Threshold Dynamics of Voter Radicalization on the Probability Simplex

Abstract

We analyse two coupled ODE models of political competition on invariant probability simplices with a conserved electorate. The baseline three-group model tracks left-radical, centrist, and right-radical voter shares. We characterise the unique interior equilibrium by a Perron--Frobenius threshold, establish global asymptotic stability in the symmetric and asymmetric cases, and exclude periodic orbits unconditionally via the Dulac criterion. A structural consequence is that the baseline model cannot produce irreversible centrist decline, history-dependent long-run floors, or multiple attractors. We then extend the model with a disengaged voter compartment and distinguish pure state shocks from permanent structural parameter shifts. The post-shock dynamics are governed by the same spectral threshold: below it the centrist state is globally asymptotically stable; above it every trajectory with a nonzero radical seed converges to the unique radicalised equilibrium. Cumulative sub-threshold structural shifts can cross the threshold and produce staircase dynamics absent from the baseline; the symmetric reduction yields closed-form expressions for the critical shock amplitude and the radicalization window.

Figures (7)

• Figure 1: Time series for three representative scenarios across the two dynamical regimes of Theorem \ref{['thm:global']}. Left ($\beta=0.22<\mu=0.25$): subcritical case. Both radical wings decay; the centrist share approaches 1. Centre ($\beta=0.40>\mu=0.20$): supercritical case, moderate polarisation. All groups persist at $P^*=0.25$, $C^*=0.50$. Right ($\beta=0.75>\mu=0.20$): supercritical case, high polarisation. Strong reactive polarisation shifts the equilibrium to $P^*=0.367$, $C^*=0.267$. Both right-panel cases belong to the same qualitative regime ($\beta>\mu$); they differ only in the degree of polarisation at equilibrium. Dashed lines show analytical predictions; numerical and analytical values agree within tolerance $10^{-8}$.
• Figure 1: The three shock regimes of the four-group model. Left: Small shock ($\Delta=0.10 < \Delta_c=0.25$): no surge, monotone recovery to $C^\infty=1$. Centre: Large state shock ($\Delta=0.55>\Delta_c$, $\Delta\beta=0$): transient populist surge followed by full recovery. The apolitical pool $A$ (dotted) drives the surge and is depleted on timescale $1/\rho=10$. Right: Structural shock ($\Delta=0.55$, $\Delta\beta=0.15$): $\beta'=0.45>\mu=0.40$; the system converges to a new equilibrium with $C^\infty=\mu/\beta'\approx0.89<1$. The state component $\Delta$ determines the transient; the structural component $\Delta\beta$ alone determines $C^\infty$. Parameters: $\beta_0=0.30$, $\mu=0.40$, $\delta=0.70$, $\rho=0.10$.
• Figure 2: Phase portraits in the feasible simplex $\Sigma_2$. Stars mark attracting equilibria. All trajectories converge to the same fixed point, confirming global stability. In Regime I the attractor is $E_0=(0,0)$; in Regimes II and III it is the interior equilibrium $E_1=(P^*,P^*)$. No limit cycles are present, consistent with the Dulac criterion (Corollary \ref{['cor:dulac']}).
• Figure 2: Staircase dynamics under four sequential shocks. Each shock has state component $\Delta_k=0.40$ and structural component $\Delta\beta_k=0.04$. The cumulative structural shift $B_k$ crosses $\mu=0.40$ at shock $k^*=3$ (since $\beta_0+3\times0.04=0.42>\mu$). Before $k^*$: the centrist share recovers (partially) after each shock. After $k^*$: each additional shock permanently lowers the long-run centrist share, producing the staircase pattern. Parameters: $\beta_0=0.30$, $\mu=0.40$, $\delta=0.70$, $\rho=0.10$.
• Figure 3: Equilibrium radical share $P^*$ (left) and centrist share $C^*$ (right) as functions of $\gamma$ for $\alpha\in\{0.05,0.10,0.15\}$ and $\mu=0.20$. The bifurcation threshold $\gamma^*=\mu-\alpha>0$ is clearly visible as the left endpoint of each curve. The hyperbolic decay $C^*=\mu/(\alpha+\gamma)$ (right panel) approaches but never reaches zero over the plotted parameter range.

Theorems & Definitions (77)

• Definition 2.1: The baseline model
• Proposition 3.1: Forward invariance of $\Sigma_2$
• Proof 1
• Remark 3.2
• Definition 3.3: Symmetric case
• Theorem 3.4: Global dynamics of the symmetric model
• Proof 2
• Corollary 3.5: Centrist share at equilibrium
• Proof 3
• Definition 3.6: Interior equilibrium