Below-threshold Bistability and Implementation Lag in a Simplex Model of Radical Vote-Share Dynamics
Abstract
We study a nonautonomous compartmental model on the probability simplex for radical vote-share dynamics under delayed policy implementation. The model distinguishes a target regime from an effective regime that adjusts with finite speed and includes a persistent alienation reservoir that can later be mobilised by radical actors. In a symmetric benchmark, persistent alienation destroys the global one-threshold picture: below the local instability threshold, a radical-free centrist--alienated equilibrium may coexist with a stable positive equilibrium, separated by a saddle branch created in a saddle-node bifurcation. By hyperbolic persistence, this coexistence survives near the symmetric baseline, while numerical exploration identifies a wider bistable region under asymmetric perturbations. Delayed implementation further separates target and effective threshold passage, producing a parameter-space lag and an additional state-space lag before a visible response in radical support. The analysis combines a Perron--Frobenius threshold for the frozen radical-free equilibrium, a geometric characterisation of positive equilibria, local adiabatic tracking under slow uniformly subcritical drift, and explicit delay bounds for transversal threshold passage. Under initial matching, the parameter-space delay scales as O(κθ−1); in a monotone ramp example the bound is nearly attained, whereas the state-space lag is substantially larger and less sensitive to implementation speed. Thus threshold restoration is preventive rather than curative: once the trajectory enters the basin of the radicalised attractor, returning below the local threshold need not restore the radical-free regime.
Versions
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BibTeX
@misc{omelchenko2026belowthreshold,
title = {{Below-threshold Bistability and Implementation Lag in a Simplex Model of Radical Vote-Share Dynamics}},
author = {Alexander Omelchenko},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27742},
eprint = {2608.27742},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27742},
}
RIS
TY - UNPB
TI - Below-threshold Bistability and Implementation Lag in a Simplex Model of Radical Vote-Share Dynamics
AU - Alexander Omelchenko
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27742
UR - https://arxiv.org/abs/2608.27742
AB - We study a nonautonomous compartmental model on the probability simplex for radical vote-share dynamics under delayed policy implementation. The model distinguishes a target regime from an effective regime that adjusts with finite speed and includes a persistent alienation reservoir that can later be mobilised by radical actors. In a symmetric benchmark, persistent alienation destroys the global one-threshold picture: below the local instability threshold, a radical-free centrist--alienated equilibrium may coexist with a stable positive equilibrium, separated by a saddle branch created in a saddle-node bifurcation. By hyperbolic persistence, this coexistence survives near the symmetric baseline, while numerical exploration identifies a wider bistable region under asymmetric perturbations. Delayed implementation further separates target and effective threshold passage, producing a parameter-space lag and an additional state-space lag before a visible response in radical support. The analysis combines a Perron--Frobenius threshold for the frozen radical-free equilibrium, a geometric characterisation of positive equilibria, local adiabatic tracking under slow uniformly subcritical drift, and explicit delay bounds for transversal threshold passage. Under initial matching, the parameter-space delay scales as O(κ_θ^−1); in a monotone ramp example the bound is nearly attained, whereas the state-space lag is substantially larger and less sensitive to implementation speed. Thus threshold restoration is preventive rather than curative: once the trajectory enters the basin of the radicalised attractor, returning below the local threshold need not restore the radical-free regime.
ER -