PETCS — An entropic and dynamical framework for the early warning of critical transitions

We present PETCS (Predictive Entropic Theory of Complex Systems), an interpretable framework for the early warning of critical transitions in stochastic systems. The method integrates three elements: (i) state reconstruction via Takens embedding, where the assumptions and data quality permit it; (ii) a local dynamical model based on stochastic differential equations in the normal forms of codimension-1 bifurcations; and (iii) a composite predictive functional combining variance, lag-1 autocorrelation, entropic measures and distance from the nominal equilibrium. Numerical simulations on saddle-node and Hopf bifurcations show that a calibrated composite functional can improve lead-time stability relative to a conventional early-warning baseline. The optimal weights depend strongly on the bifurcation class: permutation entropy and distance from equilibrium dominate in the saddle-node case, whereas variance dominates in the Hopf case with a residual entropic contribution. The framework is currently validated exclusively on synthetic data; validation on real data constitutes the natural next step.