifurcation branches loses the stability and periodic solutions appear. One of the important consequences of the obtained dynamics is that the consensus of the both opinions could be lost by maintaining the balance between the time taken for an individual to become convinced of the outsider opinion, which need be short, and the number of individuals converted to the local opinion which need be low. Finally, we provide numerical simulations to illustrate and support our theoretical results.