We study heterogeneous interactions in a time-continuous bounded confidence model for opinion formation. The key new modelling aspects are to distinguish between open-minded and closed-minded behaviour and to include an open-mindedness social norm. The investigations focus on the equilibria supported by the proposed new model; particular attention is given to a novel class of equilibria consisting of multiple connected opinion clusters, which does not occur in the absence of heterogeneity. Various rigorous stability results concerning these equilibria are established. We also incorporate the effect of media in the model and study its implications for opinion formation.
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Figure 1.
Schematic diagram of opinion interactions along the opinion continuum
Figure 2.
Evolution to equilibria of solutions to model (4) with (6) with
Figure 3.
Numerical illustration of local stability: At time
Figure 8.
Bifurcation with respect to the location of the media source. The filled circles represent the locations of equilibrium clusters, while the dashed line indicates the media bias; parameter values are
Figure 4.
Dependence of the number and stability of cluster equilibria on
Figure 5.
Extremist consensus arising as bifurcations in parameter space. Within each plot, the same non-symmetric (type NS) initial conditions are used to generate equilibria for all parameter values. The inserts show equilibria obtained from symmetric initial data (type S), where no extremist consensus can arise. (a) Dependence on
Figure 6.
Effect of a social norm of open-mindedness on opinion convergence. The initial data is of type NS (non-symmetric); parameters are set at
Figure 7.
Evolution to equilibria of the opinion model with media (19)-(20) with
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