Certified Breathing Stability Regions in Nonlinear Dynamical Systems Composite Lyapunov Certificates M-Matrix Conditions Resilience–Fragility Correspondence

Jul 16, 2026

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Unified certified lower bounds are derived for the time-to-boundary margin M governing transient stability of interconnected dissipative systems subject to slow parameter drift. The work characterizes M as the first-passage time of the joint state–parameter motion to the synchronism boundary and shows exact agreement on the one-machine–infinite-bus reduction. A composite (mixed-region) Lyapunov construction yields a positively invariant inner estimate when a test matrix is a nonsingular M-matrix, producing a drift-controlled breathing region governed by a single critical synchronising stiffness.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages10

Reading Time25 min

Abstract

Introduction

Motivation: low-inertia drift and time-to-boundary safety

Companion context: extending CCT to M

Background strands in transient-stability certificates

Contribution summary and novelty claims

arXiv :2607 .04305v1 [ ee ss . SY] 5 Jul 2026 Certified Breathing Stability Regions in Nonlinear Dynamical Systems: Composite Lyapunov Certificates, M-Matrix Conditions, and a Resilience–Fragility Correspondence Mari´an Meˇster 1 1 Department of Electric Power Engineering, FEI, Technical University of Koˇsice, Koˇsice, Slovakia. Contributing [authors: marian.mester@tuke.sk](authors: marian.mester@tuke.sk); Abstract 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 therestoring 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 kc (λ), and askc → 0 at the boundary the region breathes shut and the certified margin Mlow ≤ Mtrue 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. Keywords: transient stability, vector Lyapunov function, M-matrix, comparison principle, synchronisation, saddle-node bifurcation, region of attraction, resilience 1 Introduction Power systems are being pushed closer to their stability limits by the replacement of synchronous generation with converter-interfaced renewables, which erodes inertia and shrinks the corrective window left after a disturbance [8,9] . In this lowinertia regime the operationally decisive question is no longer only whether the system is secure at the present operating point, but how long it will remain so as that point drifts: the quantity of interest is a time to a boundary, not a binary verdict. A recent European disturbance illustrates the operational stakes: in the Czech grid incident of 4 July 2025 an ordinary single-line contingency cascaded to an island collapse, with the corrective actions arriving too slowly and the cascading risk present in routine N − 1 results yet not surfaced to the operator in real time [49] . The classical index answering a fixed-point version of this question is the critical clearing time (CCT) . The companion paper [1] generalises CCT to a time-to-boundary margin M, the first time the joint state–parameter trajectory reaches the survival boundary Σ = ∂A (δ∗ ) under slow drift, and proves M = CCT exactly on the one-machine–infinite-bus (OMIB) reduction. It also delimits the multimachine case honestly: the direct energy method is non-conservative there, the controlling unstable equilibrium being the binding obstruction to a certified multimachine margin. This places the present work within a long line of transient-stability certificates, usefully separated into five strands. (i) Direct and energyfunction methods estimate the region of attraction through a scalar energy and a controlling-UEP or PEBS threshold [10,11] . (ii) Lyapunov-family and convex certificates relax the energy functi

FAQs

What is the main quantity certified in this work?

The paper certifies a unified lower bound M on the time-to-boundary margin for transient stability under slow parameter drift.

How does the proposed certificate relate to the synchronism boundary?

M is identified as the first-passage time for the joint state–parameter trajectory to reach the synchronism boundary under the slow drift model.

When does the composite Lyapunov method guarantee a positively invariant certified region?

It guarantees a positively invariant inner estimate of the region of attraction whenever the associated test matrix is a nonsingular M-matrix.

Jul 16, 2026

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