  We study the $\beta$-model ($\beta$-NG) and the Bayesian Naming Game (BNG) as
dynamical systems. By applying linear stability analysis to the dynamical
system associated with the $\beta$-model, we demonstrate the existence of a
non-generic bifurcation with a bifurcation point $\beta_c = 1/3$. As $\beta$
passes through $\beta_c$, the stability of isolated fixed points changes,
giving rise to a one-dimensional manifold of fixed points. Notably, this
attracting invariant manifold forms an arc of an ellipse. In the context of the
BNG, we propose modeling the Bayesian learning probabilities $p_A$ and $p_B$ as
logistic functions. This modeling approach allows us to establish the existence
of fixed points without relying on the overly strong assumption that $p_A = p_B
= p$, where $p$ is a constant.
