4. Sensitivity Analysis of U/JU/JU/J Ratios
Incremental changes in the U/JU/JU/J ratio produce non-linear effects on DDD and SSS.
Critical threshold ratios exist, beyond which small increases in UUU or decreases in JJJ produce large jumps in degradation, consistent with bifurcation analysis from Section IV.C.
These thresholds define policy-relevant "tipping points", suggesting that preventing widespread vote buying or strengthening justice perception can stabilize democratic systems.
5. Implications for Policy and Civic Action
Controlling monetary incentives (UUU) or enhancing perceived justice (JJJ) are key levers to prevent systemic collapse.
Civic awareness campaigns, institutional reforms, and anti-corruption measures can shift the system toward low U/JU/JU/J ratios, preserving democratic integrity.
Conversely, ignoring the feedback between incentives and justice perception risks pushing the system past critical bifurcation points, facilitating authoritarian consolidation or social unrest.
C. Visualization of Trajectories toward Authoritarian or Anarchic Regimes
To complement the quantitative simulations of varying ratios, visualizations of system trajectories provide an intuitive understanding of how the Transactional Degradation of Democracy (TDD) framework predicts long-term political outcomes. These visualizations highlight paths toward authoritarian consolidation or anarchic instability depending on the balance of monetary incentives and perceived justice.
1. Phase Space Representation
A 2D phase space is constructed with:
X-axis: Monetary incentive (UUU) normalized to [0,1]
Y-axis: Perceived justice (JJJ) normalized to [0,1]
Each point represents a unique U/JU/JU/J ratio, with arrows indicating the direction and magnitude of change in system stability (SSS) over time.
Color-coding:
Green: Stable democratic regime (S0S \approx 0S0)
Blue: Drift toward authoritarian consolidation (S>0S > 0S>0)
Red: Drift toward anarchic instability (S<0S < 0S<0)
Interpretation: Regions with high U/JU/JU/J ratios consistently lead to authoritarian paths, whereas low U/JU/JU/J ratios are more likely to generate citizen resistance and potential unrest.
2. Trajectories of System Stability Over Time
Time-series plots illustrate the evolution of system stability (SSS) under different scenarios:
Scenario 1 (High U, Low J): SSS increases rapidly and stabilizes at a positive value, reflecting authoritarian consolidation.
Scenario 2 (Low U, High J): SSS decreases and stabilizes at a negative value, indicating potential mobilization for reform or unrest.
Scenario 3 (Moderate U and J): SSS exhibits oscillatory behavior near zero, corresponding to a bifurcation region with high sensitivity to small perturbations.
Interpretation: The temporal trajectories confirm the non-linear and path-dependent nature of transactional politics, where early-stage conditions can determine long-term systemic outcomes.
3. Multi-Level Interaction Visualization
Stacked trajectory plots integrate micro-level voter rationalization (ArA_rAr), meso-level normative reinforcement (SmesoS_\text{meso}Smeso), and macro-level democratic degradation (DDD) to visualize cascading effects across levels.
Observation: In high U/JU/JU/J scenarios, micro-level compliance quickly amplifies meso-level norm acceptance, leading to accelerated macro-level degradation. Conversely, low U/JU/JU/J maintains system resilience, as resistance at the micro-level dampens meso-level normalization.
4. Bifurcation Diagrams
