Certified Breathing Stability Regions in Nonlinear Dynamical Systems: Composite Lyapunov Certificates, M-Matrix Conditions, and a Resilience-Fragility Correspondence
We develop a unified, certified lower bound on the time-to-boundary margin M for transient stability of interconnected dissipative systems under slow parameter drift. The companion work establishes M as the first-passage time of the joint state-parameter motion to the synchronism boundary and proves M = CCT exactly on the one-machine-infinite-bus reduction, while leaving the multimachine certified margin open. Here a composite (mixed-region) Lyapunov function, formed by absorbing the restoring intra-group coupling into group energy functions and treating only the residual cross-cut coupling through the comparison principle, yields a positively invariant inner estimate of the region of attraction whenever an associated test matrix is a nonsingular M-matrix. The certified region breathes with the drift: its size is governed by a single critical synchronising stiffness k_c (lambda), and as k_c -> 0 at the boundary the region breathes shut and the certified margin M_low <= M_true vanishes. We give a nonlinear sector form of the construction, a domain-neutral resilience-fragility reading in which the coupling that certifies order is the one whose growth certifies collapse, and a constructive control corollary establishing a sharp dichotomy between damping injection and structural action. The mechanism is demonstrated identically on the WSCC nine-bus power system and on an inertial Kuramoto network, whose normalised breathing curves collapse, to leading order, onto a single profile. We present this collapse as numerical evidence for a conjectured universal form; a normal-form proof is identified as the precise open step.