Melnikov-based observability breakdown in singular bilinear periodic matrix differential systems

Singular bilinear periodic matrix systems subject to Melnikov-type oscillatory forcing arise in chemical reaction networks, constrained mechanical systems, and biological oscillator monitoring. We address the question: at what critical forcing amplitude does state reconstruction from sensor output fail irreversibly? We introduce the Singular Bilinear Melnikov Periodic Matrix System for Observability (SBMPMS-O) and establish four results. First, a second-order expansion of the observability Gramian W o ( 0 , T ; ɛ ) with Kronecker-free coefficient matrices. Second, a critical observability phase transition threshold ɛ † with closed-form lower bound ɛ † ≥ σ min ( W o ( 0 ) ) / ( ‖ W o ( 1 ) ‖ + ‖ W o ( 2 ) ‖ ) : below ɛ † the minimum singular value of W o (the order parameter) remains positive; above it, the system undergoes a continuous rank collapse analogous to a second-order phase transition. Standard descriptor-system observability tests cannot detect this nonlinear dynamical phenomenon. Third, a Kalman–Hewer equivalence theorem valid for | ɛ | < ɛ † , with provable failure beyond it. Fourth, a duality theorem identifying the blind control regime ɛ † < | ɛ | < ɛ ∗ , where the system remains steerable yet any two state trajectories differing by an unobservable mode produce identical outputs. No output-feedback observer can resolve this failure. A worked 4 × 4 instance, verified by direct numerical integration in two independent implementations, illustrates the Gramian expansion and the Kronecker-free algorithm end-to-end; over the tested forcing range this particular instance remains full rank throughout. A Kronecker-free O ( N n 3 ) algorithm is proposed; its own running time is measured directly for n ≤ 8 , and its speedup over the classical Kronecker-vectorised approach is a complexity-based projection at every dimension reported (the Kronecker baseline itself was not run), reaching a projected 481 × at n = 8 and 105,800 × at n = 50 based on the proven O ( N n 3 ) versus O ( N n 6 ) complexity ratio. The threshold ɛ † constitutes a codimension-one bifurcation of the Gramian rank, distinguishing this work from classical Gramian perturbation analysis.