The Geometry of Instability: A Geometric Framework for Rate-Induced Transitions

This work develops a geometric framework for rate-induced transitions in dynamical systems with accelerating control parameters. The central dimensionless invariant is Ψ = ρ / A, where ρ is the perturbation flux and A is the local basin width. All scalings derive directly from the local saddle-node geometry under Airy rescaling and require no fitting parameters. Deviation growth in the critical layer amplifies variance, lag-1 autocorrelation, and recovery time. The Standardized Stability Index (TSI) scales as Ψ, while the operational speed S = √(1 − Ψ²) quantifies geometric loss of responsiveness. The results yield a coordinate-invariant early-warning structure for rate-induced instability.