Convergence of the critical trajectory at the differentiation threshold

Determine whether the deterministic two-party transport–diffusion dynamics at the critical institutional-payoff weight \(\theta=\theta_c\), with \(K=2\), voter-tolerance ratio \(r=0.6\), and diffusion \(\varepsilon=0.002\), converge to a stationary state or exhibit another long-run behavior.

Background

The paper studies a two-party system at the numerically identified critical institutional-payoff weight $\theta_c$, where the leading reflection-odd growth rate vanishes. Below this threshold, asymmetric perturbations grow and the parties reach separated stationary profiles; above it, the perturbations decay back to the common centrist profile.

At the critical value itself, the deterministic trajectory remains close to its initial displacement throughout the simulated horizon $t=1600$. The reported computations therefore do not resolve whether the trajectory eventually converges, and if so, whether its limiting state is the centrist profile, a non-centrist stationary configuration, or another type of asymptotic state.