Adaptive Lag and Transition Susceptibility (v3.1.1): Clarified Nonlinear State-Dependent Permeability and Saddle-Node Threshold in Constrained Control Systems

Version 3.1.1 provides formal clarifications to the nonlinear extension of the Adaptive Lag framework introduced in v3.1. The model replaces the previous linear tracking formulation with a state-dependent permeability function

Phi(e) = Phi0 * exp(−gamma * e^2),

yielding a bounded positive adaptive gain

k(e) = Phi(e) / (R + epsilon),

with R > 0 and epsilon > 0 ensuring strictly positive and finite gain under all admissible states.

Under constant environmental drift and smooth permeability decay, the reduced one-dimensional tracking dynamics admit a saddle-node bifurcation. After nondimensionalization (gamma = 1), the analytically derived critical threshold is

nu_c = (1 / sqrt(2)) * exp(−1/2) ≈ 0.4289.

Deterministic phase diagrams and stochastic simulations (Ornstein–Uhlenbeck forcing) confirm the analytical boundary between stable tracking and divergence. A revised hazard formulation introduces state-dependent noise amplification, distinguishing constructive from destabilizing variance regimes.

This framework does not claim universal collapse prediction. It formalizes a dynamical vulnerability mechanism under bounded feedback capacity in nonlinear control systems and complex adaptive systems.

Community feedback, replication attempts, and critical evaluation are welcome.

1. Nonlinear Dynamics
2. Saddle-Node Bifurcation
3. Complex Adaptive Systems
4. Feedback Permeability
5. State-Dependent Gain
6. Adaptive Lag
7. Control Theory
8. Stochastic Forcing
9. Ornstein–Uhlenbeck Process
10. Systemic Fragility
11. Transition Thresholds
12. Feedback Saturation
13. Dynamical Stability
14. Hazard Modeling
15. Constrained Control Systems