Characterized by high PP^\ast plateaus, low TT^\ast, growing EE^\ast, and HH^\ast often above threshold.
Reinforcing loops dominate; delays and saturation produce oscillatory or irregular dynamics.
The system may display sensitive dependence on initial conditions, quasi-periodic waves of unrest, or sudden jumps (noise-induced tipping).
Between A and B there may be multistability (both equilibria coexist) or metastability (long-lived transients) --- standard signatures of systems with saddle-node or subcritical Hopf bifurcations.
2. Quantitative diagnostics & bifurcation boundaries
A. Bifurcation parameter rr
Empirically track r^t=^K(t)/^U(t)\hat r_t = \hat\rho_{\mathcal{K}}(t)/\hat\rho_U(t) (policy responsiveness ratio).
Define rcr_c as the critical value estimated from bifurcation analysis / ensemble sweep (Section III).
If r^t<rc\hat r_t < r_c: system likely in Basin A (stability).
If r^t>rc\hat r_t > r_c: system at risk of transition to Basin B.
Practical guide: calibrate rcr_c via sweep or surrogate model and update as new data arrive.
B. Eigenvalue / local linearization criteria
For an operating point xx, compute Jacobian J(x)J(x).
If \exists eigenvalue \lambda with ()>0\Re(\lambda)>0, the fixed point is unstable; crossing ()=0\Re(\lambda)=0 signals bifurcation (saddle-node or Hopf).
For operational use, monitor leading eigenvalue proxy via state-space identification (e.g., compute linear AR(1) fit to short rolling window of XX and track estimated dominant root).
C. Lyapunov exponent surrogate
Positive short-term largest Lyapunov exponent estimate from ensemble divergence indicates sensitivity and likely chaotic-like dynamics.
3. Stochastic signatures of approach to transition
Under noise, deterministic bifurcation signals are augmented by stochastic early warnings:
Critical slowing down: rising autocorrelation (lag-1) in PP and TT. Operational threshold: consistent upward trend > 10--15% relative to baseline over a 7--14 day rolling window.
Rising variance: increasing variance of PP, especially when coupled with increased skewness (heavy right tail), signals risk.
Spectral reddening: power shifts to lower-frequency fluctuations in time series spectra (longer return times).
Intermittency and burstiness: growing kurtosis in event-count time series of protests or online virality.
