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The modeling of opinion dynamics has seen much study in varying academic disciplines. Understanding the complex ways information can be disseminated is a complicated problem for mathematicians as well as social scientists. We present a nonlinear model of opinion dynamics that utilizes an environmental averaging protocol similar to the DeGroot and Freidkin-Johnsen models. Indeed, the way opinions evolve is complex and nonlinear effects ought to be considered when modelling. For this model, the nonlinearity destroys the translation invariance of the equations, as well as the convexity of the associated payout functions. The standard theory for well-posedness and convergence no longer applies and we must utilize the Brouwer topological degree and nonconvex analysis in order to achieve these results. Numerical simulations of the model reveal that the nonlinearity behaves similarly to the well-known Friedkin-Johnsen for so-called "reasonable" opinions, but better models the way agents that hold "extreme" opinions are more stubborn than their reasonable counterparts.

We study the asymptotic emergent dynamics of two models that can be thought of as extensions of the well known Schr\"odinger-Lohe model for quantum synchronization. More precisely, the interaction strength between different oscillators is determined by intrinsic parameters, following Cucker-Smale communication protocol. Unlike the original Schr\"odinger-Lohe system, where the interaction strength was assumed to be uniform, in the cases under our consideration the total mass of each quantum oscillator is allowed to vary in time. A striking consequence of this property is that these extended models yield configurations exhibiting phase, but not space, synchronization. The results are mainly based on the analysis of the ODE systems arising from the correlations, control over the well known Cucker-Smale dynamics, and the dynamics satisfied by the quantum order parameter., Comment: 18 pages, minor changes and submitted